ab-angle->ABCF B

Percentage Accurate: 54.4% → 67.2%

Time: 28.3s

Alternatives: 10

Speedup: 32.2×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot {\left(\sqrt[3]{angle\_m \cdot 0.005555555555555556}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{e^{3 \cdot \log \pi}}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= (/ angle_m 180.0) 4e+133)
    (*
     (+ a b)
     (*
      (- b a)
      (sin (* 2.0 (* PI (pow (cbrt (* angle_m 0.005555555555555556)) 3.0))))))
    (*
     (+ a b)
     (*
      (- b a)
      (sin
       (*
        2.0
        (*
         (* angle_m 0.005555555555555556)
         (cbrt (exp (* 3.0 (log PI))))))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+133) {
		tmp = (a + b) * ((b - a) * sin((2.0 * (((double) M_PI) * pow(cbrt((angle_m * 0.005555555555555556)), 3.0)))));
	} else {
		tmp = (a + b) * ((b - a) * sin((2.0 * ((angle_m * 0.005555555555555556) * cbrt(exp((3.0 * log(((double) M_PI)))))))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 4e+133) {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * (Math.PI * Math.pow(Math.cbrt((angle_m * 0.005555555555555556)), 3.0)))));
	} else {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * ((angle_m * 0.005555555555555556) * Math.cbrt(Math.exp((3.0 * Math.log(Math.PI))))))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 4e+133)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * (cbrt(Float64(angle_m * 0.005555555555555556)) ^ 3.0))))));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(angle_m * 0.005555555555555556) * cbrt(exp(Float64(3.0 * log(pi)))))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 4e+133], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[Power[N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[Exp[N[(3.0 * N[Log[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000001e133

   1. Initial program 56.5%

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

      1. associate-*l* 56.5%

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

      2. *-commutative 56.5%

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

      3. associate-*l* 56.5%

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

   3. Simplified 56.5%

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

      1. unpow2 56.5%

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

      2. unpow2 56.5%

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

      3. difference-of-squares 64.3%

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

   6. Applied egg-rr 64.3%

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

      1. pow1 64.3%

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

      2. 2-sin 64.3%

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

      3. div-inv 63.5%

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

      4. metadata-eval 63.5%

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

   8. Applied egg-rr 63.5%

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

      1. unpow1 63.5%

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

      2. associate-*l* 74.2%

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

      3. +-commutative 74.2%

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

      4. *-commutative 74.2%

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

   10. Simplified 74.2%

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

       1. *-commutative 74.2%

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

       2. metadata-eval 74.2%

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

       3. div-inv 75.0%

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

       4. add-cube-cbrt 73.9%

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

       5. pow3 74.2%

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

       6. div-inv 76.0%

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

       7. metadata-eval 76.0%

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

       8. *-commutative 76.0%

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

   12. Applied egg-rr 76.0%

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

   if 4.0000000000000001e133 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 24.9%

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

      1. associate-*l* 24.9%

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

      2. *-commutative 24.9%

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

      3. associate-*l* 24.9%

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

   3. Simplified 24.9%

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

      1. unpow2 24.9%

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

      2. unpow2 24.9%

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

      3. difference-of-squares 28.1%

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

   6. Applied egg-rr 28.1%

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

      1. pow1 28.1%

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

      2. 2-sin 28.1%

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

      3. div-inv 28.0%

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

      4. metadata-eval 28.0%

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

   8. Applied egg-rr 28.0%

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

      1. unpow1 28.0%

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

      2. associate-*l* 28.0%

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

      3. +-commutative 28.0%

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

      4. *-commutative 28.0%

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

   10. Simplified 28.0%

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

       1. add-cbrt-cube 38.4%

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

       2. pow3 38.4%

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

   12. Applied egg-rr 38.4%

       \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\color{blue}{\sqrt[3]{{\pi}^{3}}} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. add-exp-log 38.4%

