ab-angle->ABCF B

Percentage Accurate: 53.6% → 67.8%

Time: 17.0s

Alternatives: 20

Speedup: 32.2×

• 32.6% 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: 53.6% 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: 67.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{\pi}^{1.5}}\\ \mathbf{if}\;{b}^{2} \leq 0:\\ \;\;\;\;\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{t\_0 \cdot t\_0}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (pow PI 1.5))))
   (if (<= (pow b 2.0) 0.0)
     (*
      (* (* 2.0 (sin (/ PI (/ 180.0 angle)))) (+ b a))
      (* (cos (/ (* t_0 t_0) (/ 180.0 angle))) (- b a)))
     (* (- b a) (* (+ b a) (sin (* 0.011111111111111112 (* PI angle))))))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(pow(((double) M_PI), 1.5));
	double tmp;
	if (pow(b, 2.0) <= 0.0) {
		tmp = ((2.0 * sin((((double) M_PI) / (180.0 / angle)))) * (b + a)) * (cos(((t_0 * t_0) / (180.0 / angle))) * (b - a));
	} else {
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.pow(Math.PI, 1.5));
	double tmp;
	if (Math.pow(b, 2.0) <= 0.0) {
		tmp = ((2.0 * Math.sin((Math.PI / (180.0 / angle)))) * (b + a)) * (Math.cos(((t_0 * t_0) / (180.0 / angle))) * (b - a));
	} else {
		tmp = (b - a) * ((b + a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cbrt((pi ^ 1.5))
	tmp = 0.0
	if ((b ^ 2.0) <= 0.0)
		tmp = Float64(Float64(Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle)))) * Float64(b + a)) * Float64(cos(Float64(Float64(t_0 * t_0) / Float64(180.0 / angle))) * Float64(b - a)));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[Power[b, 2.0], $MachinePrecision], 0.0], N[(N[(N[(2.0 * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 b #s(literal 2 binary64)) < 0.0

   1. Initial program 71.7%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 75.6%

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

      1. add-cbrt-cube N/A

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

      2. add-sqr-sqrt N/A

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

      3. unswap-sqr N/A

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

      4. cbrt-prod N/A

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

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

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

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

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

      7. pow1 N/A

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

      8. pow1/2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. metadata-eval N/A

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

      15. cbrt-lowering-cbrt.f64 N/A

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

      16. pow1 N/A

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

      17. pow1/2 N/A

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

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

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

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

   6. Applied egg-rr 84.6%

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\color{blue}{\sqrt[3]{{\pi}^{1.5}} \cdot \sqrt[3]{{\pi}^{1.5}}}}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right) \]

   if 0.0 < (pow.f64 b #s(literal 2 binary64))

   1. Initial program 55.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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 72.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} \leq 0:\\ \;\;\;\;\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \left(\frac{\sqrt[3]{{\pi}^{1.5}} \cdot \sqrt[3]{{\pi}^{1.5}}}{\frac{180}{angle}}\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t\_0\right) \cdot \cos t\_0 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi + 1}{\frac{180}{angle}} - \frac{angle}{180}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<=
        (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_0)) (cos t_0))
        5e+131)
     (* (- b a) (* (+ b a) (sin (* 0.011111111111111112 (* PI angle)))))
     (*
      (* (* 2.0 (sin (/ PI (/ 180.0 angle)))) (+ b a))
      (* (- b a) (cos (- (/ (+ PI 1.0) (/ 180.0 angle)) (/ angle 180.0))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_0)) * cos(t_0)) <= 5e+131) {
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	} else {
		tmp = ((2.0 * sin((((double) M_PI) / (180.0 / angle)))) * (b + a)) * ((b - a) * cos((((((double) M_PI) + 1.0) / (180.0 / angle)) - (angle / 180.0))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * Math.sin(t_0)) * Math.cos(t_0)) <= 5e+131) {
		tmp = (b - a) * ((b + a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	} else {
		tmp = ((2.0 * Math.sin((Math.PI / (180.0 / angle)))) * (b + a)) * ((b - a) * Math.cos((((Math.PI + 1.0) / (180.0 / angle)) - (angle / 180.0))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * math.sin(t_0)) * math.cos(t_0)) <= 5e+131:
		tmp = (b - a) * ((b + a) * math.sin((0.011111111111111112 * (math.pi * angle))))
	else:
		tmp = ((2.0 * math.sin((math.pi / (180.0 / angle)))) * (b + a)) * ((b - a) * math.cos((((math.pi + 1.0) / (180.0 / angle)) - (angle / 180.0))))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 5e+131)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	else
		tmp = Float64(Float64(Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle)))) * Float64(b + a)) * Float64(Float64(b - a) * cos(Float64(Float64(Float64(pi + 1.0) / Float64(180.0 / angle)) - Float64(angle / 180.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * sin(t_0)) * cos(t_0)) <= 5e+131)
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (pi * angle))));
	else
		tmp = ((2.0 * sin((pi / (180.0 / angle)))) * (b + a)) * ((b - a) * cos((((pi + 1.0) / (180.0 / angle)) - (angle / 180.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 5e+131], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(N[(N[(Pi + 1.0), $MachinePrecision] / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision] - N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) < 4.99999999999999995e131

