ab-angle->ABCF B

Percentage Accurate: 54.4% → 67.3%

Time: 7.3s

Alternatives: 16

Speedup: 6.6×

• 67.9% 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: 54.4% 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: 67.3% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(2 \cdot \sin \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right)\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556, \pi \cdot \left|angle\right|, \pi \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left|angle\right| \cdot \left(\left(\sqrt[3]{\pi} \cdot 2.1450293971110255\right) \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 5e+116)
    (*
     (*
      (+ b a)
      (* (- b a) (* 2.0 (sin (* (* 0.005555555555555556 (fabs angle)) PI)))))
     (sin (fma -0.005555555555555556 (* PI (fabs angle)) (* PI 0.5))))
    (*
     (+ a b)
     (*
      (- b a)
      (sin
       (*
        (fabs angle)
        (* (* (cbrt PI) 2.1450293971110255) 0.011111111111111112))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 5e+116) {
		tmp = ((b + a) * ((b - a) * (2.0 * sin(((0.005555555555555556 * fabs(angle)) * ((double) M_PI)))))) * sin(fma(-0.005555555555555556, (((double) M_PI) * fabs(angle)), (((double) M_PI) * 0.5)));
	} else {
		tmp = (a + b) * ((b - a) * sin((fabs(angle) * ((cbrt(((double) M_PI)) * 2.1450293971110255) * 0.011111111111111112))));
	}
	return copysign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 5e+116)
		tmp = Float64(Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(2.0 * sin(Float64(Float64(0.005555555555555556 * abs(angle)) * pi))))) * sin(fma(-0.005555555555555556, Float64(pi * abs(angle)), Float64(pi * 0.5))));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(abs(angle) * Float64(Float64(cbrt(pi) * 2.1450293971110255) * 0.011111111111111112)))));
	end
	return Float64(copysign(1.0, angle) * 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[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+116], N[(N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(-0.005555555555555556 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision] + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[Abs[angle], $MachinePrecision] * N[(N[(N[Power[Pi, 1/3], $MachinePrecision] * 2.1450293971110255), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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)))) < 5.00000000000000025e116

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 57.8%

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

   3. Applied rewrites 57.8%

      \[\leadsto \left(\color{blue}{\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. div-flip-rev N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. cos-neg-rev N/A

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

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

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

      12. lower-sin.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-lft-neg-in N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

   5. Applied rewrites 57.6%

      \[\leadsto \left(\left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot 2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-0.005555555555555556, \pi \cdot angle, \pi \cdot 0.5\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 66.9%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 66.9%

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

   7. Applied rewrites 66.9%

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

   if 5.00000000000000025e116 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{90}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{90}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      8. pow1/3 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      10. pow1/3 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      11. pow-prod-up N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{\color{blue}{\frac{2}{3}}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \frac{1}{90}\right)\right) \]

      15. lower-cbrt.f64 66.8%

          \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{0.6666666666666666}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right) \cdot 0.011111111111111112\right)\right) \]

   5. Applied rewrites 66.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}\right)} \cdot 0.011111111111111112\right)\right) \]

   6. Evaluated real constant 66.8%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{2.1450293971110255}\right) \cdot \sqrt[3]{\pi}\right) \cdot 0.011111111111111112\right)\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \frac{4830176796763987}{2251799813685248}\right) \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \frac{4830176796763987}{2251799813685248}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{1}{90}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \frac{4830176796763987}{2251799813685248}\right)} \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \left(\frac{4830176796763987}{2251799813685248} \cdot \sqrt[3]{\pi}\right)\right)} \cdot \frac{1}{90}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\left(\frac{4830176796763987}{2251799813685248} \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right)}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\left(\frac{4830176796763987}{2251799813685248} \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \color{blue}{\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \sqrt[3]{\pi}\right) \cdot \frac{1}{90}\right)}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\color{blue}{\left(\sqrt[3]{\pi} \cdot \frac{4830176796763987}{2251799813685248}\right)} \cdot \frac{1}{90}\right)\right)\right) \]

      9. lower-*.f64 66.7%

         \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(angle \cdot \left(\color{blue}{\left(\sqrt[3]{\pi} \cdot 2.1450293971110255\right)} \cdot 0.011111111111111112\right)\right)\right) \]