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

       2. log-pow 47.6%

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

   14. Applied egg-rr 47.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot {\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{e^{3 \cdot \log \pi}}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot {\left(\sqrt[3]{angle\_m \cdot 0.005555555555555556}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 4.5e+202)
    (*
     (+ a b)
     (*
      (- b a)
      (sin (* 2.0 (* (* angle_m 0.005555555555555556) (cbrt (pow PI 3.0)))))))
    (*
     (+ a b)
     (*
      (- b a)
      (sin
       (* 2.0 (* PI (pow (cbrt (* angle_m 0.005555555555555556)) 3.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.5e+202) {
		tmp = (a + b) * ((b - a) * sin((2.0 * ((angle_m * 0.005555555555555556) * cbrt(pow(((double) M_PI), 3.0))))));
	} else {
		tmp = (a + b) * ((b - a) * sin((2.0 * (((double) M_PI) * pow(cbrt((angle_m * 0.005555555555555556)), 3.0)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.5e+202) {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * ((angle_m * 0.005555555555555556) * Math.cbrt(Math.pow(Math.PI, 3.0))))));
	} else {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * (Math.PI * Math.pow(Math.cbrt((angle_m * 0.005555555555555556)), 3.0)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 4.5e+202)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(Float64(angle_m * 0.005555555555555556) * cbrt((pi ^ 3.0)))))));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * (cbrt(Float64(angle_m * 0.005555555555555556)) ^ 3.0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 4.5e+202], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[Power[N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 4.49999999999999978e202

   1. Initial program 53.4%

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

      1. associate-*l* 53.4%

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

      2. *-commutative 53.4%

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

      3. associate-*l* 53.4%

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

   3. Simplified 53.4%

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

      1. unpow2 53.4%

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

      2. unpow2 53.4%

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

      3. difference-of-squares 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. pow1 59.2%