   1. Initial program 61.5%

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

      1. *-commutative N/A

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

      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 71.1%

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

   if 4.99999999999999995e131 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64)))))

   1. Initial program 53.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 78.2%

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

      1. expm1-log1p-u N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)\right)}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      2. expm1-undefine N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)} - 1}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}} - \frac{1}{\frac{180}{angle}}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}} - \frac{angle}{180}\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(\frac{e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)}}{\frac{180}{angle}}\right), \left(\frac{angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(180, angle\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(e^{\mathsf{log1p}\left(\mathsf{PI}\left(\right)\right)}\right), \left(\frac{180}{angle}\right)\right), \left(\frac{angle}{180}\right)\right)\right), \mathsf{\_.f64}\left(b, a\right)\right)\right) \]

      7. log1p-undefine N/A

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

      8. rem-exp-log N/A

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

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

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

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

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

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

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

      12. /-lowering-/.f64 78.6%

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

   6. Applied egg-rr 78.6%

      \[\leadsto \left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\cos \color{blue}{\left(\frac{1 + \pi}{\frac{180}{angle}} - \frac{angle}{180}\right)} \cdot \left(b - a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi + 1}{\frac{180}{angle}} - \frac{angle}{180}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.1e-108)
   (*
    (* (+ b a) (* 2.0 (sin (* PI (/ angle 180.0)))))
    (* (- b a) (cos (/ PI (/ 180.0 angle)))))
   (* (- b a) (* (+ b a) (sin (* 0.011111111111111112 (* PI angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.1e-108) {
		tmp = ((b + a) * (2.0 * sin((((double) M_PI) * (angle / 180.0))))) * ((b - a) * cos((((double) M_PI) / (180.0 / angle))));
	} else {
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.1e-108) {
		tmp = ((b + a) * (2.0 * Math.sin((Math.PI * (angle / 180.0))))) * ((b - a) * Math.cos((Math.PI / (180.0 / angle))));
	} else {
		tmp = (b - a) * ((b + a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.1e-108:
		tmp = ((b + a) * (2.0 * math.sin((math.pi * (angle / 180.0))))) * ((b - a) * math.cos((math.pi / (180.0 / angle))))
	else:
		tmp = (b - a) * ((b + a) * math.sin((0.011111111111111112 * (math.pi * angle))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.1e-108)
		tmp = Float64(Float64(Float64(b + a) * Float64(2.0 * sin(Float64(pi * Float64(angle / 180.0))))) * Float64(Float64(b - a) * cos(Float64(pi / Float64(180.0 / angle)))));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.1e-108)
		tmp = ((b + a) * (2.0 * sin((pi * (angle / 180.0))))) * ((b - a) * cos((pi / (180.0 / angle))));
	else
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (pi * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.1e-108], N[(N[(N[(b + a), $MachinePrecision] * N[(2.0 * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Cos[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.10000000000000007e-108

   1. Initial program 59.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 70.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. clear-num N/A

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

      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 71.7%

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

   6. Applied egg-rr 71.7%

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

   if 6.10000000000000007e-108 < b

   1. Initial program 58.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. Add Preprocessing
   3. 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 75.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.1 \cdot 10^{-108}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(\left(b - a\right) \cdot \cos \left(\frac{\pi}{\frac{180}{angle}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-134}:\\ \;\;\;\;\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2.7e-134)
   (* (* (* 2.0 (sin (/ PI (/ 180.0 angle)))) (+ b a)) (- b a))
   (* (- b a) (* (+ b a) (sin (* 0.011111111111111112 (* PI angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.7e-134) {
		tmp = ((2.0 * sin((((double) M_PI) / (180.0 / angle)))) * (b + a)) * (b - a);
	} else {
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2.7e-134) {
		tmp = ((2.0 * Math.sin((Math.PI / (180.0 / angle)))) * (b + a)) * (b - a);
	} else {
		tmp = (b - a) * ((b + a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2.7e-134:
		tmp = ((2.0 * math.sin((math.pi / (180.0 / angle)))) * (b + a)) * (b - a)
	else:
		tmp = (b - a) * ((b + a) * math.sin((0.011111111111111112 * (math.pi * angle))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2.7e-134)
		tmp = Float64(Float64(Float64(2.0 * sin(Float64(pi / Float64(180.0 / angle)))) * Float64(b + a)) * Float64(b - a));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2.7e-134)
		tmp = ((2.0 * sin((pi / (180.0 / angle)))) * (b + a)) * (b - a);
	else
		tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (pi * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2.7e-134], N[(N[(N[(2.0 * N[Sin[N[(Pi / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6999999999999998e-134