   8. Applied rewrites 66.7%

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \color{blue}{\left(angle \cdot \left(\left(\sqrt[3]{\pi} \cdot 2.1450293971110255\right) \cdot 0.011111111111111112\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := b + \left|a\right|\\ t_1 := b - \left|a\right|\\ t_2 := 0.005555555555555556 \cdot \left|angle\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 10^{+58}:\\ \;\;\;\;t\_1 \cdot \left(\left(2 \cdot t\_0\right) \cdot \left(\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot \left|angle\right|\right)\right) \cdot \sin \left(t\_2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot t\_0\right) \cdot \sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot t\_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (+ b (fabs a)))
        (t_1 (- b (fabs a)))
        (t_2 (* 0.005555555555555556 (fabs angle))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 1e+58)
      (*
       t_1
       (*
        (* 2.0 t_0)
        (*
         (cos (* -0.005555555555555556 (* PI (fabs angle))))
         (sin (* t_2 PI)))))
      (*
       (* t_1 t_0)
       (sin (* 2.0 (* (pow PI 0.6666666666666666) (* (cbrt PI) t_2)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = b + fabs(a);
	double t_1 = b - fabs(a);
	double t_2 = 0.005555555555555556 * fabs(angle);
	double tmp;
	if (fabs(angle) <= 1e+58) {
		tmp = t_1 * ((2.0 * t_0) * (cos((-0.005555555555555556 * (((double) M_PI) * fabs(angle)))) * sin((t_2 * ((double) M_PI)))));
	} else {
		tmp = (t_1 * t_0) * sin((2.0 * (pow(((double) M_PI), 0.6666666666666666) * (cbrt(((double) M_PI)) * t_2))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = b + Math.abs(a);
	double t_1 = b - Math.abs(a);
	double t_2 = 0.005555555555555556 * Math.abs(angle);
	double tmp;
	if (Math.abs(angle) <= 1e+58) {
		tmp = t_1 * ((2.0 * t_0) * (Math.cos((-0.005555555555555556 * (Math.PI * Math.abs(angle)))) * Math.sin((t_2 * Math.PI))));
	} else {
		tmp = (t_1 * t_0) * Math.sin((2.0 * (Math.pow(Math.PI, 0.6666666666666666) * (Math.cbrt(Math.PI) * t_2))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(b + abs(a))
	t_1 = Float64(b - abs(a))
	t_2 = Float64(0.005555555555555556 * abs(angle))
	tmp = 0.0
	if (abs(angle) <= 1e+58)
		tmp = Float64(t_1 * Float64(Float64(2.0 * t_0) * Float64(cos(Float64(-0.005555555555555556 * Float64(pi * abs(angle)))) * sin(Float64(t_2 * pi)))));
	else
		tmp = Float64(Float64(t_1 * t_0) * sin(Float64(2.0 * Float64((pi ^ 0.6666666666666666) * Float64(cbrt(pi) * t_2)))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(b + N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b - N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * 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], 1e+58], N[(t$95$1 * N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(N[Cos[N[(-0.005555555555555556 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$2 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[Power[Pi, 0.6666666666666666], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 9.99999999999999944e57

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 57.8%

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

   3. Applied rewrites 57.8%

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

   5. Applied rewrites 57.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 66.8%

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

   7. Applied rewrites 66.7%

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

   if 9.99999999999999944e57 < angle

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 57.8%

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

   3. Applied rewrites 57.8%

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

   5. Applied rewrites 57.5%

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

   7. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \]

      6. lift-cbrt.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      10. pow1/3 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      12. pow1/3 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      13. pow-prod-up N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      14. lower-pow.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      16. lower-*.f64 57.7%

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right) \]

