ab-angle->ABCF B

Percentage Accurate: 54.0% → 67.5%

Time: 19.4s

Alternatives: 26

Speedup: 23.3×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	return ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0);
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	return ((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{-281}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle\_m}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{0 - \log \left(\frac{180}{\pi \cdot angle\_m}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin t\_0\right) \cdot \left(\left(b - a\right) \cdot \cos t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m))))
   (*
    angle_s
    (if (<= (pow b 2.0) 1e-281)
      (*
       (+ b a)
       (*
        (* 2.0 (sin (/ (/ PI 180.0) (/ 1.0 angle_m))))
        (* (- b a) (cos (exp (- 0.0 (log (/ 180.0 (* PI angle_m)))))))))
      (* (+ b a) (* (* 2.0 (sin t_0)) (* (- b a) (cos t_0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	double tmp;
	if (pow(b, 2.0) <= 1e-281) {
		tmp = (b + a) * ((2.0 * sin(((((double) M_PI) / 180.0) / (1.0 / angle_m)))) * ((b - a) * cos(exp((0.0 - log((180.0 / (((double) M_PI) * angle_m))))))));
	} else {
		tmp = (b + a) * ((2.0 * sin(t_0)) * ((b - a) * cos(t_0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	double tmp;
	if (Math.pow(b, 2.0) <= 1e-281) {
		tmp = (b + a) * ((2.0 * Math.sin(((Math.PI / 180.0) / (1.0 / angle_m)))) * ((b - a) * Math.cos(Math.exp((0.0 - Math.log((180.0 / (Math.PI * angle_m))))))));
	} else {
		tmp = (b + a) * ((2.0 * Math.sin(t_0)) * ((b - a) * Math.cos(t_0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	tmp = 0
	if math.pow(b, 2.0) <= 1e-281:
		tmp = (b + a) * ((2.0 * math.sin(((math.pi / 180.0) / (1.0 / angle_m)))) * ((b - a) * math.cos(math.exp((0.0 - math.log((180.0 / (math.pi * angle_m))))))))
	else:
		tmp = (b + a) * ((2.0 * math.sin(t_0)) * ((b - a) * math.cos(t_0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	tmp = 0.0
	if ((b ^ 2.0) <= 1e-281)
		tmp = Float64(Float64(b + a) * Float64(Float64(2.0 * sin(Float64(Float64(pi / 180.0) / Float64(1.0 / angle_m)))) * Float64(Float64(b - a) * cos(exp(Float64(0.0 - log(Float64(180.0 / Float64(pi * angle_m)))))))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(2.0 * sin(t_0)) * Float64(Float64(b - a) * cos(t_0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	tmp = 0.0;
	if ((b ^ 2.0) <= 1e-281)
		tmp = (b + a) * ((2.0 * sin(((pi / 180.0) / (1.0 / angle_m)))) * ((b - a) * cos(exp((0.0 - log((180.0 / (pi * angle_m))))))));
	else
		tmp = (b + a) * ((2.0 * sin(t_0)) * ((b - a) * cos(t_0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 1e-281], N[(N[(b + a), $MachinePrecision] * N[(N[(2.0 * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[Exp[N[(0.0 - N[Log[N[(180.0 / N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 1e-281

   1. Initial program 55.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 66.6%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 66.6%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180 \cdot \frac{1}{angle}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{180}}{\frac{1}{angle}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180}\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 67.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.6%

      \[\leadsto \left(b + a\right) \cdot \left(\left(2 \cdot \sin \color{blue}{\left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)}\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\left({\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)}^{-1}\right)\right)\right)\right)\right) \]

      3. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)\right)\right)\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right), -1\right)\right)\right)\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)\right), -1\right)\right)\right)\right)\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{180}{angle \cdot \mathsf{PI}\left(\right)}\right)\right), -1\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), -1\right)\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), -1\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), -1\right)\right)\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 31.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), -1\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 31.5%

       \[\leadsto \left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \color{blue}{\left(e^{\log \left(\frac{180}{\pi \cdot angle}\right) \cdot -1}\right)}\right)\right) \]

   if 1e-281 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 59.3%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{-281}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{0 - \log \left(\frac{180}{\pi \cdot angle}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ t_1 := 2 \cdot \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\left(b + a\right) \cdot \left(t\_1 \cdot \left(\left(b - a\right) \cdot \cos \left(e^{0 - \log \left(\frac{\frac{180}{angle\_m}}{\pi}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(t\_1 \cdot \left(\left(b - a\right) \cdot \cos t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m))) (t_1 (* 2.0 (sin t_0))))
   (*
    angle_s
    (if (<= (pow b 2.0) 5e-307)
      (*
       (+ b a)
       (* t_1 (* (- b a) (cos (exp (- 0.0 (log (/ (/ 180.0 angle_m) PI))))))))
      (* (+ b a) (* t_1 (* (- b a) (cos t_0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	double t_1 = 2.0 * sin(t_0);
	double tmp;
	if (pow(b, 2.0) <= 5e-307) {
		tmp = (b + a) * (t_1 * ((b - a) * cos(exp((0.0 - log(((180.0 / angle_m) / ((double) M_PI))))))));
	} else {
		tmp = (b + a) * (t_1 * ((b - a) * cos(t_0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	double t_1 = 2.0 * Math.sin(t_0);
	double tmp;
	if (Math.pow(b, 2.0) <= 5e-307) {
		tmp = (b + a) * (t_1 * ((b - a) * Math.cos(Math.exp((0.0 - Math.log(((180.0 / angle_m) / Math.PI)))))));
	} else {
		tmp = (b + a) * (t_1 * ((b - a) * Math.cos(t_0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	t_1 = 2.0 * math.sin(t_0)
	tmp = 0
	if math.pow(b, 2.0) <= 5e-307:
		tmp = (b + a) * (t_1 * ((b - a) * math.cos(math.exp((0.0 - math.log(((180.0 / angle_m) / math.pi)))))))
	else:
		tmp = (b + a) * (t_1 * ((b - a) * math.cos(t_0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	t_1 = Float64(2.0 * sin(t_0))
	tmp = 0.0
	if ((b ^ 2.0) <= 5e-307)
		tmp = Float64(Float64(b + a) * Float64(t_1 * Float64(Float64(b - a) * cos(exp(Float64(0.0 - log(Float64(Float64(180.0 / angle_m) / pi))))))));
	else
		tmp = Float64(Float64(b + a) * Float64(t_1 * Float64(Float64(b - a) * cos(t_0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	t_1 = 2.0 * sin(t_0);
	tmp = 0.0;
	if ((b ^ 2.0) <= 5e-307)
		tmp = (b + a) * (t_1 * ((b - a) * cos(exp((0.0 - log(((180.0 / angle_m) / pi)))))));
	else
		tmp = (b + a) * (t_1 * ((b - a) * cos(t_0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 5e-307], N[(N[(b + a), $MachinePrecision] * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[Exp[N[(0.0 - N[Log[N[(N[(180.0 / angle$95$m), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 5.00000000000000014e-307

   1. Initial program 52.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\left({\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)}^{-1}\right)\right)\right)\right)\right) \]

      3. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right)\right)\right)\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right), -1\right)\right)\right)\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)\right), -1\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{180}{angle}\right), \mathsf{PI}\left(\right)\right)\right), -1\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(180, angle\right), \mathsf{PI}\left(\right)\right)\right), -1\right)\right)\right)\right)\right)\right) \]