   1. Initial program 60.6%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 71.7%

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

   5. Taylor expanded in angle around 0

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

   7. Simplified 71.8%

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

   if 2.6999999999999998e-134 < b

   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

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

      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 73.0%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{-134}:\\ \;\;\;\;\left(\left(2 \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;a \leq 1.85 \cdot 10^{-94}:\\ \;\;\;\;b \cdot \left(\left(b + a\right) \cdot \sin t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.011111111111111112 (* PI angle))))
   (if (<= a 1.85e-94)
     (* b (* (+ b a) (sin t_0)))
     (* (- b a) (* (+ b a) t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.011111111111111112 * (((double) M_PI) * angle);
	double tmp;
	if (a <= 1.85e-94) {
		tmp = b * ((b + a) * sin(t_0));
	} else {
		tmp = (b - a) * ((b + a) * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 0.011111111111111112 * (Math.PI * angle);
	double tmp;
	if (a <= 1.85e-94) {
		tmp = b * ((b + a) * Math.sin(t_0));
	} else {
		tmp = (b - a) * ((b + a) * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 0.011111111111111112 * (math.pi * angle)
	tmp = 0
	if a <= 1.85e-94:
		tmp = b * ((b + a) * math.sin(t_0))
	else:
		tmp = (b - a) * ((b + a) * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(0.011111111111111112 * Float64(pi * angle))
	tmp = 0.0
	if (a <= 1.85e-94)
		tmp = Float64(b * Float64(Float64(b + a) * sin(t_0)));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 0.011111111111111112 * (pi * angle);
	tmp = 0.0;
	if (a <= 1.85e-94)
		tmp = b * ((b + a) * sin(t_0));
	else
		tmp = (b - a) * ((b + a) * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.85e-94], N[(b * N[(N[(b + a), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.8499999999999999e-94

   1. Initial program 63.6%

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

      1. *-commutative N/A

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

      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 76.1%

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

   5. Taylor expanded in b around inf

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

   7. Simplified 58.5%

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

   if 1.8499999999999999e-94 < a

   1. Initial program 50.7%

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

      1. *-commutative N/A

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

      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 66.9%

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

   5. Taylor expanded in angle around 0

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

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

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

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

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

      3. PI-lowering-PI.f64 67.3%

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

   7. Simplified 67.3%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-94}:\\ \;\;\;\;b \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (- b a) (* (+ b a) (sin (* 0.011111111111111112 (* PI angle))))))

C

double code(double a, double b, double angle) {
	return (b - a) * ((b + a) * sin((0.011111111111111112 * (((double) M_PI) * angle))));
}

Java

public static double code(double a, double b, double angle) {
	return (b - a) * ((b + a) * Math.sin((0.011111111111111112 * (Math.PI * angle))));
}

Python

def code(a, b, angle):
	return (b - a) * ((b + a) * math.sin((0.011111111111111112 * (math.pi * angle))))

Julia

function code(a, b, angle)
	return Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(0.011111111111111112 * Float64(pi * angle)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b - a) * ((b + a) * sin((0.011111111111111112 * (pi * angle))));
end

Wolfram

code[a_, b_, angle_] := N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. 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 73.1%

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

5. Final simplification 73.1%

   \[\leadsto \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right) \]
6. Add Preprocessing

Alternative 7: 67.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (- b a) (* (+ b a) (sin (* angle (* PI 0.011111111111111112))))))

C

double code(double a, double b, double angle) {
	return (b - a) * ((b + a) * sin((angle * (((double) M_PI) * 0.011111111111111112))));
}

Java

public static double code(double a, double b, double angle) {
	return (b - a) * ((b + a) * Math.sin((angle * (Math.PI * 0.011111111111111112))));
}

Python

def code(a, b, angle):
	return (b - a) * ((b + a) * math.sin((angle * (math.pi * 0.011111111111111112))))

Julia

function code(a, b, angle)
	return Float64(Float64(b - a) * Float64(Float64(b + a) * sin(Float64(angle * Float64(pi * 0.011111111111111112)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b - a) * ((b + a) * sin((angle * (pi * 0.011111111111111112))));
end

Wolfram

code[a_, b_, angle_] := N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. 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 73.1%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

