ab-angle->ABCF B

Percentage Accurate: 54.6% → 68.1%

Time: 6.4s

Alternatives: 19

Speedup: 3.3×

• 67.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

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: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - a\\ t_1 := \pi \cdot \left|angle\right|\\ t_2 := a + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 3.4 \cdot 10^{+186}:\\ \;\;\;\;\left(t\_2 \cdot \left(t\_0 \cdot \left(\sin \left(\frac{t\_1}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(\left|angle\right| \cdot 0.005555555555555556\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_2 \cdot \left(t\_0 \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{\left|angle\right|}{180}\right)\right) \cdot \sin \left(-0.005555555555555556 \cdot t\_1 + \pi \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) a))
       (t_1 (* PI (fabs angle)))
       (t_2 (+ a (fabs b))))
  (*
   (copysign 1.0 angle)
   (if (<= (fabs angle) 3.4e+186)
     (*
      (* t_2 (* t_0 (* (sin (/ t_1 180.0)) 2.0)))
      (sin
       (+
        1.5707963267948966
        (* (* (fabs angle) 0.005555555555555556) PI))))
     (*
      (* (* t_2 (* t_0 2.0)) (sin (* PI (/ (fabs angle) 180.0))))
      (sin (+ (* -0.005555555555555556 t_1) (* PI 0.5))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - a;
	double t_1 = ((double) M_PI) * fabs(angle);
	double t_2 = a + fabs(b);
	double tmp;
	if (fabs(angle) <= 3.4e+186) {
		tmp = (t_2 * (t_0 * (sin((t_1 / 180.0)) * 2.0))) * sin((1.5707963267948966 + ((fabs(angle) * 0.005555555555555556) * ((double) M_PI))));
	} else {
		tmp = ((t_2 * (t_0 * 2.0)) * sin((((double) M_PI) * (fabs(angle) / 180.0)))) * sin(((-0.005555555555555556 * t_1) + (((double) M_PI) * 0.5)));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - a;
	double t_1 = Math.PI * Math.abs(angle);
	double t_2 = a + Math.abs(b);
	double tmp;
	if (Math.abs(angle) <= 3.4e+186) {
		tmp = (t_2 * (t_0 * (Math.sin((t_1 / 180.0)) * 2.0))) * Math.sin((1.5707963267948966 + ((Math.abs(angle) * 0.005555555555555556) * Math.PI)));
	} else {
		tmp = ((t_2 * (t_0 * 2.0)) * Math.sin((Math.PI * (Math.abs(angle) / 180.0)))) * Math.sin(((-0.005555555555555556 * t_1) + (Math.PI * 0.5)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - a
	t_1 = math.pi * math.fabs(angle)
	t_2 = a + math.fabs(b)
	tmp = 0
	if math.fabs(angle) <= 3.4e+186:
		tmp = (t_2 * (t_0 * (math.sin((t_1 / 180.0)) * 2.0))) * math.sin((1.5707963267948966 + ((math.fabs(angle) * 0.005555555555555556) * math.pi)))
	else:
		tmp = ((t_2 * (t_0 * 2.0)) * math.sin((math.pi * (math.fabs(angle) / 180.0)))) * math.sin(((-0.005555555555555556 * t_1) + (math.pi * 0.5)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - a)
	t_1 = Float64(pi * abs(angle))
	t_2 = Float64(a + abs(b))
	tmp = 0.0
	if (abs(angle) <= 3.4e+186)
		tmp = Float64(Float64(t_2 * Float64(t_0 * Float64(sin(Float64(t_1 / 180.0)) * 2.0))) * sin(Float64(1.5707963267948966 + Float64(Float64(abs(angle) * 0.005555555555555556) * pi))));
	else
		tmp = Float64(Float64(Float64(t_2 * Float64(t_0 * 2.0)) * sin(Float64(pi * Float64(abs(angle) / 180.0)))) * sin(Float64(Float64(-0.005555555555555556 * t_1) + Float64(pi * 0.5))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - a;
	t_1 = pi * abs(angle);
	t_2 = a + abs(b);
	tmp = 0.0;
	if (abs(angle) <= 3.4e+186)
		tmp = (t_2 * (t_0 * (sin((t_1 / 180.0)) * 2.0))) * sin((1.5707963267948966 + ((abs(angle) * 0.005555555555555556) * pi)));
	else
		tmp = ((t_2 * (t_0 * 2.0)) * sin((pi * (abs(angle) / 180.0)))) * sin(((-0.005555555555555556 * t_1) + (pi * 0.5)));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 3.4e+186], N[(N[(t$95$2 * N[(t$95$0 * N[(N[Sin[N[(t$95$1 / 180.0), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(1.5707963267948966 + N[(N[(N[Abs[angle], $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(N[Abs[angle], $MachinePrecision] / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(-0.005555555555555556 * t$95$1), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.4000000000000001e186

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 67.5%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]

   6. Evaluated real constant 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\color{blue}{1.5707963267948966} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)} \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      5. mult-flip N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      8. lower-/.f64 67.3%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      11. lower-*.f64 67.3%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   8. Applied rewrites 67.3%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{\pi \cdot angle}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   if 3.4000000000000001e186 < angle

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. difference-of-squares N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 58.4%

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

   3. Applied rewrites 58.4%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 57.8%

      \[\leadsto \left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right) + \pi \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - a\\ t_1 := a + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 3.85 \cdot 10^{+186}:\\ \;\;\;\;\left(t\_1 \cdot \left(t\_0 \cdot \left(\sin \left(\frac{\pi \cdot \left|angle\right|}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(\left|angle\right| \cdot 0.005555555555555556\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot \left(t\_0 \cdot \left(\sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(0.5 \cdot \pi - \left|\left|angle\right|\right| \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) a)) (t_1 (+ a (fabs b))))
  (*
   (copysign 1.0 angle)
   (if (<= (fabs angle) 3.85e+186)
     (*
      (* t_1 (* t_0 (* (sin (/ (* PI (fabs angle)) 180.0)) 2.0)))
      (sin
       (+
        1.5707963267948966
        (* (* (fabs angle) 0.005555555555555556) PI))))
     (*
      (*
       t_1
       (*
        t_0
        (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) 2.0)))
      (sin
       (-
        (* 0.5 PI)
        (* (fabs (fabs angle)) (* PI 0.005555555555555556)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - a;
	double t_1 = a + fabs(b);
	double tmp;
	if (fabs(angle) <= 3.85e+186) {
		tmp = (t_1 * (t_0 * (sin(((((double) M_PI) * fabs(angle)) / 180.0)) * 2.0))) * sin((1.5707963267948966 + ((fabs(angle) * 0.005555555555555556) * ((double) M_PI))));
	} else {
		tmp = (t_1 * (t_0 * (sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * 2.0))) * sin(((0.5 * ((double) M_PI)) - (fabs(fabs(angle)) * (((double) M_PI) * 0.005555555555555556))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - a;
	double t_1 = a + Math.abs(b);
	double tmp;
	if (Math.abs(angle) <= 3.85e+186) {
		tmp = (t_1 * (t_0 * (Math.sin(((Math.PI * Math.abs(angle)) / 180.0)) * 2.0))) * Math.sin((1.5707963267948966 + ((Math.abs(angle) * 0.005555555555555556) * Math.PI)));
	} else {
		tmp = (t_1 * (t_0 * (Math.sin(((0.005555555555555556 * Math.abs(angle)) * Math.PI)) * 2.0))) * Math.sin(((0.5 * Math.PI) - (Math.abs(Math.abs(angle)) * (Math.PI * 0.005555555555555556))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - a
	t_1 = a + math.fabs(b)
	tmp = 0
	if math.fabs(angle) <= 3.85e+186:
		tmp = (t_1 * (t_0 * (math.sin(((math.pi * math.fabs(angle)) / 180.0)) * 2.0))) * math.sin((1.5707963267948966 + ((math.fabs(angle) * 0.005555555555555556) * math.pi)))
	else:
		tmp = (t_1 * (t_0 * (math.sin(((0.005555555555555556 * math.fabs(angle)) * math.pi)) * 2.0))) * math.sin(((0.5 * math.pi) - (math.fabs(math.fabs(angle)) * (math.pi * 0.005555555555555556))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - a)
	t_1 = Float64(a + abs(b))
	tmp = 0.0
	if (abs(angle) <= 3.85e+186)
		tmp = Float64(Float64(t_1 * Float64(t_0 * Float64(sin(Float64(Float64(pi * abs(angle)) / 180.0)) * 2.0))) * sin(Float64(1.5707963267948966 + Float64(Float64(abs(angle) * 0.005555555555555556) * pi))));
	else
		tmp = Float64(Float64(t_1 * Float64(t_0 * Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * sin(Float64(Float64(0.5 * pi) - Float64(abs(abs(angle)) * Float64(pi * 0.005555555555555556)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - a;
	t_1 = a + abs(b);
	tmp = 0.0;
	if (abs(angle) <= 3.85e+186)
		tmp = (t_1 * (t_0 * (sin(((pi * abs(angle)) / 180.0)) * 2.0))) * sin((1.5707963267948966 + ((abs(angle) * 0.005555555555555556) * pi)));
	else
		tmp = (t_1 * (t_0 * (sin(((0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * sin(((0.5 * pi) - (abs(abs(angle)) * (pi * 0.005555555555555556))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, Block[{t$95$1 = N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 3.85e+186], N[(N[(t$95$1 * N[(t$95$0 * N[(N[Sin[N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(1.5707963267948966 + N[(N[(N[Abs[angle], $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(t$95$0 * N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.5 * Pi), $MachinePrecision] - N[(N[Abs[N[Abs[angle], $MachinePrecision]], $MachinePrecision] * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.8500000000000002e186