      9. PI-lowering-PI.f64 32.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(180, angle\right), \mathsf{PI.f64}\left(\right)\right)\right), -1\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 32.5%

      \[\leadsto \left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \color{blue}{\left(e^{\log \left(\frac{\frac{180}{angle}}{\pi}\right) \cdot -1}\right)}\right)\right) \]

   if 5.00000000000000014e-307 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 60.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{0 - \log \left(\frac{\frac{180}{angle}}{\pi}\right)}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ t_1 := 2 \cdot \sin t\_0\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(b + a\right) \cdot t\_1\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{0 - \log \left(\frac{180}{\pi \cdot angle\_m}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(t\_1 \cdot \left(\left(b - a\right) \cdot \cos t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m))) (t_1 (* 2.0 (sin t_0))))
   (*
    angle_s
    (if (<= (pow b 2.0) 5e-307)
      (*
       (* (+ b a) t_1)
       (* (- b a) (cos (exp (- 0.0 (log (/ 180.0 (* PI angle_m))))))))
      (* (+ b a) (* t_1 (* (- b a) (cos t_0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	double t_1 = 2.0 * sin(t_0);
	double tmp;
	if (pow(b, 2.0) <= 5e-307) {
		tmp = ((b + a) * t_1) * ((b - a) * cos(exp((0.0 - log((180.0 / (((double) M_PI) * angle_m)))))));
	} else {
		tmp = (b + a) * (t_1 * ((b - a) * cos(t_0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	double t_1 = 2.0 * Math.sin(t_0);
	double tmp;
	if (Math.pow(b, 2.0) <= 5e-307) {
		tmp = ((b + a) * t_1) * ((b - a) * Math.cos(Math.exp((0.0 - Math.log((180.0 / (Math.PI * angle_m)))))));
	} else {
		tmp = (b + a) * (t_1 * ((b - a) * Math.cos(t_0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	t_1 = 2.0 * math.sin(t_0)
	tmp = 0
	if math.pow(b, 2.0) <= 5e-307:
		tmp = ((b + a) * t_1) * ((b - a) * math.cos(math.exp((0.0 - math.log((180.0 / (math.pi * angle_m)))))))
	else:
		tmp = (b + a) * (t_1 * ((b - a) * math.cos(t_0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	t_1 = Float64(2.0 * sin(t_0))
	tmp = 0.0
	if ((b ^ 2.0) <= 5e-307)
		tmp = Float64(Float64(Float64(b + a) * t_1) * Float64(Float64(b - a) * cos(exp(Float64(0.0 - log(Float64(180.0 / Float64(pi * angle_m))))))));
	else
		tmp = Float64(Float64(b + a) * Float64(t_1 * Float64(Float64(b - a) * cos(t_0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	t_1 = 2.0 * sin(t_0);
	tmp = 0.0;
	if ((b ^ 2.0) <= 5e-307)
		tmp = ((b + a) * t_1) * ((b - a) * cos(exp((0.0 - log((180.0 / (pi * angle_m)))))));
	else
		tmp = (b + a) * (t_1 * ((b - a) * cos(t_0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 5e-307], N[(N[(N[(b + a), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[Exp[N[(0.0 - N[Log[N[(180.0 / N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 5.00000000000000014e-307

   1. Initial program 52.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left({\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)}^{-1}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. pow-to-exp N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(e^{\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right) \cdot -1\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right), -1\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)\right), -1\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. associate-/l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{180}{\mathsf{PI}\left(\right) \cdot angle}\right)\right), -1\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \left(\mathsf{PI}\left(\right) \cdot angle\right)\right)\right), -1\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right)\right)\right), -1\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      10. PI-lowering-PI.f64 32.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(180, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right)\right)\right), -1\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   6. Applied egg-rr 32.5%

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \color{blue}{\left(e^{\log \left(\frac{180}{\pi \cdot angle}\right) \cdot -1}\right)} \cdot \left(b - a\right)\right) \]

   if 5.00000000000000014e-307 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 60.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(e^{0 - \log \left(\frac{180}{\pi \cdot angle}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ t_1 := 2 \cdot \sin t\_0\\ t_2 := \pi \cdot \left(\pi \cdot \pi\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 10^{-204}:\\ \;\;\;\;\left(b + a\right) \cdot \left(t\_1 \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{{\left(t\_2 \cdot t\_2\right)}^{0.16666666666666666}}{\frac{180}{angle\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(t\_1 \cdot \left(\left(b - a\right) \cdot \cos t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m)))
        (t_1 (* 2.0 (sin t_0)))
        (t_2 (* PI (* PI PI))))
   (*
    angle_s
    (if (<= (pow b 2.0) 1e-204)
      (*
       (+ b a)
       (*
        t_1
        (*
         (- b a)
         (cos (/ (pow (* t_2 t_2) 0.16666666666666666) (/ 180.0 angle_m))))))
      (* (+ b a) (* t_1 (* (- b a) (cos t_0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	double t_1 = 2.0 * sin(t_0);
	double t_2 = ((double) M_PI) * (((double) M_PI) * ((double) M_PI));
	double tmp;
	if (pow(b, 2.0) <= 1e-204) {
		tmp = (b + a) * (t_1 * ((b - a) * cos((pow((t_2 * t_2), 0.16666666666666666) / (180.0 / angle_m)))));
	} else {
		tmp = (b + a) * (t_1 * ((b - a) * cos(t_0)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	double t_1 = 2.0 * Math.sin(t_0);
	double t_2 = Math.PI * (Math.PI * Math.PI);
	double tmp;
	if (Math.pow(b, 2.0) <= 1e-204) {
		tmp = (b + a) * (t_1 * ((b - a) * Math.cos((Math.pow((t_2 * t_2), 0.16666666666666666) / (180.0 / angle_m)))));
	} else {
		tmp = (b + a) * (t_1 * ((b - a) * Math.cos(t_0)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	t_1 = 2.0 * math.sin(t_0)
	t_2 = math.pi * (math.pi * math.pi)
	tmp = 0
	if math.pow(b, 2.0) <= 1e-204:
		tmp = (b + a) * (t_1 * ((b - a) * math.cos((math.pow((t_2 * t_2), 0.16666666666666666) / (180.0 / angle_m)))))
	else:
		tmp = (b + a) * (t_1 * ((b - a) * math.cos(t_0)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	t_1 = Float64(2.0 * sin(t_0))
	t_2 = Float64(pi * Float64(pi * pi))
	tmp = 0.0
	if ((b ^ 2.0) <= 1e-204)
		tmp = Float64(Float64(b + a) * Float64(t_1 * Float64(Float64(b - a) * cos(Float64((Float64(t_2 * t_2) ^ 0.16666666666666666) / Float64(180.0 / angle_m))))));
	else
		tmp = Float64(Float64(b + a) * Float64(t_1 * Float64(Float64(b - a) * cos(t_0))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	t_1 = 2.0 * sin(t_0);
	t_2 = pi * (pi * pi);
	tmp = 0.0;
	if ((b ^ 2.0) <= 1e-204)
		tmp = (b + a) * (t_1 * ((b - a) * cos((((t_2 * t_2) ^ 0.16666666666666666) / (180.0 / angle_m)))));
	else
		tmp = (b + a) * (t_1 * ((b - a) * cos(t_0)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 1e-204], N[(N[(b + a), $MachinePrecision] * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[Power[N[(t$95$2 * t$95$2), $MachinePrecision], 0.16666666666666666], $MachinePrecision] / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(t$95$1 * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 1e-204

   1. Initial program 57.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 69.5%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 69.5%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. add-cbrt-cube N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      3. pow1/3 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{3}}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      4. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      5. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left({\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 76.9%

      \[\leadsto \left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\color{blue}{{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)}^{0.16666666666666666}}}{\frac{180}{angle}}\right)\right)\right) \]

   if 1e-204 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 58.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ angle\_s \cdot \left(\left(b + a\right) \cdot \left(\left(2 \cdot \sin t\_0\right) \cdot \left(\left(b - a\right) \cdot \cos t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m))))
   (* angle_s (* (+ b a) (* (* 2.0 (sin t_0)) (* (- b a) (cos t_0)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	return angle_s * ((b + a) * ((2.0 * sin(t_0)) * ((b - a) * cos(t_0))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	return angle_s * ((b + a) * ((2.0 * Math.sin(t_0)) * ((b - a) * Math.cos(t_0))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	return angle_s * ((b + a) * ((2.0 * math.sin(t_0)) * ((b - a) * math.cos(t_0))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	return Float64(angle_s * Float64(Float64(b + a) * Float64(Float64(2.0 * sin(t_0)) * Float64(Float64(b - a) * cos(t_0)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	tmp = angle_s * ((b + a) * ((2.0 * sin(t_0)) * ((b - a) * cos(t_0))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * N[(N[(b + a), $MachinePrecision] * N[(N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 58.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

   2. pow2 N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   3. pow2 N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