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

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

   6. PI-lowering-PI.f64 72.3%

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

6. Applied egg-rr 72.3%

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

7. Final simplification 72.3%

   \[\leadsto \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \]
8. Add Preprocessing

Alternative 8: 63.0% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-84}:\\ \;\;\;\;\frac{1}{\frac{1}{b - a}} \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right) + \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.85e-84)
   (*
    (/ 1.0 (/ 1.0 (- b a)))
    (* 0.011111111111111112 (* angle (* PI (+ b a)))))
   (*
    (- b a)
    (*
     (+ b a)
     (*
      angle
      (+
       (* (* -2.2862368541380886e-7 (* angle angle)) (* PI (* PI PI)))
       (* PI 0.011111111111111112)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.85e-84) {
		tmp = (1.0 / (1.0 / (b - a))) * (0.011111111111111112 * (angle * (((double) M_PI) * (b + a))));
	} else {
		tmp = (b - a) * ((b + a) * (angle * (((-2.2862368541380886e-7 * (angle * angle)) * (((double) M_PI) * (((double) M_PI) * ((double) M_PI)))) + (((double) M_PI) * 0.011111111111111112))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.85e-84) {
		tmp = (1.0 / (1.0 / (b - a))) * (0.011111111111111112 * (angle * (Math.PI * (b + a))));
	} else {
		tmp = (b - a) * ((b + a) * (angle * (((-2.2862368541380886e-7 * (angle * angle)) * (Math.PI * (Math.PI * Math.PI))) + (Math.PI * 0.011111111111111112))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.85e-84:
		tmp = (1.0 / (1.0 / (b - a))) * (0.011111111111111112 * (angle * (math.pi * (b + a))))
	else:
		tmp = (b - a) * ((b + a) * (angle * (((-2.2862368541380886e-7 * (angle * angle)) * (math.pi * (math.pi * math.pi))) + (math.pi * 0.011111111111111112))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.85e-84)
		tmp = Float64(Float64(1.0 / Float64(1.0 / Float64(b - a))) * Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b + a)))));
	else
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(angle * Float64(Float64(Float64(-2.2862368541380886e-7 * Float64(angle * angle)) * Float64(pi * Float64(pi * pi))) + Float64(pi * 0.011111111111111112)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.85e-84)
		tmp = (1.0 / (1.0 / (b - a))) * (0.011111111111111112 * (angle * (pi * (b + a))));
	else
		tmp = (b - a) * ((b + a) * (angle * (((-2.2862368541380886e-7 * (angle * angle)) * (pi * (pi * pi))) + (pi * 0.011111111111111112))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.85e-84], N[(N[(1.0 / N[(1.0 / N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(angle * N[(N[(N[(-2.2862368541380886e-7 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.85e-84

   1. Initial program 60.2%

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

      1. *-commutative N/A

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

      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 72.1%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

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

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

      7. --lowering--.f64 72.1%

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

   6. Applied egg-rr 72.1%

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

   7. Taylor expanded in angle around 0

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

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

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

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

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

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

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

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

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

      5. +-lowering-+.f64 67.1%

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

   9. Simplified 67.1%

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

   if 1.85e-84 < b

   1. Initial program 57.6%

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

      1. *-commutative N/A

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

      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 75.1%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\color{blue}{\left(angle \cdot \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)}, \mathsf{+.f64}\left(b, a\right)\right)\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{b}, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{-1}{4374000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\left(\frac{-1}{4374000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{4374000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{1}{90} \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

      17. PI-lowering-PI.f64 73.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(b, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4374000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{1}{90}, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{+.f64}\left(b, a\right)\right)\right) \]