   9. Applied rewrites 57.7%

      \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.0% accurate, 1.2× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 2 \cdot 10^{+96}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot \left|angle\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot \left|angle\right|\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 2e+96)
    (* (+ a b) (* (- b a) (sin (* (* 0.011111111111111112 PI) (fabs angle)))))
    (*
     (* (- b a) (+ b a))
     (sin
      (*
       2.0
       (*
        (pow PI 0.6666666666666666)
        (* (cbrt PI) (* 0.005555555555555556 (fabs angle))))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 2e+96) {
		tmp = (a + b) * ((b - a) * sin(((0.011111111111111112 * ((double) M_PI)) * fabs(angle))));
	} else {
		tmp = ((b - a) * (b + a)) * sin((2.0 * (pow(((double) M_PI), 0.6666666666666666) * (cbrt(((double) M_PI)) * (0.005555555555555556 * fabs(angle))))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 2e+96) {
		tmp = (a + b) * ((b - a) * Math.sin(((0.011111111111111112 * Math.PI) * Math.abs(angle))));
	} else {
		tmp = ((b - a) * (b + a)) * Math.sin((2.0 * (Math.pow(Math.PI, 0.6666666666666666) * (Math.cbrt(Math.PI) * (0.005555555555555556 * Math.abs(angle))))));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 2e+96)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(0.011111111111111112 * pi) * abs(angle)))));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(2.0 * Float64((pi ^ 0.6666666666666666) * Float64(cbrt(pi) * Float64(0.005555555555555556 * abs(angle)))))));
	end
	return Float64(copysign(1.0, angle) * 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], 2e+96], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[Power[Pi, 0.6666666666666666], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(0.005555555555555556 * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 2.0000000000000001e96

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.0%

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

   5. Applied rewrites 67.0%

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

   if 2.0000000000000001e96 < angle

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 57.8%

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

   3. Applied rewrites 57.8%

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

   5. Applied rewrites 57.5%

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

   7. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \]

      6. lift-cbrt.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right) \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)}\right) \]

      9. lift-PI.f64 N/A

         \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      10. pow1/3 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      12. pow1/3 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      13. pow-prod-up N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      14. lower-pow.f64 N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\sqrt[3]{\pi} \cdot \left(\frac{1}{180} \cdot angle\right)\right)\right)\right) \]

      16. lower-*.f64 57.7%

          \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \left({\pi}^{0.6666666666666666} \cdot \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\right) \]

   9. Applied rewrites 57.7%

      \[\leadsto \left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(2 \cdot \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 66.9% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 10^{-26}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(b + a\right)\right) \cdot \sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot \left|angle\right|\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 1e-26)
    (* (+ a b) (* 0.011111111111111112 (* (fabs angle) (* PI (- b a)))))
    (*
     (* (- b a) (+ b a))
     (sin (* (* 0.011111111111111112 PI) (fabs angle)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 1e-26) {
		tmp = (a + b) * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (b - a))));
	} else {
		tmp = ((b - a) * (b + a)) * sin(((0.011111111111111112 * ((double) M_PI)) * fabs(angle)));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 1e-26) {
		tmp = (a + b) * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * (b - a))));
	} else {
		tmp = ((b - a) * (b + a)) * Math.sin(((0.011111111111111112 * Math.PI) * Math.abs(angle)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 1e-26:
		tmp = (a + b) * (0.011111111111111112 * (math.fabs(angle) * (math.pi * (b - a))))
	else:
		tmp = ((b - a) * (b + a)) * math.sin(((0.011111111111111112 * math.pi) * math.fabs(angle)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 1e-26)
		tmp = Float64(Float64(a + b) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(b - a)))));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(b + a)) * sin(Float64(Float64(0.011111111111111112 * pi) * abs(angle))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 1e-26)
		tmp = (a + b) * (0.011111111111111112 * (abs(angle) * (pi * (b - a))));
	else
		tmp = ((b - a) * (b + a)) * sin(((0.011111111111111112 * pi) * abs(angle)));
	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], 1e-26], N[(N[(a + b), $MachinePrecision] * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1e-26

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.8%

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

   6. Applied rewrites 41.8%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 40.3%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 40.3%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 N/A