   8. associate-*r* N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

   9. difference-of-squares N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

   10. associate-*l* N/A

       \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

4. Applied egg-rr 70.7%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

6. Applied egg-rr 70.7%

   \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
7. Add Preprocessing

Alternative 6: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \frac{\pi}{\frac{180}{angle\_m}}\\ angle\_s \cdot \left(\left(\left(b + a\right) \cdot \left(2 \cdot \sin t\_0\right)\right) \cdot \left(\left(b - a\right) \cdot \cos t\_0\right)\right) \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (/ PI (/ 180.0 angle_m))))
   (* angle_s (* (* (+ b a) (* 2.0 (sin t_0))) (* (- b a) (cos t_0))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) / (180.0 / angle_m);
	return angle_s * (((b + a) * (2.0 * sin(t_0))) * ((b - a) * cos(t_0)));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI / (180.0 / angle_m);
	return angle_s * (((b + a) * (2.0 * Math.sin(t_0))) * ((b - a) * Math.cos(t_0)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi / (180.0 / angle_m)
	return angle_s * (((b + a) * (2.0 * math.sin(t_0))) * ((b - a) * math.cos(t_0)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi / Float64(180.0 / angle_m))
	return Float64(angle_s * Float64(Float64(Float64(b + a) * Float64(2.0 * sin(t_0))) * Float64(Float64(b - a) * cos(t_0))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	t_0 = pi / (180.0 / angle_m);
	tmp = angle_s * (((b + a) * (2.0 * sin(t_0))) * ((b - a) * cos(t_0)));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * N[(N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 58.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

   2. pow2 N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   3. pow2 N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

   8. associate-*r* N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

   9. difference-of-squares N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

   10. associate-*l* N/A

       \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

4. Applied egg-rr 70.7%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]

5. Final simplification 70.7%

   \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \]
6. Add Preprocessing

Alternative 7: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (*
   (* (- b a) (cos (/ PI (/ 180.0 angle_m))))
   (* (+ b a) (* 2.0 (sin (* PI (/ angle_m 180.0))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (((b - a) * cos((((double) M_PI) / (180.0 / angle_m)))) * ((b + a) * (2.0 * sin((((double) M_PI) * (angle_m / 180.0))))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (((b - a) * Math.cos((Math.PI / (180.0 / angle_m)))) * ((b + a) * (2.0 * Math.sin((Math.PI * (angle_m / 180.0))))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (((b - a) * math.cos((math.pi / (180.0 / angle_m)))) * ((b + a) * (2.0 * math.sin((math.pi * (angle_m / 180.0))))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle_m)))) * Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi * Float64(angle_m / 180.0)))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (((b - a) * cos((pi / (180.0 / angle_m)))) * ((b + a) * (2.0 * sin((pi * (angle_m / 180.0))))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

   2. pow2 N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   3. pow2 N/A

      \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

   6. *-commutative N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

   7. associate-*r/ N/A

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

   8. associate-*r* N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

   9. difference-of-squares N/A

      \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

   10. associate-*l* N/A

       \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

4. Applied egg-rr 70.7%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
5. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{\frac{180}{angle}}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{angle}{180}\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(\color{blue}{180}, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   6. PI-lowering-PI.f64 69.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(angle, 180\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

6. Applied egg-rr 69.8%

   \[\leadsto \left(\left(2 \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \pi\right)}\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]

7. Final simplification 69.8%

   \[\leadsto \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
8. Add Preprocessing

Alternative 8: 54.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 9.8 \cdot 10^{-231}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b \cdot 2\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.62 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)}{\frac{1}{b + a}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 9.8e-231)
    (* (- b a) (* (* b 2.0) (sin (* 0.005555555555555556 (* PI angle_m)))))
    (if (<= a 1.62e+189)
      (/
       (* (- b a) (sin (* PI (* angle_m 0.011111111111111112))))
       (/ 1.0 (+ b a)))
      (* (+ b a) (* 0.011111111111111112 (* angle_m (* PI (- b a)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.8e-231) {
		tmp = (b - a) * ((b * 2.0) * sin((0.005555555555555556 * (((double) M_PI) * angle_m))));
	} else if (a <= 1.62e+189) {
		tmp = ((b - a) * sin((((double) M_PI) * (angle_m * 0.011111111111111112)))) / (1.0 / (b + a));
	} else {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - a))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.8e-231) {
		tmp = (b - a) * ((b * 2.0) * Math.sin((0.005555555555555556 * (Math.PI * angle_m))));
	} else if (a <= 1.62e+189) {
		tmp = ((b - a) * Math.sin((Math.PI * (angle_m * 0.011111111111111112)))) / (1.0 / (b + a));
	} else {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * (Math.PI * (b - a))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 9.8e-231:
		tmp = (b - a) * ((b * 2.0) * math.sin((0.005555555555555556 * (math.pi * angle_m))))
	elif a <= 1.62e+189:
		tmp = ((b - a) * math.sin((math.pi * (angle_m * 0.011111111111111112)))) / (1.0 / (b + a))
	else:
		tmp = (b + a) * (0.011111111111111112 * (angle_m * (math.pi * (b - a))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 9.8e-231)
		tmp = Float64(Float64(b - a) * Float64(Float64(b * 2.0) * sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))));
	elseif (a <= 1.62e+189)
		tmp = Float64(Float64(Float64(b - a) * sin(Float64(pi * Float64(angle_m * 0.011111111111111112)))) / Float64(1.0 / Float64(b + a)));
	else
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - a)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 9.8e-231)
		tmp = (b - a) * ((b * 2.0) * sin((0.005555555555555556 * (pi * angle_m))));
	elseif (a <= 1.62e+189)
		tmp = ((b - a) * sin((pi * (angle_m * 0.011111111111111112)))) / (1.0 / (b + a));
	else
		tmp = (b + a) * (0.011111111111111112 * (angle_m * (pi * (b - a))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 9.8e-231], N[(N[(b - a), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.62e+189], N[(N[(N[(b - a), $MachinePrecision] * N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if a < 9.80000000000000007e-231

   1. Initial program 61.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 45.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 45.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 46.3%

       \[\leadsto \left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\color{blue}{1} \cdot \left(b - a\right)\right) \]

   if 9.80000000000000007e-231 < a < 1.6200000000000001e189

   1. Initial program 59.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180 \cdot \frac{1}{angle}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{180}}{\frac{1}{angle}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180}\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 67.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.2%

      \[\leadsto \left(b + a\right) \cdot \left(\left(2 \cdot \sin \color{blue}{\left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)}\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \]

   10. Applied egg-rr 67.6%

       \[\leadsto \color{blue}{\frac{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)}{\frac{1}{b + a}}} \]

   if 1.6200000000000001e189 < a

   1. Initial program 40.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b - a\right)}\right)\right)\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} - a\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Simplified 75.7%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.8 \cdot 10^{-231}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b \cdot 2\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.62 \cdot 10^{+189}:\\ \;\;\;\;\frac{\left(b - a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{\frac{1}{b + a}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-230}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b \cdot 2\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+189}:\\ \;\;\;\;\left(b + a\right) \cdot \frac{\sin \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)}{\frac{1}{b - a}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 4.2e-230)
    (* (- b a) (* (* b 2.0) (sin (* 0.005555555555555556 (* PI angle_m)))))
    (if (<= a 1.6e+189)
      (*
       (+ b a)
       (/ (sin (* PI (* angle_m 0.011111111111111112))) (/ 1.0 (- b a))))
      (* (+ b a) (* 0.011111111111111112 (* angle_m (* PI (- b a)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.2e-230) {
		tmp = (b - a) * ((b * 2.0) * sin((0.005555555555555556 * (((double) M_PI) * angle_m))));
	} else if (a <= 1.6e+189) {
		tmp = (b + a) * (sin((((double) M_PI) * (angle_m * 0.011111111111111112))) / (1.0 / (b - a)));
	} else {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * (((double) M_PI) * (b - a))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.2e-230) {
		tmp = (b - a) * ((b * 2.0) * Math.sin((0.005555555555555556 * (Math.PI * angle_m))));
	} else if (a <= 1.6e+189) {
		tmp = (b + a) * (Math.sin((Math.PI * (angle_m * 0.011111111111111112))) / (1.0 / (b - a)));
	} else {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * (Math.PI * (b - a))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if a <= 4.2e-230:
		tmp = (b - a) * ((b * 2.0) * math.sin((0.005555555555555556 * (math.pi * angle_m))))
	elif a <= 1.6e+189:
		tmp = (b + a) * (math.sin((math.pi * (angle_m * 0.011111111111111112))) / (1.0 / (b - a)))
	else:
		tmp = (b + a) * (0.011111111111111112 * (angle_m * (math.pi * (b - a))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 4.2e-230)
		tmp = Float64(Float64(b - a) * Float64(Float64(b * 2.0) * sin(Float64(0.005555555555555556 * Float64(pi * angle_m)))));
	elseif (a <= 1.6e+189)
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(pi * Float64(angle_m * 0.011111111111111112))) / Float64(1.0 / Float64(b - a))));
	else
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b - a)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (a <= 4.2e-230)
		tmp = (b - a) * ((b * 2.0) * sin((0.005555555555555556 * (pi * angle_m))));
	elseif (a <= 1.6e+189)
		tmp = (b + a) * (sin((pi * (angle_m * 0.011111111111111112))) / (1.0 / (b - a)));
	else
		tmp = (b + a) * (0.011111111111111112 * (angle_m * (pi * (b - a))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 4.2e-230], N[(N[(b - a), $MachinePrecision] * N[(N[(b * 2.0), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.6e+189], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if a < 4.1999999999999997e-230