   7. Simplified 73.3%

      \[\leadsto \left(b - a\right) \cdot \left(\color{blue}{\left(angle \cdot \left(\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right) + 0.011111111111111112 \cdot \pi\right)\right)} \cdot \left(b + a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{-84}:\\ \;\;\;\;\frac{1}{\frac{1}{b - a}} \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(\left(-2.2862368541380886 \cdot 10^{-7} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right) + \pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b \cdot b\right)\right) \cdot \left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) + \pi \cdot 0.005555555555555556\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 2e+264)
   (* (- b a) (* (+ b a) (* 0.011111111111111112 (* PI angle))))
   (*
    (* 2.0 (* b b))
    (*
     angle
     (+
      (* (* PI (* PI PI)) (* (* angle angle) -2.8577960676726107e-8))
      (* PI 0.005555555555555556))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+264) {
		tmp = (b - a) * ((b + a) * (0.011111111111111112 * (((double) M_PI) * angle)));
	} else {
		tmp = (2.0 * (b * b)) * (angle * (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * ((angle * angle) * -2.8577960676726107e-8)) + (((double) M_PI) * 0.005555555555555556)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 2e+264) {
		tmp = (b - a) * ((b + a) * (0.011111111111111112 * (Math.PI * angle)));
	} else {
		tmp = (2.0 * (b * b)) * (angle * (((Math.PI * (Math.PI * Math.PI)) * ((angle * angle) * -2.8577960676726107e-8)) + (Math.PI * 0.005555555555555556)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 2e+264:
		tmp = (b - a) * ((b + a) * (0.011111111111111112 * (math.pi * angle)))
	else:
		tmp = (2.0 * (b * b)) * (angle * (((math.pi * (math.pi * math.pi)) * ((angle * angle) * -2.8577960676726107e-8)) + (math.pi * 0.005555555555555556)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 2e+264)
		tmp = Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(pi * angle))));
	else
		tmp = Float64(Float64(2.0 * Float64(b * b)) * Float64(angle * Float64(Float64(Float64(pi * Float64(pi * pi)) * Float64(Float64(angle * angle) * -2.8577960676726107e-8)) + Float64(pi * 0.005555555555555556))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 2e+264)
		tmp = (b - a) * ((b + a) * (0.011111111111111112 * (pi * angle)));
	else
		tmp = (2.0 * (b * b)) * (angle * (((pi * (pi * pi)) * ((angle * angle) * -2.8577960676726107e-8)) + (pi * 0.005555555555555556)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 2e+264], N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.00000000000000009e264

   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

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

      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 72.2%

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

   5. Taylor expanded in angle around 0

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

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

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

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

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

      3. PI-lowering-PI.f64 65.7%

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

   7. Simplified 65.7%

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

   if 2.00000000000000009e264 < b

   1. Initial program 62.5%

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

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 75.0%

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

   5. Simplified 75.0%

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

   6. Taylor expanded in angle around 0

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

   8. Simplified 62.5%

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

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right), 1\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\left(\left(\frac{-1}{34992000} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{34992000} \cdot {angle}^{2}\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \left({angle}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \left(angle \cdot angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left({\mathsf{PI}\left(\right)}^{3}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), 1\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right)\right)\right), 1\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \frac{1}{180}\right)\right)\right)\right), 1\right) \]

       18. PI-lowering-PI.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(b, b\right)\right), \mathsf{*.f64}\left(angle, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{34992000}, \mathsf{*.f64}\left(angle, angle\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \frac{1}{180}\right)\right)\right)\right), 1\right) \]

   11. Simplified 100.0%

       \[\leadsto \left(\left(2 \cdot \left(b \cdot b\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right) + \pi \cdot 0.005555555555555556\right)\right)}\right) \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(b \cdot b\right)\right) \cdot \left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) + \pi \cdot 0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.3e+117)
   (* (* angle 0.011111111111111112) (* PI (- (* b b) (* a a))))
   (* b (* (* angle 0.011111111111111112) (* b PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.3e+117) {
		tmp = (angle * 0.011111111111111112) * (((double) M_PI) * ((b * b) - (a * a)));
	} else {
		tmp = b * ((angle * 0.011111111111111112) * (b * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.3e+117) {
		tmp = (angle * 0.011111111111111112) * (Math.PI * ((b * b) - (a * a)));
	} else {
		tmp = b * ((angle * 0.011111111111111112) * (b * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.3e+117:
		tmp = (angle * 0.011111111111111112) * (math.pi * ((b * b) - (a * a)))
	else:
		tmp = b * ((angle * 0.011111111111111112) * (b * math.pi))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.3e+117)
		tmp = Float64(Float64(angle * 0.011111111111111112) * Float64(pi * Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(b * Float64(Float64(angle * 0.011111111111111112) * Float64(b * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.3e+117)
		tmp = (angle * 0.011111111111111112) * (pi * ((b * b) - (a * a)));
	else
		tmp = b * ((angle * 0.011111111111111112) * (b * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.3e+117], N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(Pi * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.29999999999999998e117