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

      2. 2-sin 59.2%

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

      3. div-inv 59.4%

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

      4. metadata-eval 59.4%

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

   8. Applied egg-rr 59.4%

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

      1. unpow1 59.4%

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

      2. associate-*l* 68.2%

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

      3. +-commutative 68.2%

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

      4. *-commutative 68.2%

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

   10. Simplified 68.2%

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

       1. add-cbrt-cube 68.9%

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

       2. pow3 68.9%

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

   12. Applied egg-rr 68.9%

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

   if 4.49999999999999978e202 < a

   1. Initial program 45.4%

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

      1. associate-*l* 45.4%

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

      2. *-commutative 45.4%

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

      3. associate-*l* 45.4%

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

   3. Simplified 45.4%

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

      1. unpow2 45.4%

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

      2. unpow2 45.4%

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

      3. difference-of-squares 63.9%

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

   6. Applied egg-rr 63.9%

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

      1. pow1 63.9%

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

      2. 2-sin 63.9%

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

      3. div-inv 56.5%

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

      4. metadata-eval 56.5%

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

   8. Applied egg-rr 56.5%

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

      1. unpow1 56.5%

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

      2. associate-*l* 70.2%

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

      3. +-commutative 70.2%

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

      4. *-commutative 70.2%

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

   10. Simplified 70.2%

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

       1. *-commutative 70.2%

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

       2. metadata-eval 70.2%

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

       3. div-inv 77.7%

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

       4. add-cube-cbrt 66.4%

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

       5. pow3 77.5%

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

       6. div-inv 85.0%

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

       7. metadata-eval 85.0%

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

       8. *-commutative 85.0%

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

   12. Applied egg-rr 85.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.5 \cdot 10^{+202}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{{\pi}^{3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot {\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{3}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{+205}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{1}{\frac{180}{angle\_m}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot {\left(\sqrt[3]{angle\_m \cdot 0.005555555555555556}\right)}^{3}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= a 6.2e+205)
    (* (+ a b) (* (- b a) (sin (* 2.0 (* PI (/ 1.0 (/ 180.0 angle_m)))))))
    (*
     (+ a b)
     (*
      (- b a)
      (sin
       (* 2.0 (* PI (pow (cbrt (* angle_m 0.005555555555555556)) 3.0)))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 6.2e+205) {
		tmp = (a + b) * ((b - a) * sin((2.0 * (((double) M_PI) * (1.0 / (180.0 / angle_m))))));
	} else {
		tmp = (a + b) * ((b - a) * sin((2.0 * (((double) M_PI) * pow(cbrt((angle_m * 0.005555555555555556)), 3.0)))));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (a <= 6.2e+205) {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * (Math.PI * (1.0 / (180.0 / angle_m))))));
	} else {
		tmp = (a + b) * ((b - a) * Math.sin((2.0 * (Math.PI * Math.pow(Math.cbrt((angle_m * 0.005555555555555556)), 3.0)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (a <= 6.2e+205)
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * Float64(1.0 / Float64(180.0 / angle_m)))))));
	else
		tmp = Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(2.0 * Float64(pi * (cbrt(Float64(angle_m * 0.005555555555555556)) ^ 3.0))))));
	end
	return Float64(angle_s * tmp)
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[a, 6.2e+205], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(1.0 / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + b), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[Power[N[Power[N[(angle$95$m * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 6.20000000000000035e205

   1. Initial program 53.4%

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

      1. associate-*l* 53.4%

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

      2. *-commutative 53.4%

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

      3. associate-*l* 53.4%

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

   3. Simplified 53.4%

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

      1. unpow2 53.4%

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

      2. unpow2 53.4%

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

      3. difference-of-squares 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. pow1 59.2%

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

      2. 2-sin 59.2%

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

      3. div-inv 59.4%

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

      4. metadata-eval 59.4%

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

   8. Applied egg-rr 59.4%

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

      1. unpow1 59.4%

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

      2. associate-*l* 68.2%

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

      3. +-commutative 68.2%

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

      4. *-commutative 68.2%

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

   10. Simplified 68.2%

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

       1. metadata-eval 68.2%

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

       2. associate-/r/ 69.4%

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

   12. Applied egg-rr 69.4%

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

   if 6.20000000000000035e205 < a

   1. Initial program 45.4%

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

      1. associate-*l* 45.4%

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

      2. *-commutative 45.4%

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

      3. associate-*l* 45.4%

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

   3. Simplified 45.4%

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

      1. unpow2 45.4%

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

      2. unpow2 45.4%

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

      3. difference-of-squares 63.9%

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

   6. Applied egg-rr 63.9%

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

      1. pow1 63.9%

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

      2. 2-sin 63.9%

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

      3. div-inv 56.5%

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

      4. metadata-eval 56.5%

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

   8. Applied egg-rr 56.5%

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

      1. unpow1 56.5%

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

      2. associate-*l* 70.2%

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

      3. +-commutative 70.2%

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

      4. *-commutative 70.2%

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

   10. Simplified 70.2%

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

       1. *-commutative 70.2%

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

       2. metadata-eval 70.2%

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

       3. div-inv 77.7%

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

       4. add-cube-cbrt 66.4%

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

       5. pow3 77.5%

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

       6. div-inv 85.0%

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

       7. metadata-eval 85.0%

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

       8. *-commutative 85.0%

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

   12. Applied egg-rr 85.0%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.2 \cdot 10^{+205}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \frac{1}{\frac{180}{angle}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot {\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{3}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 3.6× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* (+ a b) (* (- b a) (sin (* 2.0 (/ 1.0 (/ 180.0 (* angle_m PI)))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. associate-*l* 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*l* 52.6%