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 67.5%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]

   6. Evaluated real constant 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\color{blue}{1.5707963267948966} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)} \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      5. mult-flip N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      8. lower-/.f64 67.3%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      11. lower-*.f64 67.3%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   8. Applied rewrites 67.3%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{\pi \cdot angle}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   if 3.8500000000000002e186 < angle

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 67.5%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. mult-flip N/A

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

      14. lift-PI.f64 N/A

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

      15. sin-+PI/2-rev N/A

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

      16. cos-fabs-rev N/A

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

      17. cos-neg-rev N/A

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

      18. sin-+PI/2-rev N/A

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

      19. lower-sin.f64 N/A

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

   7. Applied rewrites 67.6%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\left(-\left|\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right|\right) + 0.5 \cdot \pi\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \pi - \left|\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right|\right)} \]

      6. lift-fabs.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. mult-flip N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. associate-*l* N/A

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

      16. fabs-mul N/A

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

      17. fabs-mul N/A

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

      18. metadata-eval N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. lift-PI.f64 N/A

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

      22. lift-PI.f64 N/A

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

   9. Applied rewrites 67.9%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \pi - \left|angle\right| \cdot \left(\pi \cdot 0.005555555555555556\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 68.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - a\\ t_1 := \pi \cdot \left|angle\right|\\ t_2 := a + \left|b\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 3.85 \cdot 10^{+186}:\\ \;\;\;\;\left(t\_2 \cdot \left(t\_0 \cdot \left(\sin \left(\frac{t\_1}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(\left|angle\right| \cdot 0.005555555555555556\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \left(t\_0 \cdot \left(\sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(-0.005555555555555556 \cdot t\_1 + \pi \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) a))
       (t_1 (* PI (fabs angle)))
       (t_2 (+ a (fabs b))))
  (*
   (copysign 1.0 angle)
   (if (<= (fabs angle) 3.85e+186)
     (*
      (* t_2 (* t_0 (* (sin (/ t_1 180.0)) 2.0)))
      (sin
       (+
        1.5707963267948966
        (* (* (fabs angle) 0.005555555555555556) PI))))
     (*
      (*
       t_2
       (*
        t_0
        (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) 2.0)))
      (sin (+ (* -0.005555555555555556 t_1) (* PI 0.5))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - a;
	double t_1 = ((double) M_PI) * fabs(angle);
	double t_2 = a + fabs(b);
	double tmp;
	if (fabs(angle) <= 3.85e+186) {
		tmp = (t_2 * (t_0 * (sin((t_1 / 180.0)) * 2.0))) * sin((1.5707963267948966 + ((fabs(angle) * 0.005555555555555556) * ((double) M_PI))));
	} else {
		tmp = (t_2 * (t_0 * (sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * 2.0))) * sin(((-0.005555555555555556 * t_1) + (((double) M_PI) * 0.5)));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - a;
	double t_1 = Math.PI * Math.abs(angle);
	double t_2 = a + Math.abs(b);
	double tmp;
	if (Math.abs(angle) <= 3.85e+186) {
		tmp = (t_2 * (t_0 * (Math.sin((t_1 / 180.0)) * 2.0))) * Math.sin((1.5707963267948966 + ((Math.abs(angle) * 0.005555555555555556) * Math.PI)));
	} else {
		tmp = (t_2 * (t_0 * (Math.sin(((0.005555555555555556 * Math.abs(angle)) * Math.PI)) * 2.0))) * Math.sin(((-0.005555555555555556 * t_1) + (Math.PI * 0.5)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - a
	t_1 = math.pi * math.fabs(angle)
	t_2 = a + math.fabs(b)
	tmp = 0
	if math.fabs(angle) <= 3.85e+186:
		tmp = (t_2 * (t_0 * (math.sin((t_1 / 180.0)) * 2.0))) * math.sin((1.5707963267948966 + ((math.fabs(angle) * 0.005555555555555556) * math.pi)))
	else:
		tmp = (t_2 * (t_0 * (math.sin(((0.005555555555555556 * math.fabs(angle)) * math.pi)) * 2.0))) * math.sin(((-0.005555555555555556 * t_1) + (math.pi * 0.5)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - a)
	t_1 = Float64(pi * abs(angle))
	t_2 = Float64(a + abs(b))
	tmp = 0.0
	if (abs(angle) <= 3.85e+186)
		tmp = Float64(Float64(t_2 * Float64(t_0 * Float64(sin(Float64(t_1 / 180.0)) * 2.0))) * sin(Float64(1.5707963267948966 + Float64(Float64(abs(angle) * 0.005555555555555556) * pi))));
	else
		tmp = Float64(Float64(t_2 * Float64(t_0 * Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * sin(Float64(Float64(-0.005555555555555556 * t_1) + Float64(pi * 0.5))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - a;
	t_1 = pi * abs(angle);
	t_2 = a + abs(b);
	tmp = 0.0;
	if (abs(angle) <= 3.85e+186)
		tmp = (t_2 * (t_0 * (sin((t_1 / 180.0)) * 2.0))) * sin((1.5707963267948966 + ((abs(angle) * 0.005555555555555556) * pi)));
	else
		tmp = (t_2 * (t_0 * (sin(((0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * sin(((-0.005555555555555556 * t_1) + (pi * 0.5)));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 3.85e+186], N[(N[(t$95$2 * N[(t$95$0 * N[(N[Sin[N[(t$95$1 / 180.0), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(1.5707963267948966 + N[(N[(N[Abs[angle], $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 * N[(t$95$0 * N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(-0.005555555555555556 * t$95$1), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.8500000000000002e186

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 67.5%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]

   6. Evaluated real constant 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\color{blue}{1.5707963267948966} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \pi\right)} \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      5. mult-flip N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      6. associate-*l/ N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      8. lower-/.f64 67.3%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{angle \cdot \pi}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      10. *-commutative N/A

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\frac{884279719003555}{562949953421312} + \left(angle \cdot \frac{1}{180}\right) \cdot \pi\right) \]

      11. lower-*.f64 67.3%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\frac{\color{blue}{\pi \cdot angle}}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   8. Applied rewrites 67.3%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \color{blue}{\left(\frac{\pi \cdot angle}{180}\right)} \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   if 3.8500000000000002e186 < angle

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 68.1%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right) + \pi \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 68.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := -0.005555555555555556 \cdot \left(\pi \cdot \left|angle\right|\right)\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 6.2 \cdot 10^{+193}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(1.5707963267948966 - t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(t\_0 + \pi \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* -0.005555555555555556 (* PI (fabs angle)))))
  (*
   (copysign 1.0 angle)
   (if (<= (fabs angle) 6.2e+193)
     (*
      (*
       (* (- b a) 2.0)
       (sin (* (* PI 0.005555555555555556) (fabs angle))))
      (* (+ b a) (sin (- 1.5707963267948966 t_0))))
     (*
      (*
       (+ a b)
       (*
        (- b a)
        (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) 2.0)))
      (sin (+ t_0 (* PI 0.5))))))))