   1. Initial program 60.9%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 58.2%

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

   5. Simplified 58.2%

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

   if 4.29999999999999998e117 < b

   1. Initial program 51.7%

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

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 54.2%

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

   5. Simplified 54.2%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 49.0%

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

   8. Simplified 49.0%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. PI-lowering-PI.f64 73.6%

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

   10. Applied egg-rr 73.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;\left(angle \cdot 0.011111111111111112\right) \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.9% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\left(\pi \cdot 0.011111111111111112\right) \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8e+153)
   (* (* PI 0.011111111111111112) (* angle (- (* b b) (* a a))))
   (* b (* (* angle 0.011111111111111112) (* b PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8e+153) {
		tmp = (((double) M_PI) * 0.011111111111111112) * (angle * ((b * b) - (a * a)));
	} else {
		tmp = b * ((angle * 0.011111111111111112) * (b * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8e+153) {
		tmp = (Math.PI * 0.011111111111111112) * (angle * ((b * b) - (a * a)));
	} else {
		tmp = b * ((angle * 0.011111111111111112) * (b * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 8e+153:
		tmp = (math.pi * 0.011111111111111112) * (angle * ((b * b) - (a * a)))
	else:
		tmp = b * ((angle * 0.011111111111111112) * (b * math.pi))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8e+153)
		tmp = Float64(Float64(pi * 0.011111111111111112) * Float64(angle * Float64(Float64(b * b) - Float64(a * a))));
	else
		tmp = Float64(b * Float64(Float64(angle * 0.011111111111111112) * Float64(b * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 8e+153)
		tmp = (pi * 0.011111111111111112) * (angle * ((b * b) - (a * a)));
	else
		tmp = b * ((angle * 0.011111111111111112) * (b * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8e+153], N[(N[(Pi * 0.011111111111111112), $MachinePrecision] * N[(angle * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8e153

   1. Initial program 61.5%

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

      1. *-commutative N/A

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

      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 70.8%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip-- N/A

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

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

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

      7. --lowering--.f64 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. difference-of-squares N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      13. --lowering--.f64 N/A

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 58.4%

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

   9. Simplified 58.4%

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

   if 8e153 < b

   1. Initial program 45.8%

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

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 43.0%

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

   8. Simplified 43.0%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. PI-lowering-PI.f64 73.3%

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

   10. Applied egg-rr 73.3%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+153}:\\ \;\;\;\;\left(\pi \cdot 0.011111111111111112\right) \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.2% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+92}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \cdot -0.011111111111111112\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.1e+92)
   (* b (* (* angle 0.011111111111111112) (* b PI)))
   (* (* a (* a (* PI angle))) -0.011111111111111112)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.1e+92) {
		tmp = b * ((angle * 0.011111111111111112) * (b * ((double) M_PI)));
	} else {
		tmp = (a * (a * (((double) M_PI) * angle))) * -0.011111111111111112;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.1e+92) {
		tmp = b * ((angle * 0.011111111111111112) * (b * Math.PI));
	} else {
		tmp = (a * (a * (Math.PI * angle))) * -0.011111111111111112;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.1e+92:
		tmp = b * ((angle * 0.011111111111111112) * (b * math.pi))
	else:
		tmp = (a * (a * (math.pi * angle))) * -0.011111111111111112
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.1e+92)
		tmp = Float64(b * Float64(Float64(angle * 0.011111111111111112) * Float64(b * pi)));
	else
		tmp = Float64(Float64(a * Float64(a * Float64(pi * angle))) * -0.011111111111111112);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.1e+92)
		tmp = b * ((angle * 0.011111111111111112) * (b * pi));
	else
		tmp = (a * (a * (pi * angle))) * -0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.1e+92], N[(b * N[(N[(angle * 0.011111111111111112), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.09999999999999986e92

   1. Initial program 60.1%

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

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.0%

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

   5. Simplified 46.0%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 42.2%

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

   8. Simplified 42.2%

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. PI-lowering-PI.f64 47.2%

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

   10. Applied egg-rr 47.2%

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

   if 2.09999999999999986e92 < a

   1. Initial program 55.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 57.5%

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

   5. Simplified 57.5%

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

   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 63.0%

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

   8. Simplified 63.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 73.0%

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

   10. Applied egg-rr 73.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.1 \cdot 10^{+92}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot 0.011111111111111112\right) \cdot \left(b \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \cdot -0.011111111111111112\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+92}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \cdot -0.011111111111111112\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.3e+92)
   (* 0.011111111111111112 (* (* b PI) (* b angle)))
   (* (* a (* a (* PI angle))) -0.011111111111111112)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.3e+92) {
		tmp = 0.011111111111111112 * ((b * ((double) M_PI)) * (b * angle));
	} else {
		tmp = (a * (a * (((double) M_PI) * angle))) * -0.011111111111111112;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.3e+92) {
		tmp = 0.011111111111111112 * ((b * Math.PI) * (b * angle));
	} else {
		tmp = (a * (a * (Math.PI * angle))) * -0.011111111111111112;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.3e+92:
		tmp = 0.011111111111111112 * ((b * math.pi) * (b * angle))
	else:
		tmp = (a * (a * (math.pi * angle))) * -0.011111111111111112
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.3e+92)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * pi) * Float64(b * angle)));
	else
		tmp = Float64(Float64(a * Float64(a * Float64(pi * angle))) * -0.011111111111111112);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.3e+92)
		tmp = 0.011111111111111112 * ((b * pi) * (b * angle));
	else
		tmp = (a * (a * (pi * angle))) * -0.011111111111111112;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.3e+92], N[(0.011111111111111112 * N[(N[(b * Pi), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.29999999999999998e92