   1. Initial program 61.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 45.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 45.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 46.3%

       \[\leadsto \left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(\color{blue}{1} \cdot \left(b - a\right)\right) \]

   if 4.1999999999999997e-230 < a < 1.6e189

   1. Initial program 59.5%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180 \cdot \frac{1}{angle}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{180}}{\frac{1}{angle}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180}\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 67.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 67.2%

      \[\leadsto \left(b + a\right) \cdot \left(\left(2 \cdot \sin \color{blue}{\left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)}\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \]

   10. Applied egg-rr 67.6%

       \[\leadsto \left(b + a\right) \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)}{\frac{1}{b - a}}} \]

   if 1.6e189 < a

   1. Initial program 40.0%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 72.5%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b - a\right)}\right)\right)\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} - a\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 75.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Simplified 75.7%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.2 \cdot 10^{-230}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b \cdot 2\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+189}:\\ \;\;\;\;\left(b + a\right) \cdot \frac{\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)}{\frac{1}{b - a}}\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{+166}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{2}{\frac{\frac{180}{angle\_m}}{\pi}}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 2.35e+166)
    (* (- b a) (* (+ b a) (sin (* (* PI angle_m) 0.011111111111111112))))
    (* b (* (- b a) (sin (/ 2.0 (/ (/ 180.0 angle_m) PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.35e+166) {
		tmp = (b - a) * ((b + a) * sin(((((double) M_PI) * angle_m) * 0.011111111111111112)));
	} else {
		tmp = b * ((b - a) * sin((2.0 / ((180.0 / angle_m) / ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.35e+166) {
		tmp = (b - a) * ((b + a) * Math.sin(((Math.PI * angle_m) * 0.011111111111111112)));
	} else {
		tmp = b * ((b - a) * Math.sin((2.0 / ((180.0 / angle_m) / Math.PI))));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 2.35e+166:
		tmp = (b - a) * ((b + a) * math.sin(((math.pi * angle_m) * 0.011111111111111112)))
	else:
		tmp = b * ((b - a) * math.sin((2.0 / ((180.0 / angle_m) / math.pi))))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 2.35e+166)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(Float64(pi * angle_m) * 0.011111111111111112))));
	else
		tmp = Float64(b * Float64(Float64(b - a) * sin(Float64(2.0 / Float64(Float64(180.0 / angle_m) / pi)))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.35e+166)
		tmp = (b - a) * ((b + a) * sin(((pi * angle_m) * 0.011111111111111112)));
	else
		tmp = b * ((b - a) * sin((2.0 / ((180.0 / angle_m) / pi))));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 2.35e+166], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 / N[(N[(180.0 / angle$95$m), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 2.35e166

   1. Initial program 58.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(\left(\color{blue}{2} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {\color{blue}{a}}^{2}\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot \color{blue}{a}\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\color{blue}{b \cdot b} - a \cdot a\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)} \]

      10. difference-of-squares N/A

          \[\leadsto \left(\cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(b - a\right)}\right) \]

   4. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{\left(b - a\right) \cdot \left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\right)} \]

   if 2.35e166 < b

   1. Initial program 52.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 85.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 74.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 74.4%

      \[\leadsto \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot b\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   9. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{b \cdot \left(\sin \left(\frac{2}{\frac{\frac{180}{angle}}{\pi}}\right) \cdot \left(b - a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.35 \cdot 10^{+166}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{2}{\frac{\frac{180}{angle}}{\pi}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{2}{\frac{\frac{180}{angle\_m}}{\pi}}\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (+ b a) (* (- b a) (sin (/ 2.0 (/ (/ 180.0 angle_m) PI)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b + a) * ((b - a) * sin((2.0 / ((180.0 / angle_m) / ((double) M_PI))))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((b + a) * ((b - a) * Math.sin((2.0 / ((180.0 / angle_m) / Math.PI)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((b + a) * ((b - a) * math.sin((2.0 / ((180.0 / angle_m) / math.pi)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(2.0 / Float64(Float64(180.0 / angle_m) / pi))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((b + a) * ((b - a) * sin((2.0 / ((180.0 / angle_m) / pi)))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 / N[(N[(180.0 / angle$95$m), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(b \cdot b - {a}^{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   2. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(b \cdot b - a \cdot a\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   3. difference-of-squares N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\left(b - a\right) \cdot \left(b + a\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   5. flip3-- N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{b}^{3} - {a}^{3}}{b \cdot b + \left(a \cdot a + b \cdot a\right)} \cdot \left(b + a\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   6. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{{b}^{3} - {a}^{3}}{b \cdot b + \left(a \cdot a + b \cdot a\right)} \cdot \frac{{b}^{3} + {a}^{3}}{b \cdot b + \left(a \cdot a - b \cdot a\right)}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   7. frac-times N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\left({b}^{3} - {a}^{3}\right) \cdot \left({b}^{3} + {a}^{3}\right)}{\left(b \cdot b + \left(a \cdot a + b \cdot a\right)\right) \cdot \left(b \cdot b + \left(a \cdot a - b \cdot a\right)\right)}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\left({b}^{3} - {a}^{3}\right) \cdot \left({b}^{3} + {a}^{3}\right)\right), \left(\left(b \cdot b + \left(a \cdot a + b \cdot a\right)\right) \cdot \left(b \cdot b + \left(a \cdot a - b \cdot a\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right) \]

4. Applied egg-rr 11.8%

   \[\leadsto \left(\left(2 \cdot \color{blue}{\frac{\left(b \cdot \left(b \cdot b\right) - a \cdot \left(a \cdot a\right)\right) \cdot \left(a \cdot \left(a \cdot a\right) + b \cdot \left(b \cdot b\right)\right)}{\left(b \cdot b + a \cdot \left(b + a\right)\right) \cdot \left(b \cdot b + a \cdot \left(a - b\right)\right)}}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

6. Applied egg-rr 58.9%

   \[\leadsto \color{blue}{\left(b \cdot b - a \cdot a\right) \cdot \sin \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)} \]
7. Step-by-step derivation

   1. difference-of-squares N/A

      \[\leadsto \left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \color{blue}{\left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \]

   2. associate-*l* N/A

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)}\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(b - a\right)} \cdot \sin \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \color{blue}{\sin \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \sin \color{blue}{\left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)}\right)\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{sin.f64}\left(\left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right)\right) \]

   8. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{sin.f64}\left(\left(2 \cdot \frac{1}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

   9. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{sin.f64}\left(\left(\frac{2}{\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{\frac{180}{angle}}{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{180}{angle}\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(180, angle\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   13. PI-lowering-PI.f64 69.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(180, angle\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 69.4%

   \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{2}{\frac{\frac{180}{angle}}{\pi}}\right)\right)} \]
9. Add Preprocessing

Alternative 12: 64.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle\_m \cdot \left(0.011111111111111112 \cdot t\_0 + \left(2 \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot t\_0\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (- b a))))
   (*
    angle_s
    (if (<= (/ angle_m 180.0) 5e+64)
      (*
       (+ b a)
       (*
        angle_m
        (+
         (* 0.011111111111111112 t_0)
         (*
          (* 2.0 (* angle_m angle_m))
          (* (- b a) (* (* PI (* PI PI)) -1.1431184270690443e-7))))))
      (* (* (+ b a) t_0) (* angle_m 0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 5e+64) {
		tmp = (b + a) * (angle_m * ((0.011111111111111112 * t_0) + ((2.0 * (angle_m * angle_m)) * ((b - a) * ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * -1.1431184270690443e-7)))));
	} else {
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (b - a);
	double tmp;
	if ((angle_m / 180.0) <= 5e+64) {
		tmp = (b + a) * (angle_m * ((0.011111111111111112 * t_0) + ((2.0 * (angle_m * angle_m)) * ((b - a) * ((Math.PI * (Math.PI * Math.PI)) * -1.1431184270690443e-7)))));
	} else {
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (b - a)
	tmp = 0
	if (angle_m / 180.0) <= 5e+64:
		tmp = (b + a) * (angle_m * ((0.011111111111111112 * t_0) + ((2.0 * (angle_m * angle_m)) * ((b - a) * ((math.pi * (math.pi * math.pi)) * -1.1431184270690443e-7)))))
	else:
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(b - a))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e+64)
		tmp = Float64(Float64(b + a) * Float64(angle_m * Float64(Float64(0.011111111111111112 * t_0) + Float64(Float64(2.0 * Float64(angle_m * angle_m)) * Float64(Float64(b - a) * Float64(Float64(pi * Float64(pi * pi)) * -1.1431184270690443e-7))))));
	else
		tmp = Float64(Float64(Float64(b + a) * t_0) * Float64(angle_m * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (b - a);
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e+64)
		tmp = (b + a) * (angle_m * ((0.011111111111111112 * t_0) + ((2.0 * (angle_m * angle_m)) * ((b - a) * ((pi * (pi * pi)) * -1.1431184270690443e-7)))));
	else
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e+64], N[(N[(b + a), $MachinePrecision] * N[(angle$95$m * N[(N[(0.011111111111111112 * t$95$0), $MachinePrecision] + N[(N[(2.0 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * -1.1431184270690443e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e64