C

double code(double a, double b, double angle) {
	double t_0 = -0.005555555555555556 * (((double) M_PI) * fabs(angle));
	double tmp;
	if (fabs(angle) <= 6.2e+193) {
		tmp = (((b - a) * 2.0) * sin(((((double) M_PI) * 0.005555555555555556) * fabs(angle)))) * ((b + a) * sin((1.5707963267948966 - t_0)));
	} else {
		tmp = ((a + b) * ((b - a) * (sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * 2.0))) * sin((t_0 + (((double) M_PI) * 0.5)));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = -0.005555555555555556 * (Math.PI * Math.abs(angle));
	double tmp;
	if (Math.abs(angle) <= 6.2e+193) {
		tmp = (((b - a) * 2.0) * Math.sin(((Math.PI * 0.005555555555555556) * Math.abs(angle)))) * ((b + a) * Math.sin((1.5707963267948966 - t_0)));
	} else {
		tmp = ((a + b) * ((b - a) * (Math.sin(((0.005555555555555556 * Math.abs(angle)) * Math.PI)) * 2.0))) * Math.sin((t_0 + (Math.PI * 0.5)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = -0.005555555555555556 * (math.pi * math.fabs(angle))
	tmp = 0
	if math.fabs(angle) <= 6.2e+193:
		tmp = (((b - a) * 2.0) * math.sin(((math.pi * 0.005555555555555556) * math.fabs(angle)))) * ((b + a) * math.sin((1.5707963267948966 - t_0)))
	else:
		tmp = ((a + b) * ((b - a) * (math.sin(((0.005555555555555556 * math.fabs(angle)) * math.pi)) * 2.0))) * math.sin((t_0 + (math.pi * 0.5)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(-0.005555555555555556 * Float64(pi * abs(angle)))
	tmp = 0.0
	if (abs(angle) <= 6.2e+193)
		tmp = Float64(Float64(Float64(Float64(b - a) * 2.0) * sin(Float64(Float64(pi * 0.005555555555555556) * abs(angle)))) * Float64(Float64(b + a) * sin(Float64(1.5707963267948966 - t_0))));
	else
		tmp = Float64(Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * sin(Float64(t_0 + Float64(pi * 0.5))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = -0.005555555555555556 * (pi * abs(angle));
	tmp = 0.0;
	if (abs(angle) <= 6.2e+193)
		tmp = (((b - a) * 2.0) * sin(((pi * 0.005555555555555556) * abs(angle)))) * ((b + a) * sin((1.5707963267948966 - t_0)));
	else
		tmp = ((a + b) * ((b - a) * (sin(((0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * sin((t_0 + (pi * 0.5)));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(-0.005555555555555556 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 6.2e+193], N[(N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(1.5707963267948966 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(t$95$0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 6.1999999999999997e193

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 67.5%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]

   6. Evaluated real constant 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\color{blue}{1.5707963267948966} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   8. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(1.5707963267948966 - -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

   if 6.1999999999999997e193 < angle

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

   5. Applied rewrites 68.1%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right) + \pi \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.9% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(1.5707963267948966 - -0.005555555555555556 \cdot \left(\pi \cdot \left|angle\right|\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(\sin \left(0.5 \cdot \pi - \left|0.005555555555555556 \cdot \left(\left|angle\right| \cdot \pi\right)\right|\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (*
 (copysign 1.0 angle)
 (if (<= (fabs angle) 5e+199)
   (*
    (*
     (* (- b a) 2.0)
     (sin (* (* PI 0.005555555555555556) (fabs angle))))
    (*
     (+ b a)
     (sin
      (-
       1.5707963267948966
       (* -0.005555555555555556 (* PI (fabs angle)))))))
   (*
    0.011111111111111112
    (*
     (fabs angle)
     (*
      PI
      (*
       (sin
        (-
         (* 0.5 PI)
         (fabs (* 0.005555555555555556 (* (fabs angle) PI)))))
       (* (+ a b) (- b a)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 5e+199) {
		tmp = (((b - a) * 2.0) * sin(((((double) M_PI) * 0.005555555555555556) * fabs(angle)))) * ((b + a) * sin((1.5707963267948966 - (-0.005555555555555556 * (((double) M_PI) * fabs(angle))))));
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (sin(((0.5 * ((double) M_PI)) - fabs((0.005555555555555556 * (fabs(angle) * ((double) M_PI)))))) * ((a + b) * (b - a)))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 5e+199) {
		tmp = (((b - a) * 2.0) * Math.sin(((Math.PI * 0.005555555555555556) * Math.abs(angle)))) * ((b + a) * Math.sin((1.5707963267948966 - (-0.005555555555555556 * (Math.PI * Math.abs(angle))))));
	} else {
		tmp = 0.011111111111111112 * (Math.abs(angle) * (Math.PI * (Math.sin(((0.5 * Math.PI) - Math.abs((0.005555555555555556 * (Math.abs(angle) * Math.PI))))) * ((a + b) * (b - a)))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 5e+199:
		tmp = (((b - a) * 2.0) * math.sin(((math.pi * 0.005555555555555556) * math.fabs(angle)))) * ((b + a) * math.sin((1.5707963267948966 - (-0.005555555555555556 * (math.pi * math.fabs(angle))))))
	else:
		tmp = 0.011111111111111112 * (math.fabs(angle) * (math.pi * (math.sin(((0.5 * math.pi) - math.fabs((0.005555555555555556 * (math.fabs(angle) * math.pi))))) * ((a + b) * (b - a)))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 5e+199)
		tmp = Float64(Float64(Float64(Float64(b - a) * 2.0) * sin(Float64(Float64(pi * 0.005555555555555556) * abs(angle)))) * Float64(Float64(b + a) * sin(Float64(1.5707963267948966 - Float64(-0.005555555555555556 * Float64(pi * abs(angle)))))));
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(sin(Float64(Float64(0.5 * pi) - abs(Float64(0.005555555555555556 * Float64(abs(angle) * pi))))) * Float64(Float64(a + b) * Float64(b - a))))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 5e+199)
		tmp = (((b - a) * 2.0) * sin(((pi * 0.005555555555555556) * abs(angle)))) * ((b + a) * sin((1.5707963267948966 - (-0.005555555555555556 * (pi * abs(angle))))));
	else
		tmp = 0.011111111111111112 * (abs(angle) * (pi * (sin(((0.5 * pi) - abs((0.005555555555555556 * (abs(angle) * pi))))) * ((a + b) * (b - a)))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 5e+199], N[(N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[Sin[N[(1.5707963267948966 - N[(-0.005555555555555556 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(N[Sin[N[(N[(0.5 * Pi), $MachinePrecision] - N[Abs[N[(0.005555555555555556 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.9999999999999998e199

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 67.5%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]

   6. Evaluated real constant 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\color{blue}{1.5707963267948966} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   8. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right) \cdot \left(\left(b + a\right) \cdot \sin \left(1.5707963267948966 - -0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

   if 4.9999999999999998e199 < angle

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 67.5%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. mult-flip N/A

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

      14. lift-PI.f64 N/A

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

      15. sin-+PI/2-rev N/A

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

      16. cos-fabs-rev N/A

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

      17. cos-neg-rev N/A

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

      18. sin-+PI/2-rev N/A

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

      19. lower-sin.f64 N/A

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

   7. Applied rewrites 67.6%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\left(-\left|\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right|\right) + 0.5 \cdot \pi\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\sin \left(\frac{1}{2} \cdot \pi - \left|\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right|\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left|\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right|\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left|\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right|\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left|\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right|\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left|\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right|\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \left|\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right|\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)\right)\right) \]

   10. Applied rewrites 54.6%

       \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\sin \left(0.5 \cdot \pi - \left|0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right|\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.9% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(1.5707963267948966 + \left(\left|angle\right| \cdot 0.005555555555555556\right) \cdot \pi\right)\right) \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (*
 (copysign 1.0 angle)
 (*
  (*
   (+ a b)
   (*
    (- b a)
    (* (sin (* (* 0.005555555555555556 (fabs angle)) PI)) 2.0)))
  (sin
   (+
    1.5707963267948966
    (* (* (fabs angle) 0.005555555555555556) PI))))))

C

double code(double a, double b, double angle) {
	return copysign(1.0, angle) * (((a + b) * ((b - a) * (sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI))) * 2.0))) * sin((1.5707963267948966 + ((fabs(angle) * 0.005555555555555556) * ((double) M_PI)))));
}