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

       5. lower--.f64 62.3%

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

   12. Applied rewrites 62.3%

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

   if 1e-26 < angle

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 57.8%

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

   3. Applied rewrites 57.8%

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

   5. Applied rewrites 57.5%

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

   7. Applied rewrites 58.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 57.7%

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

   9. Applied rewrites 57.7%

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

Alternative 5: 66.8% accurate, 1.8× speedup

Math

\[\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot 2.1450293971110255\right) \cdot \sqrt[3]{\pi}\right) \cdot 0.011111111111111112\right)\right) \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (+ a b)
  (*
   (- b a)
   (sin (* (* (* angle 2.1450293971110255) (cbrt PI)) 0.011111111111111112)))))

C

double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * sin((((angle * 2.1450293971110255) * cbrt(((double) M_PI))) * 0.011111111111111112)));
}

Java

public static double code(double a, double b, double angle) {
	return (a + b) * ((b - a) * Math.sin((((angle * 2.1450293971110255) * Math.cbrt(Math.PI)) * 0.011111111111111112)));
}

Julia

function code(a, b, angle)
	return Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(Float64(angle * 2.1450293971110255) * cbrt(pi)) * 0.011111111111111112))))
end

Wolfram

code[a_, b_, angle_] := N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(N[(angle * 2.1450293971110255), $MachinePrecision] * N[Power[Pi, 1/3], $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

3. Applied rewrites 66.8%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{90}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{1}{90}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   8. pow1/3 N/A

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   9. lift-PI.f64 N/A

      \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   10. pow1/3 N/A

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   11. pow-prod-up N/A

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   12. lower-pow.f64 N/A

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{\color{blue}{\frac{2}{3}}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \frac{1}{90}\right)\right) \]

   14. lift-PI.f64 N/A

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \frac{1}{90}\right)\right) \]

   15. lower-cbrt.f64 66.8%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot {\pi}^{0.6666666666666666}\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right) \cdot 0.011111111111111112\right)\right) \]

5. Applied rewrites 66.8%

   \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\color{blue}{\left(\left(angle \cdot {\pi}^{0.6666666666666666}\right) \cdot \sqrt[3]{\pi}\right)} \cdot 0.011111111111111112\right)\right) \]

6. Evaluated real constant 66.8%

   \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(\left(angle \cdot \color{blue}{2.1450293971110255}\right) \cdot \sqrt[3]{\pi}\right) \cdot 0.011111111111111112\right)\right) \]
7. Add Preprocessing

Alternative 6: 66.8% accurate, 2.4× speedup

Math

\[\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(0.011111111111111112 \cdot \pi\right) \cdot angle\right)\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

3. Applied rewrites 66.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 67.0%

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

5. Applied rewrites 67.0%

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

Alternative 7: 66.8% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(a + b\right)\right) \cdot \sin \left(\left(\left|angle\right| \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 5e-16)
    (* (+ a b) (* 0.011111111111111112 (* (fabs angle) (* PI (- b a)))))
    (*
     (* (- b a) (+ a b))
     (sin (* (* (fabs angle) PI) 0.011111111111111112))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 5e-16) {
		tmp = (a + b) * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (b - a))));
	} else {
		tmp = ((b - a) * (a + b)) * sin(((fabs(angle) * ((double) M_PI)) * 0.011111111111111112));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 5e-16) {
		tmp = (a + b) * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * (b - a))));
	} else {
		tmp = ((b - a) * (a + b)) * Math.sin(((Math.abs(angle) * Math.PI) * 0.011111111111111112));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 5e-16:
		tmp = (a + b) * (0.011111111111111112 * (math.fabs(angle) * (math.pi * (b - a))))
	else:
		tmp = ((b - a) * (a + b)) * math.sin(((math.fabs(angle) * math.pi) * 0.011111111111111112))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 5e-16)
		tmp = Float64(Float64(a + b) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(b - a)))));
	else
		tmp = Float64(Float64(Float64(b - a) * Float64(a + b)) * sin(Float64(Float64(abs(angle) * pi) * 0.011111111111111112)));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 5e-16)
		tmp = (a + b) * (0.011111111111111112 * (abs(angle) * (pi * (b - a))));
	else
		tmp = ((b - a) * (a + b)) * sin(((abs(angle) * pi) * 0.011111111111111112));
	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-16], N[(N[(a + b), $MachinePrecision] * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - a), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 5.0000000000000004e-16

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.8%

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

   6. Applied rewrites 41.8%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 40.3%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 40.3%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 N/A