   1. Initial program 62.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 77.4%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180 \cdot \frac{1}{angle}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\left(\frac{\frac{\mathsf{PI}\left(\right)}{180}}{\frac{1}{angle}}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180}\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \left(\frac{1}{angle}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 77.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), \mathsf{/.f64}\left(1, angle\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 77.1%

      \[\leadsto \left(b + a\right) \cdot \left(\left(2 \cdot \sin \color{blue}{\left(\frac{\frac{\pi}{180}}{\frac{1}{angle}}\right)}\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right) \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(angle \cdot \left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \color{blue}{\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right) + 2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{1}{90} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right), \color{blue}{\left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)}\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right), \left(\color{blue}{2} \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(b - a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right)\right) \]

       5. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b - a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right)\right) \]

       6. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \left(2 \cdot \left({angle}^{2} \cdot \left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \left(\left(2 \cdot {angle}^{2}\right) \cdot \color{blue}{\left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\left(2 \cdot {angle}^{2}\right), \color{blue}{\left(\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right) + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left({angle}^{2}\right)\right), \left(\color{blue}{\frac{-1}{11664000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(angle \cdot angle\right)\right), \left(\frac{-1}{11664000} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, a\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\frac{-1}{11664000} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)} + \frac{-1}{34992000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(b - a\right)\right)\right)\right)\right)\right)\right) \]

   11. Simplified 69.9%

       \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right) + \left(2 \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)\right)} \]

   if 5e64 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 40.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 37.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 37.5%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b \cdot b - a \cdot a\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      2. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b + a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      8. PI-lowering-PI.f64 39.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

   7. Applied egg-rr 39.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \cdot \left(angle \cdot 0.011111111111111112\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{+64}:\\ \;\;\;\;\left(b + a\right) \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b - a\right)\right) + \left(2 \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -1.1431184270690443 \cdot 10^{-7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.6% accurate, 12.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 26500000000:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot t\_0\right)\right)\\ \mathbf{elif}\;angle\_m \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112 + \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot -2.2862368541380886 \cdot 10^{-7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot t\_0\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (- b a))))
   (*
    angle_s
    (if (<= angle_m 26500000000.0)
      (* (+ b a) (* 0.011111111111111112 (* angle_m t_0)))
      (if (<= angle_m 2.3e+67)
        (*
         b
         (*
          (- b a)
          (*
           angle_m
           (+
            (* PI 0.011111111111111112)
            (*
             (* PI (* PI PI))
             (* (* angle_m angle_m) -2.2862368541380886e-7))))))
        (* (* (+ b a) t_0) (* angle_m 0.011111111111111112)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (b - a);
	double tmp;
	if (angle_m <= 26500000000.0) {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * t_0));
	} else if (angle_m <= 2.3e+67) {
		tmp = b * ((b - a) * (angle_m * ((((double) M_PI) * 0.011111111111111112) + ((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * ((angle_m * angle_m) * -2.2862368541380886e-7)))));
	} else {
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (b - a);
	double tmp;
	if (angle_m <= 26500000000.0) {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * t_0));
	} else if (angle_m <= 2.3e+67) {
		tmp = b * ((b - a) * (angle_m * ((Math.PI * 0.011111111111111112) + ((Math.PI * (Math.PI * Math.PI)) * ((angle_m * angle_m) * -2.2862368541380886e-7)))));
	} else {
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (b - a)
	tmp = 0
	if angle_m <= 26500000000.0:
		tmp = (b + a) * (0.011111111111111112 * (angle_m * t_0))
	elif angle_m <= 2.3e+67:
		tmp = b * ((b - a) * (angle_m * ((math.pi * 0.011111111111111112) + ((math.pi * (math.pi * math.pi)) * ((angle_m * angle_m) * -2.2862368541380886e-7)))))
	else:
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(b - a))
	tmp = 0.0
	if (angle_m <= 26500000000.0)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(angle_m * t_0)));
	elseif (angle_m <= 2.3e+67)
		tmp = Float64(b * Float64(Float64(b - a) * Float64(angle_m * Float64(Float64(pi * 0.011111111111111112) + Float64(Float64(pi * Float64(pi * pi)) * Float64(Float64(angle_m * angle_m) * -2.2862368541380886e-7))))));
	else
		tmp = Float64(Float64(Float64(b + a) * t_0) * Float64(angle_m * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (b - a);
	tmp = 0.0;
	if (angle_m <= 26500000000.0)
		tmp = (b + a) * (0.011111111111111112 * (angle_m * t_0));
	elseif (angle_m <= 2.3e+67)
		tmp = b * ((b - a) * (angle_m * ((pi * 0.011111111111111112) + ((pi * (pi * pi)) * ((angle_m * angle_m) * -2.2862368541380886e-7)))));
	else
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 26500000000.0], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle$95$m, 2.3e+67], N[(b * N[(N[(b - a), $MachinePrecision] * N[(angle$95$m * N[(N[(Pi * 0.011111111111111112), $MachinePrecision] + N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if angle < 2.65e10

   1. Initial program 64.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b - a\right)}\right)\right)\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} - a\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 74.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Simplified 74.4%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 2.65e10 < angle < 2.2999999999999999e67

   1. Initial program 20.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 30.1%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 18.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 18.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot b\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   9. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{b \cdot \left(\sin \left(\frac{2}{\frac{\frac{180}{angle}}{\pi}}\right) \cdot \left(b - a\right)\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right) + \frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right), \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right), \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{90}\right), \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \left(\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\left(\frac{-1}{4374000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       12. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       15. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       18. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       19. PI-lowering-PI.f64 11.5%

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{90}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   12. Simplified 11.5%

       \[\leadsto b \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112 + \left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)} \cdot \left(b - a\right)\right) \]

   if 2.2999999999999999e67 < angle

   1. Initial program 40.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 37.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 37.5%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b \cdot b - a \cdot a\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      2. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b + a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      8. PI-lowering-PI.f64 39.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

   7. Applied egg-rr 39.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \cdot \left(angle \cdot 0.011111111111111112\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 26500000000:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 2.3 \cdot 10^{+67}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112 + \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot -2.2862368541380886 \cdot 10^{-7}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ \begin{array}{l} t_0 := \pi \cdot \left(b - a\right)\\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot t\_0\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ \end{array} \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (- b a))))
   (*
    angle_s
    (if (<= angle_m 3e-25)
      (* (+ b a) (* 0.011111111111111112 (* angle_m t_0)))
      (* (* (+ b a) t_0) (* angle_m 0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (b - a);
	double tmp;
	if (angle_m <= 3e-25) {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * t_0));
	} else {
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double t_0 = Math.PI * (b - a);
	double tmp;
	if (angle_m <= 3e-25) {
		tmp = (b + a) * (0.011111111111111112 * (angle_m * t_0));
	} else {
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	t_0 = math.pi * (b - a)
	tmp = 0
	if angle_m <= 3e-25:
		tmp = (b + a) * (0.011111111111111112 * (angle_m * t_0))
	else:
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112)
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	t_0 = Float64(pi * Float64(b - a))
	tmp = 0.0
	if (angle_m <= 3e-25)
		tmp = Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(angle_m * t_0)));
	else
		tmp = Float64(Float64(Float64(b + a) * t_0) * Float64(angle_m * 0.011111111111111112));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	t_0 = pi * (b - a);
	tmp = 0.0;
	if (angle_m <= 3e-25)
		tmp = (b + a) * (0.011111111111111112 * (angle_m * t_0));
	else
		tmp = ((b + a) * t_0) * (angle_m * 0.011111111111111112);
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]}, N[(angle$95$s * If[LessEqual[angle$95$m, 3e-25], N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + a), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.9999999999999998e-25