Java

public static double code(double a, double b, double angle) {
	return Math.copySign(1.0, angle) * (((a + b) * ((b - a) * (Math.sin(((0.005555555555555556 * Math.abs(angle)) * Math.PI)) * 2.0))) * Math.sin((1.5707963267948966 + ((Math.abs(angle) * 0.005555555555555556) * Math.PI))));
}

Python

def code(a, b, angle):
	return math.copysign(1.0, angle) * (((a + b) * ((b - a) * (math.sin(((0.005555555555555556 * math.fabs(angle)) * math.pi)) * 2.0))) * math.sin((1.5707963267948966 + ((math.fabs(angle) * 0.005555555555555556) * math.pi))))

Julia

function code(a, b, angle)
	return Float64(copysign(1.0, angle) * Float64(Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * sin(Float64(1.5707963267948966 + Float64(Float64(abs(angle) * 0.005555555555555556) * pi)))))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (sign(angle) * abs(1.0)) * (((a + b) * ((b - a) * (sin(((0.005555555555555556 * abs(angle)) * pi)) * 2.0))) * sin((1.5707963267948966 + ((abs(angle) * 0.005555555555555556) * pi))));
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(1.5707963267948966 + N[(N[(N[Abs[angle], $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift--.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. difference-of-squares N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-*.f64 68.3%

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

3. Applied rewrites 68.1%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. mult-flip N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 67.6%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 67.6%

       \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

   13. lift-/.f64 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. lower-*.f64 67.5%

       \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

5. Applied rewrites 67.5%

   \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]

6. Evaluated real constant 67.5%

   \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\color{blue}{1.5707963267948966} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]
7. Add Preprocessing

Alternative 7: 67.8% accurate, 3.3× speedup

Math

\[\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot 1 \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (*
 (*
  (+ a b)
  (* (- b a) (* (sin (* (* 0.005555555555555556 angle) PI)) 2.0)))
 1.0))

C

double code(double a, double b, double angle) {
	return ((a + b) * ((b - a) * (sin(((0.005555555555555556 * angle) * ((double) M_PI))) * 2.0))) * 1.0;
}

Java

public static double code(double a, double b, double angle) {
	return ((a + b) * ((b - a) * (Math.sin(((0.005555555555555556 * angle) * Math.PI)) * 2.0))) * 1.0;
}

Python

def code(a, b, angle):
	return ((a + b) * ((b - a) * (math.sin(((0.005555555555555556 * angle) * math.pi)) * 2.0))) * 1.0

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * 2.0))) * 1.0)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a + b) * ((b - a) * (sin(((0.005555555555555556 * angle) * pi)) * 2.0))) * 1.0;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lift--.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. difference-of-squares N/A

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

   11. associate-*l* N/A

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

   12. lower-*.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-*.f64 68.3%

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

3. Applied rewrites 68.1%

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

4. Taylor expanded in angle around 0

   \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{1} \]
5. Step-by-step derivation

6. Applied rewrites 66.4%

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

Alternative 8: 67.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 4.1 \cdot 10^{+64}:\\ \;\;\;\;t\_0 \cdot \left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot \left|angle\right|\right)\right) \cdot \left(\left|b\right| + \left|a\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(\sin 1.5707963267948966 \cdot \left(\left(\left|a\right| + \left|b\right|\right) \cdot t\_0\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (fabs b) (fabs a))))
  (*
   (copysign 1.0 angle)
   (if (<= (fabs angle) 4.1e+64)
     (*
      t_0
      (*
       (sin (* 0.011111111111111112 (* PI (fabs angle))))
       (+ (fabs b) (fabs a))))
     (*
      0.011111111111111112
      (*
       (fabs angle)
       (*
        PI
        (*
         (sin 1.5707963267948966)
         (* (+ (fabs a) (fabs b)) t_0)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double tmp;
	if (fabs(angle) <= 4.1e+64) {
		tmp = t_0 * (sin((0.011111111111111112 * (((double) M_PI) * fabs(angle)))) * (fabs(b) + fabs(a)));
	} else {
		tmp = 0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (sin(1.5707963267948966) * ((fabs(a) + fabs(b)) * t_0))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.abs(b) - Math.abs(a);
	double tmp;
	if (Math.abs(angle) <= 4.1e+64) {
		tmp = t_0 * (Math.sin((0.011111111111111112 * (Math.PI * Math.abs(angle)))) * (Math.abs(b) + Math.abs(a)));
	} else {
		tmp = 0.011111111111111112 * (Math.abs(angle) * (Math.PI * (Math.sin(1.5707963267948966) * ((Math.abs(a) + Math.abs(b)) * t_0))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.fabs(b) - math.fabs(a)
	tmp = 0
	if math.fabs(angle) <= 4.1e+64:
		tmp = t_0 * (math.sin((0.011111111111111112 * (math.pi * math.fabs(angle)))) * (math.fabs(b) + math.fabs(a)))
	else:
		tmp = 0.011111111111111112 * (math.fabs(angle) * (math.pi * (math.sin(1.5707963267948966) * ((math.fabs(a) + math.fabs(b)) * t_0))))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(abs(b) - abs(a))
	tmp = 0.0
	if (abs(angle) <= 4.1e+64)
		tmp = Float64(t_0 * Float64(sin(Float64(0.011111111111111112 * Float64(pi * abs(angle)))) * Float64(abs(b) + abs(a))));
	else
		tmp = Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(sin(1.5707963267948966) * Float64(Float64(abs(a) + abs(b)) * t_0)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = abs(b) - abs(a);
	tmp = 0.0;
	if (abs(angle) <= 4.1e+64)
		tmp = t_0 * (sin((0.011111111111111112 * (pi * abs(angle)))) * (abs(b) + abs(a)));
	else
		tmp = 0.011111111111111112 * (abs(angle) * (pi * (sin(1.5707963267948966) * ((abs(a) + abs(b)) * t_0))));
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 4.1e+64], N[(t$95$0 * N[(N[Sin[N[(0.011111111111111112 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(N[Sin[1.5707963267948966], $MachinePrecision] * N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.0999999999999998e64

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.6%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.6%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.6%

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 68.6%

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

   5. Applied rewrites 68.6%

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

   if 4.0999999999999998e64 < angle

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lift--.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. difference-of-squares N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 68.3%

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

   3. Applied rewrites 68.1%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. mult-flip N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 67.6%

         \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot \color{blue}{0.5} + \pi \cdot \frac{angle}{180}\right) \]

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.6%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\frac{angle}{180} \cdot \pi}\right) \]

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 67.5%

          \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\pi \cdot 0.5 + \color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \pi\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \color{blue}{\sin \left(\pi \cdot 0.5 + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)} \]

   6. Evaluated real constant 67.5%

      \[\leadsto \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \sin \left(\color{blue}{1.5707963267948966} + \left(angle \cdot 0.005555555555555556\right) \cdot \pi\right) \]

   7. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\sin \frac{884279719003555}{562949953421312} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sin \frac{884279719003555}{562949953421312} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\sin \frac{884279719003555}{562949953421312} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\sin \frac{884279719003555}{562949953421312} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)\right) \]

      4. lower-PI.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\color{blue}{\sin \frac{884279719003555}{562949953421312}} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\sin \frac{884279719003555}{562949953421312} \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\sin \frac{884279719003555}{562949953421312} \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\sin \frac{884279719003555}{562949953421312} \cdot \left(\left(a + b\right) \cdot \color{blue}{\left(b - a\right)}\right)\right)\right)\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \frac{1}{90} \cdot \left(angle \cdot \left(\pi \cdot \left(\sin \frac{884279719003555}{562949953421312} \cdot \left(\left(a + b\right) \cdot \left(\color{blue}{b} - a\right)\right)\right)\right)\right) \]

      9. lower--.f64 54.9%

         \[\leadsto 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\sin 1.5707963267948966 \cdot \left(\left(a + b\right) \cdot \left(b - \color{blue}{a}\right)\right)\right)\right)\right) \]