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

       5. lower--.f64 62.3%

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

   12. Applied rewrites 62.3%

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

   if 5.0000000000000004e-16 < angle

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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 57.6%

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

Alternative 8: 66.8% accurate, 2.3× speedup

Math

\[\left(b - \left|a\right|\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b + \left|a\right|\right)\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

3. Applied rewrites 66.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 66.8%

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

   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 66.8%

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

   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 66.8%

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

   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 66.8%

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

5. Applied rewrites 66.8%

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

Alternative 9: 63.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| - \left|a\right|\\ \mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 4.3 \cdot 10^{+122}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \left(\left|b\right| + \left|a\right|\right)}{t\_0} \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- (fabs b) (fabs a))))
   (*
    (copysign 1.0 angle)
    (if (<= (fabs angle) 4.3e+122)
      (*
       (+ (fabs a) (fabs b))
       (* 0.011111111111111112 (* (fabs angle) (* PI t_0))))
      (*
       (/ (* t_0 (+ (fabs b) (fabs a))) t_0)
       (* 0.011111111111111112 (* (fabs angle) (* (fabs b) PI))))))))

C

double code(double a, double b, double angle) {
	double t_0 = fabs(b) - fabs(a);
	double tmp;
	if (fabs(angle) <= 4.3e+122) {
		tmp = (fabs(a) + fabs(b)) * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * t_0)));
	} else {
		tmp = ((t_0 * (fabs(b) + fabs(a))) / t_0) * (0.011111111111111112 * (fabs(angle) * (fabs(b) * ((double) M_PI))));
	}
	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) <= 4.3e+122) {
		tmp = (Math.abs(a) + Math.abs(b)) * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * t_0)));
	} else {
		tmp = ((t_0 * (Math.abs(b) + Math.abs(a))) / t_0) * (0.011111111111111112 * (Math.abs(angle) * (Math.abs(b) * Math.PI)));
	}
	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) <= 4.3e+122:
		tmp = (math.fabs(a) + math.fabs(b)) * (0.011111111111111112 * (math.fabs(angle) * (math.pi * t_0)))
	else:
		tmp = ((t_0 * (math.fabs(b) + math.fabs(a))) / t_0) * (0.011111111111111112 * (math.fabs(angle) * (math.fabs(b) * math.pi)))
	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) <= 4.3e+122)
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * t_0))));
	else
		tmp = Float64(Float64(Float64(t_0 * Float64(abs(b) + abs(a))) / t_0) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(abs(b) * pi))));
	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) <= 4.3e+122)
		tmp = (abs(a) + abs(b)) * (0.011111111111111112 * (abs(angle) * (pi * t_0)));
	else
		tmp = ((t_0 * (abs(b) + abs(a))) / t_0) * (0.011111111111111112 * (abs(angle) * (abs(b) * pi)));
	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], 4.3e+122], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.29999999999999971e122

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.8%

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

   6. Applied rewrites 41.8%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 40.3%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 40.3%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 N/A

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

       5. lower--.f64 62.3%

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

   12. Applied rewrites 62.3%

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

   if 4.29999999999999971e122 < angle

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.8%

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

   6. Applied rewrites 41.8%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 40.3%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 40.3%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. flip-+ N/A

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

       4. lower-unsound--.f64 N/A

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

       5. lower-unsound-/.f64 N/A

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

       6. lower-unsound--.f32 N/A

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

       7. lower--.f32 N/A

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

       8. lower-unsound-*.f32 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-unsound-*.f32 N/A