   1. Initial program 60.1%

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

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.0%

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

   5. Simplified 46.0%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 42.2%

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

   8. Simplified 42.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
   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 47.2%

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

   10. Applied egg-rr 47.2%

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

   if 2.29999999999999998e92 < a

   1. Initial program 55.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 57.5%

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

   5. Simplified 57.5%

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

   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 63.0%

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

   8. Simplified 63.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 73.0%

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

   10. Applied egg-rr 73.0%

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

Alternative 14: 46.1% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 6.8 \cdot 10^{+92}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 6.8e+92)
   (* 0.011111111111111112 (* (* b PI) (* b angle)))
   (* (* a (* PI angle)) (* a -0.011111111111111112))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.8e+92) {
		tmp = 0.011111111111111112 * ((b * ((double) M_PI)) * (b * angle));
	} else {
		tmp = (a * (((double) M_PI) * angle)) * (a * -0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 6.8e+92) {
		tmp = 0.011111111111111112 * ((b * Math.PI) * (b * angle));
	} else {
		tmp = (a * (Math.PI * angle)) * (a * -0.011111111111111112);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 6.8e+92:
		tmp = 0.011111111111111112 * ((b * math.pi) * (b * angle))
	else:
		tmp = (a * (math.pi * angle)) * (a * -0.011111111111111112)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 6.8e+92)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * pi) * Float64(b * angle)));
	else
		tmp = Float64(Float64(a * Float64(pi * angle)) * Float64(a * -0.011111111111111112));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 6.8e+92)
		tmp = 0.011111111111111112 * ((b * pi) * (b * angle));
	else
		tmp = (a * (pi * angle)) * (a * -0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 6.8e+92], N[(0.011111111111111112 * N[(N[(b * Pi), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.7999999999999996e92

   1. Initial program 60.1%

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

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.0%

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

   5. Simplified 46.0%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 42.2%

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

   8. Simplified 42.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
   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 47.2%

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

   10. Applied egg-rr 47.2%

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

   if 6.7999999999999996e92 < a

   1. Initial program 55.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 57.5%

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

   5. Simplified 57.5%

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

   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 63.0%

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

   8. Simplified 63.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. PI-lowering-PI.f64 73.0%

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

   10. Applied egg-rr 73.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.8 \cdot 10^{+92}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a \cdot -0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.7% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{+91}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.6e+91)
   (* 0.011111111111111112 (* (* b PI) (* b angle)))
   (* (* PI angle) (* (* a a) -0.011111111111111112))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.6e+91) {
		tmp = 0.011111111111111112 * ((b * ((double) M_PI)) * (b * angle));
	} else {
		tmp = (((double) M_PI) * angle) * ((a * a) * -0.011111111111111112);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.6e+91) {
		tmp = 0.011111111111111112 * ((b * Math.PI) * (b * angle));
	} else {
		tmp = (Math.PI * angle) * ((a * a) * -0.011111111111111112);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 4.6e+91:
		tmp = 0.011111111111111112 * ((b * math.pi) * (b * angle))
	else:
		tmp = (math.pi * angle) * ((a * a) * -0.011111111111111112)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.6e+91)
		tmp = Float64(0.011111111111111112 * Float64(Float64(b * pi) * Float64(b * angle)));
	else
		tmp = Float64(Float64(pi * angle) * Float64(Float64(a * a) * -0.011111111111111112));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.6e+91)
		tmp = 0.011111111111111112 * ((b * pi) * (b * angle));
	else
		tmp = (pi * angle) * ((a * a) * -0.011111111111111112);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 4.6e+91], N[(0.011111111111111112 * N[(N[(b * Pi), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * angle), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.59999999999999982e91

   1. Initial program 60.1%

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

   3. Taylor expanded in b around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 46.0%

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

   5. Simplified 46.0%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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 42.2%

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

   8. Simplified 42.2%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
   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 47.2%

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

   10. Applied egg-rr 47.2%

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

   if 4.59999999999999982e91 < a

   1. Initial program 55.2%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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 57.5%

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

   5. Simplified 57.5%

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

   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 63.0%

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

   8. Simplified 63.0%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{+91}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot angle\right) \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (- b a) (* (+ b a) (* 0.011111111111111112 (* PI angle)))))

C

double code(double a, double b, double angle) {
	return (b - a) * ((b + a) * (0.011111111111111112 * (((double) M_PI) * angle)));
}

Java

public static double code(double a, double b, double angle) {
	return (b - a) * ((b + a) * (0.011111111111111112 * (Math.PI * angle)));
}