   9. Applied rewrites 54.9%

      \[\leadsto \color{blue}{0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(\sin 1.5707963267948966 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|angle\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2.9 \cdot 10^{+283}:\\ \;\;\;\;\left(\left|b\right| - \left|a\right|\right) \cdot \left(\sin \left(0.011111111111111112 \cdot t\_0\right) \cdot \left(\left|b\right| + \left|a\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|a\right| \cdot \left|a\right|\right) \cdot \left(t\_0 \cdot -0.011111111111111112\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* PI (fabs angle))))
  (*
   (copysign 1.0 angle)
   (if (<= (fabs angle) 2.9e+283)
     (*
      (- (fabs b) (fabs a))
      (* (sin (* 0.011111111111111112 t_0)) (+ (fabs b) (fabs a))))
     (*
      (* (* (fabs a) (fabs a)) (* t_0 -0.011111111111111112))
      1.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * fabs(angle);
	double tmp;
	if (fabs(angle) <= 2.9e+283) {
		tmp = (fabs(b) - fabs(a)) * (sin((0.011111111111111112 * t_0)) * (fabs(b) + fabs(a)));
	} else {
		tmp = ((fabs(a) * fabs(a)) * (t_0 * -0.011111111111111112)) * 1.0;
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * Math.abs(angle);
	double tmp;
	if (Math.abs(angle) <= 2.9e+283) {
		tmp = (Math.abs(b) - Math.abs(a)) * (Math.sin((0.011111111111111112 * t_0)) * (Math.abs(b) + Math.abs(a)));
	} else {
		tmp = ((Math.abs(a) * Math.abs(a)) * (t_0 * -0.011111111111111112)) * 1.0;
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * math.fabs(angle)
	tmp = 0
	if math.fabs(angle) <= 2.9e+283:
		tmp = (math.fabs(b) - math.fabs(a)) * (math.sin((0.011111111111111112 * t_0)) * (math.fabs(b) + math.fabs(a)))
	else:
		tmp = ((math.fabs(a) * math.fabs(a)) * (t_0 * -0.011111111111111112)) * 1.0
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * abs(angle))
	tmp = 0.0
	if (abs(angle) <= 2.9e+283)
		tmp = Float64(Float64(abs(b) - abs(a)) * Float64(sin(Float64(0.011111111111111112 * t_0)) * Float64(abs(b) + abs(a))));
	else
		tmp = Float64(Float64(Float64(abs(a) * abs(a)) * Float64(t_0 * -0.011111111111111112)) * 1.0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * abs(angle);
	tmp = 0.0;
	if (abs(angle) <= 2.9e+283)
		tmp = (abs(b) - abs(a)) * (sin((0.011111111111111112 * t_0)) * (abs(b) + abs(a)));
	else
		tmp = ((abs(a) * abs(a)) * (t_0 * -0.011111111111111112)) * 1.0;
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 2.9e+283], N[(N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[N[(0.011111111111111112 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * -0.011111111111111112), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.8999999999999999e283

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 68.6%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 68.6%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 68.6%

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 68.6%

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

   5. Applied rewrites 68.6%

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

   if 2.8999999999999999e283 < angle

   1. Initial program 54.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 36.9%

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

   4. Applied rewrites 36.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lower-PI.f64 35.9%

         \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.9%

       \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 35.9%

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right) \cdot 1 \]

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 35.9%

           \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot -0.011111111111111112\right)\right) \cdot 1 \]

   12. Applied rewrites 35.9%

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot \color{blue}{-0.011111111111111112}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.2% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2.9 \cdot 10^{+283}:\\ \;\;\;\;\left(a + \left|b\right|\right) \cdot \left(\left(\left|b\right| - a\right) \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot \left|angle\right|\right) \cdot -0.011111111111111112\right)\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (*
 (copysign 1.0 angle)
 (if (<= (fabs angle) 2.9e+283)
   (*
    (+ a (fabs b))
    (*
     (- (fabs b) a)
     (sin (* (* (fabs angle) PI) 0.011111111111111112))))
   (* (* (* a a) (* (* PI (fabs angle)) -0.011111111111111112)) 1.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 2.9e+283) {
		tmp = (a + fabs(b)) * ((fabs(b) - a) * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112)));
	} else {
		tmp = ((a * a) * ((((double) M_PI) * fabs(angle)) * -0.011111111111111112)) * 1.0;
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 2.9e+283) {
		tmp = (a + Math.abs(b)) * ((Math.abs(b) - a) * Math.sin(((Math.abs(angle) * Math.PI) * 0.011111111111111112)));
	} else {
		tmp = ((a * a) * ((Math.PI * Math.abs(angle)) * -0.011111111111111112)) * 1.0;
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 2.9e+283:
		tmp = (a + math.fabs(b)) * ((math.fabs(b) - a) * math.sin(((math.fabs(angle) * math.pi) * 0.011111111111111112)))
	else:
		tmp = ((a * a) * ((math.pi * math.fabs(angle)) * -0.011111111111111112)) * 1.0
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 2.9e+283)
		tmp = Float64(Float64(a + abs(b)) * Float64(Float64(abs(b) - a) * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112))));
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(pi * abs(angle)) * -0.011111111111111112)) * 1.0);
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 2.9e+283)
		tmp = (a + abs(b)) * ((abs(b) - a) * sin(((abs(angle) * pi) * 0.011111111111111112)));
	else
		tmp = ((a * a) * ((pi * abs(angle)) * -0.011111111111111112)) * 1.0;
	end
	tmp_2 = (sign(angle) * abs(1.0)) * tmp;
end

Wolfram

code[a_, b_, angle_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[angle], $MachinePrecision], 2.9e+283], N[(N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[(N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.8999999999999999e283

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 68.6%

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

   if 2.8999999999999999e283 < angle

   1. Initial program 54.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 36.9%

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

   4. Applied rewrites 36.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lower-PI.f64 35.9%

         \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.9%

       \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 35.9%

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right) \cdot 1 \]

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 35.9%

           \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot -0.011111111111111112\right)\right) \cdot 1 \]

   12. Applied rewrites 35.9%

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot \color{blue}{-0.011111111111111112}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 66.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\\ t_1 := \left|b\right| - \left|a\right|\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot angle\right) \cdot \left|a\right|\right) \cdot \left|a\right|\right) \cdot -0.011111111111111112\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+122}:\\ \;\;\;\;\left(t\_1 \cdot \left(\left|a\right| + \left|b\right|\right)\right) \cdot \sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left|b\right|\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0)))
       (t_1 (- (fabs b) (fabs a))))
  (if (<= t_0 (- INFINITY))
    (*
     (* (* (* (* PI angle) (fabs a)) (fabs a)) -0.011111111111111112)
     1.0)
    (if (<= t_0 2e+122)
      (*
       (* t_1 (+ (fabs a) (fabs b)))
       (sin (* (* 0.011111111111111112 PI) angle)))
      (*
       t_1
       (* (sin (* 0.011111111111111112 (* PI angle))) (fabs b)))))))

C

double code(double a, double b, double angle) {
	double t_0 = pow(fabs(b), 2.0) - pow(fabs(a), 2.0);
	double t_1 = fabs(b) - fabs(a);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((((((double) M_PI) * angle) * fabs(a)) * fabs(a)) * -0.011111111111111112) * 1.0;
	} else if (t_0 <= 2e+122) {
		tmp = (t_1 * (fabs(a) + fabs(b))) * sin(((0.011111111111111112 * ((double) M_PI)) * angle));
	} else {
		tmp = t_1 * (sin((0.011111111111111112 * (((double) M_PI) * angle))) * fabs(b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0);
	double t_1 = Math.abs(b) - Math.abs(a);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((Math.PI * angle) * Math.abs(a)) * Math.abs(a)) * -0.011111111111111112) * 1.0;
	} else if (t_0 <= 2e+122) {
		tmp = (t_1 * (Math.abs(a) + Math.abs(b))) * Math.sin(((0.011111111111111112 * Math.PI) * angle));
	} else {
		tmp = t_1 * (Math.sin((0.011111111111111112 * (Math.PI * angle))) * Math.abs(b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0)
	t_1 = math.fabs(b) - math.fabs(a)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = ((((math.pi * angle) * math.fabs(a)) * math.fabs(a)) * -0.011111111111111112) * 1.0
	elif t_0 <= 2e+122:
		tmp = (t_1 * (math.fabs(a) + math.fabs(b))) * math.sin(((0.011111111111111112 * math.pi) * angle))
	else:
		tmp = t_1 * (math.sin((0.011111111111111112 * (math.pi * angle))) * math.fabs(b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))
	t_1 = Float64(abs(b) - abs(a))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(pi * angle) * abs(a)) * abs(a)) * -0.011111111111111112) * 1.0);
	elseif (t_0 <= 2e+122)
		tmp = Float64(Float64(t_1 * Float64(abs(a) + abs(b))) * sin(Float64(Float64(0.011111111111111112 * pi) * angle)));
	else
		tmp = Float64(t_1 * Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * abs(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (abs(b) ^ 2.0) - (abs(a) ^ 2.0);
	t_1 = abs(b) - abs(a);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = ((((pi * angle) * abs(a)) * abs(a)) * -0.011111111111111112) * 1.0;
	elseif (t_0 <= 2e+122)
		tmp = (t_1 * (abs(a) + abs(b))) * sin(((0.011111111111111112 * pi) * angle));
	else
		tmp = t_1 * (sin((0.011111111111111112 * (pi * angle))) * abs(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+122], N[(N[(t$95$1 * N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -inf.0