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

3. Simplified 52.6%

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

   1. unpow2 52.6%

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

   2. unpow2 52.6%

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

   3. difference-of-squares 59.7%

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

6. Applied egg-rr 59.7%

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

   1. pow1 59.7%

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

   2. 2-sin 59.7%

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

   3. div-inv 59.1%

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

   4. metadata-eval 59.1%

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

8. Applied egg-rr 59.1%

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

   1. unpow1 59.1%

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

   2. associate-*l* 68.4%

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

   3. +-commutative 68.4%

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

   4. *-commutative 68.4%

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

10. Simplified 68.4%

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

    1. *-commutative 68.4%

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

    2. metadata-eval 68.4%

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

    3. div-inv 69.1%

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

    4. associate-*r/ 70.0%

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

    5. clear-num 70.5%

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

12. Applied egg-rr 70.5%

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

13. Final simplification 70.5%

    \[\leadsto \left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(2 \cdot \frac{1}{\frac{180}{angle \cdot \pi}}\right)\right) \]
14. Add Preprocessing

Alternative 5: 67.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. associate-*l* 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*l* 52.6%

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

3. Simplified 52.6%

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

   1. unpow2 52.6%

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

   2. unpow2 52.6%

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

   3. difference-of-squares 59.7%

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

6. Applied egg-rr 59.7%

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

   1. pow1 59.7%

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

   2. 2-sin 59.7%

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

   3. div-inv 59.1%

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

   4. metadata-eval 59.1%

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

8. Applied egg-rr 59.1%

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

   1. unpow1 59.1%

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

   2. associate-*l* 68.4%

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

   3. +-commutative 68.4%

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

   4. *-commutative 68.4%

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

10. Simplified 68.4%

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

    1. *-commutative 68.4%

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

    2. metadata-eval 68.4%

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

    3. div-inv 69.1%

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

    4. associate-*r/ 70.0%

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

12. Applied egg-rr 70.0%

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

13. Final simplification 70.0%

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

Alternative 6: 67.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (* (+ a b) (* (- b a) (sin (* (* angle_m PI) 0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((a + b) * ((b - a) * sin(((angle_m * ((double) M_PI)) * 0.011111111111111112))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((a + b) * ((b - a) * Math.sin(((angle_m * Math.PI) * 0.011111111111111112))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((a + b) * ((b - a) * math.sin(((angle_m * math.pi) * 0.011111111111111112))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(a + b) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m * pi) * 0.011111111111111112)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((a + b) * ((b - a) * sin(((angle_m * pi) * 0.011111111111111112))));
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. associate-*l* 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*l* 52.6%

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

3. Simplified 52.6%

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

   1. unpow2 52.6%

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

   2. unpow2 52.6%

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

   3. difference-of-squares 59.7%

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

6. Applied egg-rr 59.7%

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

   1. pow1 59.7%

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

   2. 2-sin 59.7%

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

   3. div-inv 59.1%

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

   4. metadata-eval 59.1%

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

8. Applied egg-rr 59.1%

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

   1. unpow1 59.1%

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

   2. associate-*l* 68.4%

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

   3. +-commutative 68.4%

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

   4. *-commutative 68.4%

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

10. Simplified 68.4%

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

11. Taylor expanded in angle around inf 69.7%

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

12. Final simplification 69.7%

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

Alternative 7: 62.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (+ a b) (* (- b a) (* PI (* angle_m 0.011111111111111112))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((a + b) * ((b - a) * (((double) M_PI) * (angle_m * 0.011111111111111112))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((a + b) * ((b - a) * (Math.PI * (angle_m * 0.011111111111111112))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((a + b) * ((b - a) * (math.pi * (angle_m * 0.011111111111111112))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(pi * Float64(angle_m * 0.011111111111111112)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((a + b) * ((b - a) * (pi * (angle_m * 0.011111111111111112))));
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. associate-*l* 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*l* 52.6%