   1. Initial program 63.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)\right) \cdot \left(\color{blue}{\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)} \cdot \left(b - a\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(b + a\right), \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\color{blue}{\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)}\right)\right) \]

   6. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b - a\right)}\right)\right)\right)\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} - a\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 74.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Simplified 74.7%

      \[\leadsto \left(b + a\right) \cdot \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)} \]

   if 2.9999999999999998e-25 < angle

   1. Initial program 43.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 36.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 36.0%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b \cdot b - a \cdot a\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      2. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b + a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      8. PI-lowering-PI.f64 37.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

   7. Applied egg-rr 37.4%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \cdot \left(angle \cdot 0.011111111111111112\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3 \cdot 10^{-25}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+156}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot \left(angle\_m \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 1.6e+156)
    (* (* (+ b a) (* PI (- b a))) (* angle_m 0.011111111111111112))
    (* (* b 0.011111111111111112) (* PI (* b angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.6e+156) {
		tmp = ((b + a) * (((double) M_PI) * (b - a))) * (angle_m * 0.011111111111111112);
	} else {
		tmp = (b * 0.011111111111111112) * (((double) M_PI) * (b * angle_m));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.6e+156) {
		tmp = ((b + a) * (Math.PI * (b - a))) * (angle_m * 0.011111111111111112);
	} else {
		tmp = (b * 0.011111111111111112) * (Math.PI * (b * angle_m));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 1.6e+156:
		tmp = ((b + a) * (math.pi * (b - a))) * (angle_m * 0.011111111111111112)
	else:
		tmp = (b * 0.011111111111111112) * (math.pi * (b * angle_m))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 1.6e+156)
		tmp = Float64(Float64(Float64(b + a) * Float64(pi * Float64(b - a))) * Float64(angle_m * 0.011111111111111112));
	else
		tmp = Float64(Float64(b * 0.011111111111111112) * Float64(pi * Float64(b * angle_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.6e+156)
		tmp = ((b + a) * (pi * (b - a))) * (angle_m * 0.011111111111111112);
	else
		tmp = (b * 0.011111111111111112) * (pi * (b * angle_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 1.6e+156], N[(N[(N[(b + a), $MachinePrecision] * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 1.60000000000000001e156

   1. Initial program 59.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 55.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 55.8%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b \cdot b - a \cdot a\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      2. difference-of-squares N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b + a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \left(\left(b - a\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\left(b - a\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      8. PI-lowering-PI.f64 56.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

   7. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \pi\right)\right)} \cdot \left(angle \cdot 0.011111111111111112\right) \]

   if 1.60000000000000001e156 < b

   1. Initial program 51.1%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 43.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 65.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   8. Simplified 65.4%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{angle}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot b\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(b \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \left(b \cdot angle\right)\right)\right) \]

      8. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   10. Applied egg-rr 75.4%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{1}{90} \cdot \left(b \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot b\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot b\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b \cdot angle\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

       7. *-lowering-*.f64 78.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot b\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+156}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+150}:\\ \;\;\;\;angle\_m \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 1.6e+150)
    (* angle_m (* 0.011111111111111112 (* PI (- (* b b) (* a a)))))
    (* (* b 0.011111111111111112) (* PI (* b angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.6e+150) {
		tmp = angle_m * (0.011111111111111112 * (((double) M_PI) * ((b * b) - (a * a))));
	} else {
		tmp = (b * 0.011111111111111112) * (((double) M_PI) * (b * angle_m));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.6e+150) {
		tmp = angle_m * (0.011111111111111112 * (Math.PI * ((b * b) - (a * a))));
	} else {
		tmp = (b * 0.011111111111111112) * (Math.PI * (b * angle_m));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 1.6e+150:
		tmp = angle_m * (0.011111111111111112 * (math.pi * ((b * b) - (a * a))))
	else:
		tmp = (b * 0.011111111111111112) * (math.pi * (b * angle_m))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 1.6e+150)
		tmp = Float64(angle_m * Float64(0.011111111111111112 * Float64(pi * Float64(Float64(b * b) - Float64(a * a)))));
	else
		tmp = Float64(Float64(b * 0.011111111111111112) * Float64(pi * Float64(b * angle_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.6e+150)
		tmp = angle_m * (0.011111111111111112 * (pi * ((b * b) - (a * a))));
	else
		tmp = (b * 0.011111111111111112) * (pi * (b * angle_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 1.6e+150], N[(angle$95$m * N[(0.011111111111111112 * N[(Pi * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 1.60000000000000008e150

   1. Initial program 59.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 55.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(\frac{1}{90} \cdot \color{blue}{angle}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90}\right) \cdot \color{blue}{angle} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \frac{1}{90}\right), \color{blue}{angle}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right), \frac{1}{90}\right), angle\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \frac{1}{90}\right), angle\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot b - a \cdot a\right)\right), \frac{1}{90}\right), angle\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right)\right), \frac{1}{90}\right), angle\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \frac{1}{90}\right), angle\right) \]

      9. *-lowering-*.f64 55.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \frac{1}{90}\right), angle\right) \]

   7. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot 0.011111111111111112\right) \cdot angle} \]

   if 1.60000000000000008e150 < b

   1. Initial program 51.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 44.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 44.5%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 64.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   8. Simplified 64.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{angle}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot b\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(b \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \left(b \cdot angle\right)\right)\right) \]

      8. *-lowering-*.f64 74.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   10. Applied egg-rr 74.1%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{1}{90} \cdot \left(b \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot b\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot b\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b \cdot angle\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

       7. *-lowering-*.f64 77.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 77.0%

       \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot b\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+150}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 54.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle\_m \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 1.8e+150)
    (* 0.011111111111111112 (* PI (* angle_m (- (* b b) (* a a)))))
    (* (* b 0.011111111111111112) (* PI (* b angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.8e+150) {
		tmp = 0.011111111111111112 * (((double) M_PI) * (angle_m * ((b * b) - (a * a))));
	} else {
		tmp = (b * 0.011111111111111112) * (((double) M_PI) * (b * angle_m));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.8e+150) {
		tmp = 0.011111111111111112 * (Math.PI * (angle_m * ((b * b) - (a * a))));
	} else {
		tmp = (b * 0.011111111111111112) * (Math.PI * (b * angle_m));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 1.8e+150:
		tmp = 0.011111111111111112 * (math.pi * (angle_m * ((b * b) - (a * a))))
	else:
		tmp = (b * 0.011111111111111112) * (math.pi * (b * angle_m))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 1.8e+150)
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(angle_m * Float64(Float64(b * b) - Float64(a * a)))));
	else
		tmp = Float64(Float64(b * 0.011111111111111112) * Float64(pi * Float64(b * angle_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.8e+150)
		tmp = 0.011111111111111112 * (pi * (angle_m * ((b * b) - (a * a))));
	else
		tmp = (b * 0.011111111111111112) * (pi * (b * angle_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 1.8e+150], N[(0.011111111111111112 * N[(Pi * N[(angle$95$m * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 1.79999999999999993e150

   1. Initial program 59.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 55.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot angle\right) \cdot \color{blue}{\frac{1}{90}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot angle\right), \color{blue}{\frac{1}{90}}\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)\right), \frac{1}{90}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)\right), \frac{1}{90}\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)\right), \frac{1}{90}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(b \cdot b - a \cdot a\right), angle\right)\right), \frac{1}{90}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot b\right), \left(a \cdot a\right)\right), angle\right)\right), \frac{1}{90}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right), angle\right)\right), \frac{1}{90}\right) \]

      9. *-lowering-*.f64 55.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right), angle\right)\right), \frac{1}{90}\right) \]

   7. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot angle\right)\right) \cdot 0.011111111111111112} \]

   if 1.79999999999999993e150 < b

   1. Initial program 51.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 44.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 44.5%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 64.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   8. Simplified 64.7%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{angle}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot b\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(b \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \left(b \cdot angle\right)\right)\right) \]

      8. *-lowering-*.f64 74.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   10. Applied egg-rr 74.1%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{1}{90} \cdot \left(b \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot b\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot b\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b \cdot angle\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

       7. *-lowering-*.f64 77.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 77.0%