   1. Initial program 54.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 36.9%

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

   4. Applied rewrites 36.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lower-PI.f64 35.9%

         \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.9%

       \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 35.9%

          \[\leadsto \left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right) \cdot 1 \]

       4. lift-pow.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 39.5%

           \[\leadsto \left(\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 39.5%

           \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   12. Applied rewrites 39.5%

       \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   if -inf.0 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 2e122

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 58.6%

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

   5. Applied rewrites 58.6%

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

   if 2e122 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.7%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 38.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 42.4%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.4%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 42.4%

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

   8. Applied rewrites 42.4%

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

Alternative 12: 66.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\\ t_1 := \left|b\right| - \left|a\right|\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+304}:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot angle\right) \cdot \left|a\right|\right) \cdot \left|a\right|\right) \cdot -0.011111111111111112\right) \cdot 1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\left(t\_1 \cdot \left(\left|a\right| + \left|b\right|\right)\right) \cdot \sin \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot t\_1\right) \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0)))
       (t_1 (- (fabs b) (fabs a))))
  (if (<= t_0 -2e+304)
    (*
     (* (* (* (* PI angle) (fabs a)) (fabs a)) -0.011111111111111112)
     1.0)
    (if (<= t_0 5e-13)
      (*
       (* t_1 (+ (fabs a) (fabs b)))
       (sin (* (* 0.011111111111111112 angle) PI)))
      (*
       (* (sin (* 0.011111111111111112 (* PI angle))) t_1)
       (fabs b))))))

C

double code(double a, double b, double angle) {
	double t_0 = pow(fabs(b), 2.0) - pow(fabs(a), 2.0);
	double t_1 = fabs(b) - fabs(a);
	double tmp;
	if (t_0 <= -2e+304) {
		tmp = ((((((double) M_PI) * angle) * fabs(a)) * fabs(a)) * -0.011111111111111112) * 1.0;
	} else if (t_0 <= 5e-13) {
		tmp = (t_1 * (fabs(a) + fabs(b))) * sin(((0.011111111111111112 * angle) * ((double) M_PI)));
	} else {
		tmp = (sin((0.011111111111111112 * (((double) M_PI) * angle))) * t_1) * fabs(b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0);
	double t_1 = Math.abs(b) - Math.abs(a);
	double tmp;
	if (t_0 <= -2e+304) {
		tmp = ((((Math.PI * angle) * Math.abs(a)) * Math.abs(a)) * -0.011111111111111112) * 1.0;
	} else if (t_0 <= 5e-13) {
		tmp = (t_1 * (Math.abs(a) + Math.abs(b))) * Math.sin(((0.011111111111111112 * angle) * Math.PI));
	} else {
		tmp = (Math.sin((0.011111111111111112 * (Math.PI * angle))) * t_1) * Math.abs(b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0)
	t_1 = math.fabs(b) - math.fabs(a)
	tmp = 0
	if t_0 <= -2e+304:
		tmp = ((((math.pi * angle) * math.fabs(a)) * math.fabs(a)) * -0.011111111111111112) * 1.0
	elif t_0 <= 5e-13:
		tmp = (t_1 * (math.fabs(a) + math.fabs(b))) * math.sin(((0.011111111111111112 * angle) * math.pi))
	else:
		tmp = (math.sin((0.011111111111111112 * (math.pi * angle))) * t_1) * math.fabs(b)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))
	t_1 = Float64(abs(b) - abs(a))
	tmp = 0.0
	if (t_0 <= -2e+304)
		tmp = Float64(Float64(Float64(Float64(Float64(pi * angle) * abs(a)) * abs(a)) * -0.011111111111111112) * 1.0);
	elseif (t_0 <= 5e-13)
		tmp = Float64(Float64(t_1 * Float64(abs(a) + abs(b))) * sin(Float64(Float64(0.011111111111111112 * angle) * pi)));
	else
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * t_1) * abs(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (abs(b) ^ 2.0) - (abs(a) ^ 2.0);
	t_1 = abs(b) - abs(a);
	tmp = 0.0;
	if (t_0 <= -2e+304)
		tmp = ((((pi * angle) * abs(a)) * abs(a)) * -0.011111111111111112) * 1.0;
	elseif (t_0 <= 5e-13)
		tmp = (t_1 * (abs(a) + abs(b))) * sin(((0.011111111111111112 * angle) * pi));
	else
		tmp = (sin((0.011111111111111112 * (pi * angle))) * t_1) * abs(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+304], N[(N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e-13], N[(N[(t$95$1 * N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < -1.9999999999999999e304

   1. Initial program 54.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 36.9%

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

   4. Applied rewrites 36.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lower-PI.f64 35.9%

         \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.9%

       \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 35.9%

          \[\leadsto \left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right) \cdot 1 \]

       4. lift-pow.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 39.5%

           \[\leadsto \left(\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 39.5%

           \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   12. Applied rewrites 39.5%

       \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   if -1.9999999999999999e304 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))) < 4.9999999999999999e-13

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 58.5%

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

   5. Applied rewrites 58.5%

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

   if 4.9999999999999999e-13 < (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.7%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 38.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 42.4%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.4%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 42.4%

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

   8. Applied rewrites 42.4%

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

Alternative 13: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;2 \cdot \left({\left(\left|b\right|\right)}^{2} - {\left(\left|a\right|\right)}^{2}\right) \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\left(\left(\left(\left(\pi \cdot angle\right) \cdot \left|a\right|\right) \cdot \left|a\right|\right) \cdot -0.011111111111111112\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(\left|b\right| - \left|a\right|\right)\right) \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<= (* 2.0 (- (pow (fabs b) 2.0) (pow (fabs a) 2.0))) -1e-80)
  (*
   (* (* (* (* PI angle) (fabs a)) (fabs a)) -0.011111111111111112)
   1.0)
  (*
   (*
    (sin (* 0.011111111111111112 (* PI angle)))
    (- (fabs b) (fabs a)))
   (fabs b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(fabs(b), 2.0) - pow(fabs(a), 2.0))) <= -1e-80) {
		tmp = ((((((double) M_PI) * angle) * fabs(a)) * fabs(a)) * -0.011111111111111112) * 1.0;
	} else {
		tmp = (sin((0.011111111111111112 * (((double) M_PI) * angle))) * (fabs(b) - fabs(a))) * fabs(b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(Math.abs(a), 2.0))) <= -1e-80) {
		tmp = ((((Math.PI * angle) * Math.abs(a)) * Math.abs(a)) * -0.011111111111111112) * 1.0;
	} else {
		tmp = (Math.sin((0.011111111111111112 * (Math.PI * angle))) * (Math.abs(b) - Math.abs(a))) * Math.abs(b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if (2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(math.fabs(a), 2.0))) <= -1e-80:
		tmp = ((((math.pi * angle) * math.fabs(a)) * math.fabs(a)) * -0.011111111111111112) * 1.0
	else:
		tmp = (math.sin((0.011111111111111112 * (math.pi * angle))) * (math.fabs(b) - math.fabs(a))) * math.fabs(b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= -1e-80)
		tmp = Float64(Float64(Float64(Float64(Float64(pi * angle) * abs(a)) * abs(a)) * -0.011111111111111112) * 1.0);
	else
		tmp = Float64(Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * Float64(abs(b) - abs(a))) * abs(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if ((2.0 * ((abs(b) ^ 2.0) - (abs(a) ^ 2.0))) <= -1e-80)
		tmp = ((((pi * angle) * abs(a)) * abs(a)) * -0.011111111111111112) * 1.0;
	else
		tmp = (sin((0.011111111111111112 * (pi * angle))) * (abs(b) - abs(a))) * abs(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(2.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-80], N[(N[(N[(N[(N[(Pi * angle), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -9.9999999999999996e-81