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

       11. lower-*.f32 N/A

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

       12. difference-of-squares-rev N/A

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

       13. lift-+.f64 N/A

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

       14. lift--.f64 N/A

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

       15. *-commutative N/A

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

       16. lift-*.f64 40.5%

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

   11. Applied rewrites 40.5%

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

Alternative 10: 62.3% accurate, 3.1× speedup

Math

\[\mathsf{copysign}\left(1, angle\right) \cdot \begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 4.5 \cdot 10^{+122}:\\ \;\;\;\;\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot \left(\left|b\right| - \left|a\right|\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 + \frac{\left|a\right|}{\left|b\right|}\right) \cdot \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(\left|angle\right| \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (*
  (copysign 1.0 angle)
  (if (<= (fabs angle) 4.5e+122)
    (*
     (+ (fabs a) (fabs b))
     (* 0.011111111111111112 (* (fabs angle) (* PI (- (fabs b) (fabs a))))))
    (*
     (* (+ 1.0 (/ (fabs a) (fabs b))) (fabs b))
     (* 0.011111111111111112 (* (fabs angle) (* (fabs b) PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 4.5e+122) {
		tmp = (fabs(a) + fabs(b)) * (0.011111111111111112 * (fabs(angle) * (((double) M_PI) * (fabs(b) - fabs(a)))));
	} else {
		tmp = ((1.0 + (fabs(a) / fabs(b))) * fabs(b)) * (0.011111111111111112 * (fabs(angle) * (fabs(b) * ((double) M_PI))));
	}
	return copysign(1.0, angle) * tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 4.5e+122) {
		tmp = (Math.abs(a) + Math.abs(b)) * (0.011111111111111112 * (Math.abs(angle) * (Math.PI * (Math.abs(b) - Math.abs(a)))));
	} else {
		tmp = ((1.0 + (Math.abs(a) / Math.abs(b))) * Math.abs(b)) * (0.011111111111111112 * (Math.abs(angle) * (Math.abs(b) * Math.PI)));
	}
	return Math.copySign(1.0, angle) * tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 4.5e+122:
		tmp = (math.fabs(a) + math.fabs(b)) * (0.011111111111111112 * (math.fabs(angle) * (math.pi * (math.fabs(b) - math.fabs(a)))))
	else:
		tmp = ((1.0 + (math.fabs(a) / math.fabs(b))) * math.fabs(b)) * (0.011111111111111112 * (math.fabs(angle) * (math.fabs(b) * math.pi)))
	return math.copysign(1.0, angle) * tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 4.5e+122)
		tmp = Float64(Float64(abs(a) + abs(b)) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(pi * Float64(abs(b) - abs(a))))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(abs(a) / abs(b))) * abs(b)) * Float64(0.011111111111111112 * Float64(abs(angle) * Float64(abs(b) * pi))));
	end
	return Float64(copysign(1.0, angle) * tmp)
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 4.5e+122)
		tmp = (abs(a) + abs(b)) * (0.011111111111111112 * (abs(angle) * (pi * (abs(b) - abs(a)))));
	else
		tmp = ((1.0 + (abs(a) / abs(b))) * abs(b)) * (0.011111111111111112 * (abs(angle) * (abs(b) * pi)));
	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], 4.5e+122], N[(N[(N[Abs[a], $MachinePrecision] + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * N[(N[Abs[b], $MachinePrecision] - N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[Abs[a], $MachinePrecision] / N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 4.49999999999999997e122

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.8%

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

   6. Applied rewrites 41.8%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 40.3%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 40.3%

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

   10. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 N/A

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

       5. lower--.f64 62.3%

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

   12. Applied rewrites 62.3%

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

   if 4.49999999999999997e122 < angle

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.8%

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

   6. Applied rewrites 41.8%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 40.3%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 40.3%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. sum-to-mult N/A

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

       4. lower-unsound-*.f64 N/A

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

       5. lower-unsound-+.f64 N/A

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

       6. lower-unsound-/.f64 42.6%

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

   11. Applied rewrites 42.6%

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

Alternative 11: 62.3% accurate, 6.6× speedup

Math

\[\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

3. Applied rewrites 66.8%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 41.8%

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

6. Applied rewrites 41.8%

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

7. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 40.3%

      \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

9. Applied rewrites 40.3%

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

10. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-PI.f64 N/A

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

    5. lower--.f64 62.3%

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

12. Applied rewrites 62.3%

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

Alternative 12: 62.3% accurate, 6.6× speedup

Math

\[\left(\left(0.011111111111111112 \cdot \left(angle \cdot \left(b - a\right)\right)\right) \cdot \left(b + a\right)\right) \cdot \pi \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 51.3%

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

4. Applied rewrites 51.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 51.4%

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares-rev N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 54.7%