       \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot b\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+150}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.6% accurate, 26.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-32}:\\ \;\;\;\;\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(\left(a \cdot a\right) \cdot \left(0 - \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 5.2e-32)
    (* (* angle_m 0.011111111111111112) (* (* a a) (- 0.0 PI)))
    (* b (* (- b a) (* (* PI angle_m) 0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.2e-32) {
		tmp = (angle_m * 0.011111111111111112) * ((a * a) * (0.0 - ((double) M_PI)));
	} else {
		tmp = b * ((b - a) * ((((double) M_PI) * angle_m) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.2e-32) {
		tmp = (angle_m * 0.011111111111111112) * ((a * a) * (0.0 - Math.PI));
	} else {
		tmp = b * ((b - a) * ((Math.PI * angle_m) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 5.2e-32:
		tmp = (angle_m * 0.011111111111111112) * ((a * a) * (0.0 - math.pi))
	else:
		tmp = b * ((b - a) * ((math.pi * angle_m) * 0.011111111111111112))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 5.2e-32)
		tmp = Float64(Float64(angle_m * 0.011111111111111112) * Float64(Float64(a * a) * Float64(0.0 - pi)));
	else
		tmp = Float64(b * Float64(Float64(b - a) * Float64(Float64(pi * angle_m) * 0.011111111111111112)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 5.2e-32)
		tmp = (angle_m * 0.011111111111111112) * ((a * a) * (0.0 - pi));
	else
		tmp = b * ((b - a) * ((pi * angle_m) * 0.011111111111111112));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 5.2e-32], N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(0.0 - Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b - a), $MachinePrecision] * N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 5.1999999999999995e-32

   1. Initial program 60.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 56.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}, \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{neg}\left({a}^{2} \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(0 - {a}^{2} \cdot \mathsf{PI}\left(\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left({a}^{2} \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{angle}, \frac{1}{90}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({a}^{2}\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({a}^{2}\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(a \cdot a\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

      8. *-lowering-*.f64 40.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \frac{1}{90}\right)\right) \]

   8. Simplified 40.9%

      \[\leadsto \color{blue}{\left(0 - \pi \cdot \left(a \cdot a\right)\right)} \cdot \left(angle \cdot 0.011111111111111112\right) \]

   if 5.1999999999999995e-32 < b

   1. Initial program 51.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 51.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 51.5%

      \[\leadsto \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot b\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   9. Applied egg-rr 52.8%

      \[\leadsto \color{blue}{b \cdot \left(\sin \left(\frac{2}{\frac{\frac{180}{angle}}{\pi}}\right) \cdot \left(b - a\right)\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right), \mathsf{\_.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       4. PI-lowering-PI.f64 49.3%

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   12. Simplified 49.3%

       \[\leadsto b \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-32}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\left(a \cdot a\right) \cdot \left(0 - \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.7% accurate, 26.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\pi \cdot angle\_m\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 5.5e-32)
    (* (* PI angle_m) (* (* a a) -0.011111111111111112))
    (* b (* (- b a) (* (* PI angle_m) 0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.5e-32) {
		tmp = (((double) M_PI) * angle_m) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = b * ((b - a) * ((((double) M_PI) * angle_m) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.5e-32) {
		tmp = (Math.PI * angle_m) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = b * ((b - a) * ((Math.PI * angle_m) * 0.011111111111111112));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 5.5e-32:
		tmp = (math.pi * angle_m) * ((a * a) * -0.011111111111111112)
	else:
		tmp = b * ((b - a) * ((math.pi * angle_m) * 0.011111111111111112))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 5.5e-32)
		tmp = Float64(Float64(pi * angle_m) * Float64(Float64(a * a) * -0.011111111111111112));
	else
		tmp = Float64(b * Float64(Float64(b - a) * Float64(Float64(pi * angle_m) * 0.011111111111111112)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 5.5e-32)
		tmp = (pi * angle_m) * ((a * a) * -0.011111111111111112);
	else
		tmp = b * ((b - a) * ((pi * angle_m) * 0.011111111111111112));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 5.5e-32], N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b - a), $MachinePrecision] * N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 5.50000000000000024e-32

   1. Initial program 60.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 56.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{90} \cdot {a}^{2}\right), \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left({a}^{2}\right)\right), \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot a\right)\right), \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      7. PI-lowering-PI.f64 40.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right) \]

   8. Simplified 40.8%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \pi\right)} \]

   if 5.50000000000000024e-32 < b

   1. Initial program 51.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 51.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 51.5%

      \[\leadsto \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \left(2 \cdot b\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(b \cdot 2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto b \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\right)\right)} \]

   9. Applied egg-rr 52.8%

      \[\leadsto \color{blue}{b \cdot \left(\sin \left(\frac{2}{\frac{\frac{180}{angle}}{\pi}}\right) \cdot \left(b - a\right)\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right), \mathsf{\_.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

       4. PI-lowering-PI.f64 49.3%

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   12. Simplified 49.3%

       \[\leadsto b \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)} \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b - a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 44.7% accurate, 26.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\pi \cdot angle\_m\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot \left(b \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 5.5e-32)
    (* (* PI angle_m) (* (* a a) -0.011111111111111112))
    (* 0.011111111111111112 (* (* PI (- b a)) (* b angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.5e-32) {
		tmp = (((double) M_PI) * angle_m) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * ((((double) M_PI) * (b - a)) * (b * angle_m));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.5e-32) {
		tmp = (Math.PI * angle_m) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * ((Math.PI * (b - a)) * (b * angle_m));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 5.5e-32:
		tmp = (math.pi * angle_m) * ((a * a) * -0.011111111111111112)
	else:
		tmp = 0.011111111111111112 * ((math.pi * (b - a)) * (b * angle_m))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 5.5e-32)
		tmp = Float64(Float64(pi * angle_m) * Float64(Float64(a * a) * -0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(Float64(pi * Float64(b - a)) * Float64(b * angle_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 5.5e-32)
		tmp = (pi * angle_m) * ((a * a) * -0.011111111111111112);
	else
		tmp = 0.011111111111111112 * ((pi * (b - a)) * (b * angle_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 5.5e-32], N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 5.50000000000000024e-32

   1. Initial program 60.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 56.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 56.5%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{90} \cdot {a}^{2}\right), \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left({a}^{2}\right)\right), \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot a\right)\right), \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      7. PI-lowering-PI.f64 40.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right) \]

   8. Simplified 40.8%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \pi\right)} \]

   if 5.50000000000000024e-32 < b

   1. Initial program 51.8%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      2. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - {a}^{2}\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      3. pow2 N/A

         \[\leadsto \left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)} \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot 2\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \color{blue}{\left(\left(b \cdot b - a \cdot a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)} \]

      9. difference-of-squares N/A

         \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(2 \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot \left(\left(b + a\right) \cdot \color{blue}{\left(\left(b - a\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)}\right) \]

   4. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(2 \cdot \left(b \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(2 \cdot b\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)}, \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)}\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. PI-lowering-PI.f64 51.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, b\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 51.5%

      \[\leadsto \color{blue}{\left(\left(2 \cdot b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot \left(\cos \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(angle \cdot b\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(angle \cdot b\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b - a\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b - a\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b - a\right)}\right)\right)\right) \]

      6. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} - a\right)\right)\right)\right) \]

      7. --lowering--.f64 48.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(b, \color{blue}{a}\right)\right)\right)\right) \]

   10. Simplified 48.2%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot \left(b - a\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-32}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(\pi \cdot \left(b - a\right)\right) \cdot \left(b \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 43.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;\left(\pi \cdot angle\_m\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 1.25e+104)
    (* (* PI angle_m) (* (* a a) -0.011111111111111112))
    (* (* b 0.011111111111111112) (* PI (* b angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.25e+104) {
		tmp = (((double) M_PI) * angle_m) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = (b * 0.011111111111111112) * (((double) M_PI) * (b * angle_m));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.25e+104) {
		tmp = (Math.PI * angle_m) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = (b * 0.011111111111111112) * (Math.PI * (b * angle_m));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 1.25e+104:
		tmp = (math.pi * angle_m) * ((a * a) * -0.011111111111111112)
	else:
		tmp = (b * 0.011111111111111112) * (math.pi * (b * angle_m))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 1.25e+104)
		tmp = Float64(Float64(pi * angle_m) * Float64(Float64(a * a) * -0.011111111111111112));
	else
		tmp = Float64(Float64(b * 0.011111111111111112) * Float64(pi * Float64(b * angle_m)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.25e+104)
		tmp = (pi * angle_m) * ((a * a) * -0.011111111111111112);
	else
		tmp = (b * 0.011111111111111112) * (pi * (b * angle_m));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 1.25e+104], N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 1.2499999999999999e104