   1. Initial program 54.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 36.9%

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

   4. Applied rewrites 36.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lower-PI.f64 35.9%

         \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.9%

       \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 35.9%

          \[\leadsto \left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right) \cdot 1 \]

       4. lift-pow.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 39.5%

           \[\leadsto \left(\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 39.5%

           \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   12. Applied rewrites 39.5%

       \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   if -9.9999999999999996e-81 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.7%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 38.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 42.4%

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 42.4%

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 42.4%

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

   8. Applied rewrites 42.4%

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

Alternative 14: 59.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left({b}^{2} - {a}^{2}\right)\\ t_1 := \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* 2.0 (- (pow b 2.0) (pow a 2.0))))
       (t_1
        (* (* (* (* (* PI angle) a) a) -0.011111111111111112) 1.0)))
  (if (<= t_0 -1e-80)
    t_1
    (if (<= t_0 INFINITY)
      (* (sin (* 0.011111111111111112 (* PI angle))) (* b b))
      t_1))))

C

double code(double a, double b, double angle) {
	double t_0 = 2.0 * (pow(b, 2.0) - pow(a, 2.0));
	double t_1 = ((((((double) M_PI) * angle) * a) * a) * -0.011111111111111112) * 1.0;
	double tmp;
	if (t_0 <= -1e-80) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = sin((0.011111111111111112 * (((double) M_PI) * angle))) * (b * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = 2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0));
	double t_1 = ((((Math.PI * angle) * a) * a) * -0.011111111111111112) * 1.0;
	double tmp;
	if (t_0 <= -1e-80) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sin((0.011111111111111112 * (Math.PI * angle))) * (b * b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = 2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))
	t_1 = ((((math.pi * angle) * a) * a) * -0.011111111111111112) * 1.0
	tmp = 0
	if t_0 <= -1e-80:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = math.sin((0.011111111111111112 * (math.pi * angle))) * (b * b)
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0)))
	t_1 = Float64(Float64(Float64(Float64(Float64(pi * angle) * a) * a) * -0.011111111111111112) * 1.0)
	tmp = 0.0
	if (t_0 <= -1e-80)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(sin(Float64(0.011111111111111112 * Float64(pi * angle))) * Float64(b * b));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = 2.0 * ((b ^ 2.0) - (a ^ 2.0));
	t_1 = ((((pi * angle) * a) * a) * -0.011111111111111112) * 1.0;
	tmp = 0.0;
	if (t_0 <= -1e-80)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = sin((0.011111111111111112 * (pi * angle))) * (b * b);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-80], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[Sin[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < -9.9999999999999996e-81 or +inf.0 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.6%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 36.9%

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

   4. Applied rewrites 36.9%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

      5. lower-PI.f64 35.9%

         \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. Applied rewrites 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   8. Taylor expanded in angle around 0

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.9%

       \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 35.9%

          \[\leadsto \left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right) \cdot 1 \]

       4. lift-pow.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 39.5%

           \[\leadsto \left(\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 39.5%

           \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   12. Applied rewrites 39.5%

       \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

   if -9.9999999999999996e-81 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < +inf.0

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 2-sin N/A

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

      10. count-2 N/A

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

   3. Applied rewrites 58.7%

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

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2}} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \]
   5. Step-by-step derivation

      1. lower-pow.f64 36.1%

         \[\leadsto {b}^{\color{blue}{2}} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \]

   6. Applied rewrites 36.1%

      \[\leadsto \color{blue}{{b}^{2}} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right) \cdot {b}^{2}} \]

      3. lower-*.f64 36.1%

         \[\leadsto \color{blue}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right) \cdot {b}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)} \cdot {b}^{2} \]

      5. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)} \cdot {b}^{2} \]

      6. lower-*.f64 36.1%

         \[\leadsto \sin \color{blue}{\left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)} \cdot {b}^{2} \]

      7. lift-*.f64 N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right) \cdot {b}^{2} \]

      8. *-commutative N/A

         \[\leadsto \sin \left(\frac{1}{90} \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \cdot {b}^{2} \]

      9. lower-*.f64 36.1%

         \[\leadsto \sin \left(0.011111111111111112 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \cdot {b}^{2} \]

      10. lift-pow.f64 N/A

          \[\leadsto \sin \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right)\right) \cdot {b}^{\color{blue}{2}} \]

      11. unpow2 N/A

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

      12. lower-*.f64 36.1%

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

   8. Applied rewrites 36.1%

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

Alternative 15: 39.5% accurate, 17.4× speedup

Math

\[\left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (* (* (* (* PI angle) a) a) -0.011111111111111112) 1.0))

C

double code(double a, double b, double angle) {
	return ((((((double) M_PI) * angle) * a) * a) * -0.011111111111111112) * 1.0;
}

Java

public static double code(double a, double b, double angle) {
	return ((((Math.PI * angle) * a) * a) * -0.011111111111111112) * 1.0;
}

Python

def code(a, b, angle):
	return ((((math.pi * angle) * a) * a) * -0.011111111111111112) * 1.0

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(Float64(Float64(pi * angle) * a) * a) * -0.011111111111111112) * 1.0)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((((pi * angle) * a) * a) * -0.011111111111111112) * 1.0;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

2. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-PI.f64 36.9%

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

4. Applied rewrites 36.9%

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

5. Taylor expanded in angle around 0

   \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-PI.f64 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Applied rewrites 35.9%

   \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

8. Taylor expanded in angle around 0

   \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 35.9%

    \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 35.9%

       \[\leadsto \left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot -0.011111111111111112\right) \cdot 1 \]

    4. lift-pow.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. *-commutative N/A

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

    7. unpow2 N/A

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

    8. associate-*r* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-*.f64 39.5%

        \[\leadsto \left(\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

    11. lift-*.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 39.5%

        \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]

12. Applied rewrites 39.5%

    \[\leadsto \left(\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot a\right) \cdot -0.011111111111111112\right) \cdot 1 \]
13. Add Preprocessing

Alternative 16: 35.9% accurate, 17.4× speedup

Math

\[\left(\left(\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot angle\right) \cdot 1 \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (* (* (* -0.011111111111111112 (* a a)) PI) angle) 1.0))

C

double code(double a, double b, double angle) {
	return (((-0.011111111111111112 * (a * a)) * ((double) M_PI)) * angle) * 1.0;
}

Java

public static double code(double a, double b, double angle) {
	return (((-0.011111111111111112 * (a * a)) * Math.PI) * angle) * 1.0;
}

Python

def code(a, b, angle):
	return (((-0.011111111111111112 * (a * a)) * math.pi) * angle) * 1.0

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(Float64(-0.011111111111111112 * Float64(a * a)) * pi) * angle) * 1.0)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (((-0.011111111111111112 * (a * a)) * pi) * angle) * 1.0;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \color{blue}{\left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. lower-PI.f64 36.9%

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

4. Applied rewrites 36.9%

   \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

5. Taylor expanded in angle around 0

   \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-PI.f64 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Applied rewrites 35.9%

   \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

8. Taylor expanded in angle around 0

   \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 35.9%

    \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right) \cdot 1 \]

    2. lift-pow.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 1 \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)\right) \cdot 1 \]

    4. associate-*r* N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\pi}\right)\right) \cdot 1 \]

    5. lift-*.f64 N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \pi\right)\right) \cdot 1 \]

    6. *-commutative N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(\pi \cdot angle\right)\right) \cdot 1 \]

    7. associate-*r* N/A

       \[\leadsto \left(\left(\left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \pi\right) \cdot angle\right) \cdot 1 \]

    8. lower-*.f64 N/A

       \[\leadsto \left(\left(\left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \pi\right) \cdot angle\right) \cdot 1 \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\left(\left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \pi\right) \cdot angle\right) \cdot 1 \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\left(\left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \pi\right) \cdot angle\right) \cdot 1 \]

    11. unpow2 N/A

        \[\leadsto \left(\left(\left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot angle\right) \cdot 1 \]

    12. lower-*.f64 35.9%

        \[\leadsto \left(\left(\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot angle\right) \cdot 1 \]

12. Applied rewrites 35.9%

    \[\leadsto \left(\left(\left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \pi\right) \cdot angle\right) \cdot 1 \]
13. Add Preprocessing