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 54.7%

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

6. Applied rewrites 54.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 62.3%

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-+.f64 62.3%

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

8. Applied rewrites 62.3%

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

Alternative 13: 62.2% accurate, 6.6× speedup

Math

\[0.011111111111111112 \cdot \left(\left(angle \cdot \left(b - a\right)\right) \cdot \left(\left(b + a\right) \cdot \pi\right)\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 51.3%

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

4. Applied rewrites 51.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 51.4%

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares-rev N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 N/A

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

   15. lift--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 54.7%

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 54.7%

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

6. Applied rewrites 54.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 62.2%

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 62.2%

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

8. Applied rewrites 62.2%

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

Alternative 14: 58.3% accurate, 4.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.72 \cdot 10^{+111}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(angle \cdot \left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(\left(angle \cdot \left|b\right|\right) \cdot \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs b) 1.72e+111)
   (* 0.011111111111111112 (* (* angle (* (- (fabs b) a) (+ (fabs b) a))) PI))
   (* (+ a (fabs b)) (* 0.011111111111111112 (* (* angle (fabs b)) PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 1.72e+111) {
		tmp = 0.011111111111111112 * ((angle * ((fabs(b) - a) * (fabs(b) + a))) * ((double) M_PI));
	} else {
		tmp = (a + fabs(b)) * (0.011111111111111112 * ((angle * fabs(b)) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 1.72e+111) {
		tmp = 0.011111111111111112 * ((angle * ((Math.abs(b) - a) * (Math.abs(b) + a))) * Math.PI);
	} else {
		tmp = (a + Math.abs(b)) * (0.011111111111111112 * ((angle * Math.abs(b)) * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 1.72e+111:
		tmp = 0.011111111111111112 * ((angle * ((math.fabs(b) - a) * (math.fabs(b) + a))) * math.pi)
	else:
		tmp = (a + math.fabs(b)) * (0.011111111111111112 * ((angle * math.fabs(b)) * math.pi))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 1.72e+111)
		tmp = Float64(0.011111111111111112 * Float64(Float64(angle * Float64(Float64(abs(b) - a) * Float64(abs(b) + a))) * pi));
	else
		tmp = Float64(Float64(a + abs(b)) * Float64(0.011111111111111112 * Float64(Float64(angle * abs(b)) * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 1.72e+111)
		tmp = 0.011111111111111112 * ((angle * ((abs(b) - a) * (abs(b) + a))) * pi);
	else
		tmp = (a + abs(b)) * (0.011111111111111112 * ((angle * abs(b)) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.72e+111], N[(0.011111111111111112 * N[(N[(angle * N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(0.011111111111111112 * N[(N[(angle * N[Abs[b], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.72000000000000003e111

   1. Initial program 54.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 51.3%

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

   4. Applied rewrites 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 51.4%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares-rev N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 54.7%

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 54.7%

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

   6. Applied rewrites 54.7%

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

   if 1.72000000000000003e111 < b

   1. Initial program 54.4%

      \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   2. 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--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. difference-of-squares N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

   3. Applied rewrites 66.8%

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

   4. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-PI.f64 41.8%

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

   6. Applied rewrites 41.8%

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

   7. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 40.3%

         \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

   9. Applied rewrites 40.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 40.3%

          \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]

   11. Applied rewrites 40.3%

       \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 39.8% accurate, 6.3× speedup

Math

\[\left(\left|a\right| + \left|b\right|\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\left|b\right| \cdot \pi\right)\right)\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

3. Applied rewrites 66.8%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 41.8%

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

6. Applied rewrites 41.8%

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

7. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 40.3%

      \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

9. Applied rewrites 40.3%

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

Alternative 16: 38.6% accurate, 9.4× speedup

Math

\[b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
2. 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--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. difference-of-squares N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

3. Applied rewrites 66.8%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-PI.f64 41.8%

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

6. Applied rewrites 41.8%

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

7. Taylor expanded in angle around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-PI.f64 40.3%

      \[\leadsto \left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right) \]

9. Applied rewrites 40.3%

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

10. Taylor expanded in a around 0

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

12. Applied rewrites 38.6%

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

Reproduce

herbie shell --seed 2025189 
(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)))))