   1. Initial program 59.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 55.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{90} \cdot {a}^{2}\right), \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left({a}^{2}\right)\right), \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot a\right)\right), \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      7. PI-lowering-PI.f64 40.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right) \]

   8. Simplified 40.0%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \pi\right)} \]

   if 1.2499999999999999e104 < b

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 47.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 55.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   8. Simplified 55.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{angle}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot b\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(b \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \left(b \cdot angle\right)\right)\right) \]

      8. *-lowering-*.f64 61.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   10. Applied egg-rr 61.4%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \frac{1}{90} \cdot \left(b \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{90} \cdot b\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{90} \cdot b\right), \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot angle\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b \cdot angle\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

       7. *-lowering-*.f64 63.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{90}, b\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   12. Applied egg-rr 63.4%

       \[\leadsto \color{blue}{\left(0.011111111111111112 \cdot b\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 43.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+102}:\\ \;\;\;\;\left(\pi \cdot angle\_m\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\_m\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 9e+102)
    (* (* PI angle_m) (* (* a a) -0.011111111111111112))
    (* 0.011111111111111112 (* PI (* b (* b angle_m)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 9e+102) {
		tmp = (((double) M_PI) * angle_m) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (((double) M_PI) * (b * (b * angle_m)));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 9e+102) {
		tmp = (Math.PI * angle_m) * ((a * a) * -0.011111111111111112);
	} else {
		tmp = 0.011111111111111112 * (Math.PI * (b * (b * angle_m)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 9e+102:
		tmp = (math.pi * angle_m) * ((a * a) * -0.011111111111111112)
	else:
		tmp = 0.011111111111111112 * (math.pi * (b * (b * angle_m)))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 9e+102)
		tmp = Float64(Float64(pi * angle_m) * Float64(Float64(a * a) * -0.011111111111111112));
	else
		tmp = Float64(0.011111111111111112 * Float64(pi * Float64(b * Float64(b * angle_m))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 9e+102)
		tmp = (pi * angle_m) * ((a * a) * -0.011111111111111112);
	else
		tmp = 0.011111111111111112 * (pi * (b * (b * angle_m)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 9e+102], N[(N[(Pi * angle$95$m), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(Pi * N[(b * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 9.00000000000000042e102

   1. Initial program 59.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 55.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 55.9%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{90} \cdot {a}^{2}\right), \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left({a}^{2}\right)\right), \left(\color{blue}{angle} \cdot \mathsf{PI}\left(\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \left(a \cdot a\right)\right), \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      7. PI-lowering-PI.f64 40.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{90}, \mathsf{*.f64}\left(a, a\right)\right), \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right) \]

   8. Simplified 40.0%

      \[\leadsto \color{blue}{\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot \pi\right)} \]

   if 9.00000000000000042e102 < b

   1. Initial program 53.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

      12. *-lowering-*.f64 47.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 55.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

   8. Simplified 55.0%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{angle}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot b\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(b \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \left(b \cdot angle\right)\right)\right) \]

      8. *-lowering-*.f64 61.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

   10. Applied egg-rr 61.4%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(b \cdot angle\right) \cdot \color{blue}{\left(b \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(b \cdot angle\right) \cdot b\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\left(b \cdot angle\right) \cdot b\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot angle\right), b\right), \mathsf{PI}\left(\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), b\right), \mathsf{PI}\left(\right)\right)\right) \]

       6. PI-lowering-PI.f64 61.5%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), b\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

   12. Applied egg-rr 61.5%

       \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\left(b \cdot angle\right) \cdot b\right) \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{+102}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 39.0% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot angle\_m\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* b (* b (* PI angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (b * (b * (((double) M_PI) * angle_m))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (b * (b * (Math.PI * angle_m))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (b * (b * (math.pi * angle_m))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(b * Float64(b * Float64(pi * angle_m)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (b * (b * (pi * angle_m))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(b * N[(b * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

   12. *-lowering-*.f64 54.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

5. Simplified 54.5%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 36.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

8. Simplified 36.7%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot b\right) \cdot \color{blue}{b}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot b\right), \color{blue}{b}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), b\right), b\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot angle\right), b\right), b\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), angle\right), b\right), b\right)\right) \]

   7. PI-lowering-PI.f64 38.4%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), angle\right), b\right), b\right)\right) \]

10. Applied egg-rr 38.4%

    \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\left(\pi \cdot angle\right) \cdot b\right) \cdot b\right)} \]

11. Final simplification 38.4%

    \[\leadsto 0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot angle\right)\right)\right) \]
12. Add Preprocessing

Alternative 24: 39.0% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\_m\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* PI (* b (* b angle_m))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (((double) M_PI) * (b * (b * angle_m))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (Math.PI * (b * (b * angle_m))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (math.pi * (b * (b * angle_m))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(pi * Float64(b * Float64(b * angle_m)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (pi * (b * (b * angle_m))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(Pi * N[(b * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

   12. *-lowering-*.f64 54.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

5. Simplified 54.5%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 36.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

8. Simplified 36.7%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{angle}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot b\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(b \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \left(b \cdot angle\right)\right)\right) \]

   8. *-lowering-*.f64 38.4%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

10. Applied egg-rr 38.4%

    \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(b \cdot angle\right) \cdot \color{blue}{\left(b \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(b \cdot angle\right) \cdot b\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\left(b \cdot angle\right) \cdot b\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot angle\right), b\right), \mathsf{PI}\left(\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), b\right), \mathsf{PI}\left(\right)\right)\right) \]

    6. PI-lowering-PI.f64 38.4%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), b\right), \mathsf{PI.f64}\left(\right)\right)\right) \]

12. Applied egg-rr 38.4%

    \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(\left(b \cdot angle\right) \cdot b\right) \cdot \pi\right)} \]

13. Final simplification 38.4%

    \[\leadsto 0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot angle\right)\right)\right) \]
14. Add Preprocessing

Alternative 25: 38.9% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(\left(b \cdot angle\_m\right) \cdot \left(b \cdot \pi\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* (* b angle_m) (* b PI)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((b * angle_m) * (b * ((double) M_PI))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * ((b * angle_m) * (b * Math.PI)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * ((b * angle_m) * (b * math.pi)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(Float64(b * angle_m) * Float64(b * pi))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * ((b * angle_m) * (b * pi)));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(N[(b * angle$95$m), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

   12. *-lowering-*.f64 54.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

5. Simplified 54.5%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 36.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

8. Simplified 36.7%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{angle}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot b\right) \cdot angle\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \left(\left(\mathsf{PI}\left(\right) \cdot b\right) \cdot \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot b\right), \color{blue}{\left(b \cdot angle\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\left(b \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI}\left(\right)\right), \left(\color{blue}{b} \cdot angle\right)\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \left(b \cdot angle\right)\right)\right) \]

   8. *-lowering-*.f64 38.4%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right), \mathsf{*.f64}\left(b, \color{blue}{angle}\right)\right)\right) \]

10. Applied egg-rr 38.4%

    \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)} \]

11. Final simplification 38.4%

    \[\leadsto 0.011111111111111112 \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \pi\right)\right) \]
12. Add Preprocessing

Alternative 26: 35.1% accurate, 46.6× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* PI (* b b))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * (b * b))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * (b * b))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * (b * b))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(b * b)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * (b * b))));
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * N[(0.011111111111111112 * N[(angle$95$m * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{90} \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \color{blue}{\left(\frac{1}{90} \cdot angle\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right), \color{blue}{\left(\frac{1}{90} \cdot angle\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\color{blue}{\frac{1}{90}} \cdot angle\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({b}^{2} - {a}^{2}\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left({b}^{2}\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\left(b \cdot b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left({a}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(a \cdot a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(\frac{1}{90} \cdot angle\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \left(angle \cdot \color{blue}{\frac{1}{90}}\right)\right) \]

   12. *-lowering-*.f64 54.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(a, a\right)\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{\frac{1}{90}}\right)\right) \]

5. Simplified 54.5%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{b}^{2}}\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\left({b}^{2}\right)}\right)\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\color{blue}{b}}^{2}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(b \cdot \color{blue}{b}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 36.7%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(b, \color{blue}{b}\right)\right)\right)\right) \]

8. Simplified 36.7%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))