Alternative 17: 35.9% accurate, 17.4× speedup

Math

\[\left(\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right) \cdot 1 \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (* (* PI angle) (* -0.011111111111111112 (* a a))) 1.0))

C

double code(double a, double b, double angle) {
	return ((((double) M_PI) * angle) * (-0.011111111111111112 * (a * a))) * 1.0;
}

Java

public static double code(double a, double b, double angle) {
	return ((Math.PI * angle) * (-0.011111111111111112 * (a * a))) * 1.0;
}

Python

def code(a, b, angle):
	return ((math.pi * angle) * (-0.011111111111111112 * (a * a))) * 1.0

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(pi * angle) * Float64(-0.011111111111111112 * Float64(a * a))) * 1.0)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((pi * angle) * (-0.011111111111111112 * (a * a))) * 1.0;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(Pi * angle), $MachinePrecision] * N[(-0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \color{blue}{\left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. lower-PI.f64 36.9%

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

4. Applied rewrites 36.9%

   \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

5. Taylor expanded in angle around 0

   \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-PI.f64 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Applied rewrites 35.9%

   \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

8. Taylor expanded in angle around 0

   \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 35.9%

    \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right) \cdot 1 \]

    2. lift-pow.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 1 \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)\right) \cdot 1 \]

    4. associate-*r* N/A

       \[\leadsto \left(\left(\frac{-1}{90} \cdot {a}^{2}\right) \cdot \left(angle \cdot \color{blue}{\pi}\right)\right) \cdot 1 \]

    5. *-commutative N/A

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\frac{-1}{90} \cdot \color{blue}{{a}^{2}}\right)\right) \cdot 1 \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\frac{-1}{90} \cdot \color{blue}{{a}^{2}}\right)\right) \cdot 1 \]

    7. lift-*.f64 N/A

       \[\leadsto \left(\left(angle \cdot \pi\right) \cdot \left(\frac{-1}{90} \cdot {\color{blue}{a}}^{2}\right)\right) \cdot 1 \]

    8. *-commutative N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\frac{-1}{90} \cdot {\color{blue}{a}}^{2}\right)\right) \cdot 1 \]

    9. lower-*.f64 N/A

       \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\frac{-1}{90} \cdot {\color{blue}{a}}^{2}\right)\right) \cdot 1 \]

    10. lower-*.f64 N/A

        \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\frac{-1}{90} \cdot {a}^{\color{blue}{2}}\right)\right) \cdot 1 \]

    11. unpow2 N/A

        \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(\frac{-1}{90} \cdot \left(a \cdot a\right)\right)\right) \cdot 1 \]

    12. lower-*.f64 35.9%

        \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right) \cdot 1 \]

12. Applied rewrites 35.9%

    \[\leadsto \left(\left(\pi \cdot angle\right) \cdot \left(-0.011111111111111112 \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot 1 \]
13. Add Preprocessing

Alternative 18: 35.9% accurate, 17.4× speedup

Math

\[\left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot -0.011111111111111112\right)\right) \cdot 1 \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (* (* a a) (* (* PI angle) -0.011111111111111112)) 1.0))

C

double code(double a, double b, double angle) {
	return ((a * a) * ((((double) M_PI) * angle) * -0.011111111111111112)) * 1.0;
}

Java

public static double code(double a, double b, double angle) {
	return ((a * a) * ((Math.PI * angle) * -0.011111111111111112)) * 1.0;
}

Python

def code(a, b, angle):
	return ((a * a) * ((math.pi * angle) * -0.011111111111111112)) * 1.0

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(a * a) * Float64(Float64(pi * angle) * -0.011111111111111112)) * 1.0)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * a) * ((pi * angle) * -0.011111111111111112)) * 1.0;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(a * a), $MachinePrecision] * N[(N[(Pi * angle), $MachinePrecision] * -0.011111111111111112), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \color{blue}{\left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. lower-PI.f64 36.9%

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

4. Applied rewrites 36.9%

   \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

5. Taylor expanded in angle around 0

   \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-PI.f64 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Applied rewrites 35.9%

   \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

8. Taylor expanded in angle around 0

   \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 35.9%

    \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right) \cdot 1 \]

    2. *-commutative N/A

       \[\leadsto \left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{-1}{90}\right) \cdot 1 \]

    3. lift-pow.f64 N/A

       \[\leadsto \left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{-1}{90}\right) \cdot 1 \]

    4. lift-*.f64 N/A

       \[\leadsto \left(\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{-1}{90}\right) \cdot 1 \]

    5. associate-*l* N/A

       \[\leadsto \left({a}^{2} \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \cdot 1 \]

    6. lower-*.f64 N/A

       \[\leadsto \left({a}^{2} \cdot \left(\left(angle \cdot \pi\right) \cdot \color{blue}{\frac{-1}{90}}\right)\right) \cdot 1 \]

    7. unpow2 N/A

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right)\right) \cdot 1 \]

    8. lower-*.f64 N/A

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right)\right) \cdot 1 \]

    9. lower-*.f64 35.9%

       \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right) \cdot 1 \]

    10. lift-*.f64 N/A

        \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \frac{-1}{90}\right)\right) \cdot 1 \]

    11. *-commutative N/A

        \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot \frac{-1}{90}\right)\right) \cdot 1 \]

    12. lower-*.f64 35.9%

        \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot -0.011111111111111112\right)\right) \cdot 1 \]

12. Applied rewrites 35.9%

    \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot \color{blue}{-0.011111111111111112}\right)\right) \cdot 1 \]
13. Add Preprocessing

Alternative 19: 35.9% accurate, 17.4× speedup

Math

\[\left(-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot 1 \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (* -0.011111111111111112 (* (* (* a a) PI) angle)) 1.0))

C

double code(double a, double b, double angle) {
	return (-0.011111111111111112 * (((a * a) * ((double) M_PI)) * angle)) * 1.0;
}

Java

public static double code(double a, double b, double angle) {
	return (-0.011111111111111112 * (((a * a) * Math.PI) * angle)) * 1.0;
}

Python

def code(a, b, angle):
	return (-0.011111111111111112 * (((a * a) * math.pi) * angle)) * 1.0

Julia

function code(a, b, angle)
	return Float64(Float64(-0.011111111111111112 * Float64(Float64(Float64(a * a) * pi) * angle)) * 1.0)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (-0.011111111111111112 * (((a * a) * pi) * angle)) * 1.0;
end

Wolfram

code[a_, b_, angle_] := N[(N[(-0.011111111111111112 * N[(N[(N[(a * a), $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 54.6%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

2. Taylor expanded in a around inf

   \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \color{blue}{\left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   7. lower-PI.f64 36.9%

      \[\leadsto \left(-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

4. Applied rewrites 36.9%

   \[\leadsto \color{blue}{\left(-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

5. Taylor expanded in angle around 0

   \[\leadsto \left(\frac{-1}{90} \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   5. lower-PI.f64 35.9%

      \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

7. Applied rewrites 35.9%

   \[\leadsto \left(-0.011111111111111112 \cdot \color{blue}{\left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

8. Taylor expanded in angle around 0

   \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
9. Step-by-step derivation

10. Applied rewrites 35.9%

    \[\leadsto \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \color{blue}{1} \]
11. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 1 \]

    2. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)\right) \cdot 1 \]

    3. lift-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 1 \]

    4. *-commutative N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left({a}^{2} \cdot \left(\pi \cdot angle\right)\right)\right) \cdot 1 \]

    5. associate-*r* N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left(\left({a}^{2} \cdot \pi\right) \cdot angle\right)\right) \cdot 1 \]

    6. lower-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left(\left({a}^{2} \cdot \pi\right) \cdot angle\right)\right) \cdot 1 \]

    7. lower-*.f64 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left(\left({a}^{2} \cdot \pi\right) \cdot angle\right)\right) \cdot 1 \]

    8. unpow2 N/A

       \[\leadsto \left(\frac{-1}{90} \cdot \left(\left(\left(a \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot 1 \]

    9. lower-*.f64 35.9%

       \[\leadsto \left(-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot 1 \]

12. Applied rewrites 35.9%

    \[\leadsto \left(-0.011111111111111112 \cdot \left(\left(\left(a \cdot a\right) \cdot \pi\right) \cdot angle\right)\right) \cdot 1 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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)))))