Python

def code(a, b, angle):
	return (b - a) * ((b + a) * (0.011111111111111112 * (math.pi * angle)))

Julia

function code(a, b, angle)
	return Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(0.011111111111111112 * Float64(pi * angle))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b - a) * ((b + a) * (0.011111111111111112 * (pi * angle)));
end

Wolfram

code[a_, b_, angle_] := N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. 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 73.1%

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

5. Taylor expanded in angle around 0

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

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

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

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

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

   3. PI-lowering-PI.f64 65.6%

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

7. Simplified 65.6%

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

8. Final simplification 65.6%

   \[\leadsto \left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right) \]
9. Add Preprocessing

Alternative 17: 62.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \left(b - a\right) \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* (- b a) (* (* PI (+ b a)) (* angle 0.011111111111111112))))

C

double code(double a, double b, double angle) {
	return (b - a) * ((((double) M_PI) * (b + a)) * (angle * 0.011111111111111112));
}

Java

public static double code(double a, double b, double angle) {
	return (b - a) * ((Math.PI * (b + a)) * (angle * 0.011111111111111112));
}

Python

def code(a, b, angle):
	return (b - a) * ((math.pi * (b + a)) * (angle * 0.011111111111111112))

Julia

function code(a, b, angle)
	return Float64(Float64(b - a) * Float64(Float64(pi * Float64(b + a)) * Float64(angle * 0.011111111111111112)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (b - a) * ((pi * (b + a)) * (angle * 0.011111111111111112));
end

Wolfram

code[a_, b_, angle_] := N[(N[(b - a), $MachinePrecision] * N[(N[(Pi * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. Add Preprocessing
3. 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 73.1%

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

5. Taylor expanded in angle around 0

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   6. +-lowering-+.f64 65.5%

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

7. Simplified 65.5%

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

8. Final simplification 65.5%

   \[\leadsto \left(b - a\right) \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right) \]
9. Add Preprocessing

Alternative 18: 62.8% accurate, 32.2× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* PI (* (+ b a) (* (- b a) (* angle 0.011111111111111112)))))

C

double code(double a, double b, double angle) {
	return ((double) M_PI) * ((b + a) * ((b - a) * (angle * 0.011111111111111112)));
}

Java

public static double code(double a, double b, double angle) {
	return Math.PI * ((b + a) * ((b - a) * (angle * 0.011111111111111112)));
}

Python

def code(a, b, angle):
	return math.pi * ((b + a) * ((b - a) * (angle * 0.011111111111111112)))

Julia

function code(a, b, angle)
	return Float64(pi * Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(angle * 0.011111111111111112))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = pi * ((b + a) * ((b - a) * (angle * 0.011111111111111112)));
end

Wolfram

code[a_, b_, angle_] := N[(Pi * N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
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 55.1%

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

5. Simplified 55.1%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. difference-of-squares N/A

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. PI-lowering-PI.f64 65.5%

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

7. Applied egg-rr 65.5%

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

8. Final simplification 65.5%

   \[\leadsto \pi \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right) \]
9. Add Preprocessing

Alternative 19: 39.1% accurate, 46.6× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* (* b PI) (* b angle))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((b * ((double) M_PI)) * (b * angle));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * ((b * Math.PI) * (b * angle));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * ((b * math.pi) * (b * angle))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(Float64(b * pi) * Float64(b * angle)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * ((b * pi) * (b * angle));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(N[(b * Pi), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in b around inf

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 40.5%

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

5. Simplified 40.5%

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

6. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
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 37.9%

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

8. Simplified 37.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
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 42.9%

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

10. Applied egg-rr 42.9%

    \[\leadsto 0.011111111111111112 \cdot \color{blue}{\left(\left(b \cdot \pi\right) \cdot \left(b \cdot angle\right)\right)} \]
11. Add Preprocessing

Alternative 20: 35.7% accurate, 46.6× speedup

Math

\[\begin{array}{l} \\ 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (* 0.011111111111111112 (* angle (* PI (* b b)))))

C

double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (((double) M_PI) * (b * b)));
}

Java

public static double code(double a, double b, double angle) {
	return 0.011111111111111112 * (angle * (Math.PI * (b * b)));
}

Python

def code(a, b, angle):
	return 0.011111111111111112 * (angle * (math.pi * (b * b)))

Julia

function code(a, b, angle)
	return Float64(0.011111111111111112 * Float64(angle * Float64(pi * Float64(b * b))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 0.011111111111111112 * (angle * (pi * (b * b)));
end

Wolfram

code[a_, b_, angle_] := N[(0.011111111111111112 * N[(angle * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in b around inf

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 40.5%

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

5. Simplified 40.5%

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

6. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)} \]
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 37.9%

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

8. Simplified 37.9%

   \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(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)))))