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

3. Simplified 52.6%

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

   1. unpow2 52.6%

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

   2. unpow2 52.6%

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

   3. difference-of-squares 59.7%

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

6. Applied egg-rr 59.7%

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

   1. pow1 59.7%

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

   2. 2-sin 59.7%

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

   3. div-inv 59.1%

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

   4. metadata-eval 59.1%

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

8. Applied egg-rr 59.1%

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

   1. unpow1 59.1%

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

   2. associate-*l* 68.4%

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

   3. +-commutative 68.4%

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

   4. *-commutative 68.4%

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

10. Simplified 68.4%

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

    1. add-cbrt-cube 70.6%

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

    2. pow3 70.6%

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

12. Applied egg-rr 70.6%

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

13. Taylor expanded in angle around 0 64.6%

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

    1. associate-*r* 64.6%

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

15. Simplified 64.6%

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

16. Final simplification 64.6%

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

Alternative 8: 62.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(\left(a + b\right) \cdot \left(\left(b - a\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (+ a b) (* (- b a) (* (* angle_m PI) 0.011111111111111112)))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((a + b) * ((b - a) * ((angle_m * ((double) M_PI)) * 0.011111111111111112)));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * ((a + b) * ((b - a) * ((angle_m * Math.PI) * 0.011111111111111112)));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * ((a + b) * ((b - a) * ((angle_m * math.pi) * 0.011111111111111112)))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(Float64(a + b) * Float64(Float64(b - a) * Float64(Float64(angle_m * pi) * 0.011111111111111112))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * ((a + b) * ((b - a) * ((angle_m * pi) * 0.011111111111111112)));
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. associate-*l* 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*l* 52.6%

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

3. Simplified 52.6%

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

   1. unpow2 52.6%

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

   2. unpow2 52.6%

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

   3. difference-of-squares 59.7%

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

6. Applied egg-rr 59.7%

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

   1. pow1 59.7%

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

   2. 2-sin 59.7%

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

   3. div-inv 59.1%

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

   4. metadata-eval 59.1%

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

8. Applied egg-rr 59.1%

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

   1. unpow1 59.1%

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

   2. associate-*l* 68.4%

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

   3. +-commutative 68.4%

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

   4. *-commutative 68.4%

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

10. Simplified 68.4%

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

11. Taylor expanded in angle around 0 64.6%

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

12. Final simplification 64.6%

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

Alternative 9: 54.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\right) \cdot \left(\left(a + b\right) \cdot \pi\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* (- b a) (* (+ a b) PI))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((b - a) * ((a + b) * ((double) M_PI)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * ((b - a) * ((a + b) * Math.PI))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * ((b - a) * ((a + b) * math.pi))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a) * Float64(Float64(a + b) * pi)))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * ((b - a) * ((a + b) * pi))));
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. associate-*l* 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*l* 52.6%

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

3. Simplified 52.6%

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

   1. unpow2 52.6%

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

   2. unpow2 52.6%

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

   3. difference-of-squares 59.7%

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

6. Applied egg-rr 59.7%

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

7. Taylor expanded in angle around 0 55.6%

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

8. Taylor expanded in angle around 0 55.9%

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

   1. associate-*r* 55.9%

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

   2. sub-neg 55.9%

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

   3. distribute-lft-in 50.3%

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

10. Applied egg-rr 50.3%

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

    1. distribute-lft-out 55.9%

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

    2. +-commutative 55.9%

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

    3. unsub-neg 55.9%

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

12. Simplified 55.9%

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

13. Final simplification 55.9%

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

Alternative 10: 54.6% accurate, 32.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* 0.011111111111111112 (* angle_m (* PI (* (+ a b) (- b a)))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (((double) M_PI) * ((a + b) * (b - a)))));
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	return angle_s * (0.011111111111111112 * (angle_m * (Math.PI * ((a + b) * (b - a)))));
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	return angle_s * (0.011111111111111112 * (angle_m * (math.pi * ((a + b) * (b - a)))))

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	return Float64(angle_s * Float64(0.011111111111111112 * Float64(angle_m * Float64(pi * Float64(Float64(a + b) * Float64(b - a))))))
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp = code(angle_s, a, b, angle_m)
	tmp = angle_s * (0.011111111111111112 * (angle_m * (pi * ((a + b) * (b - a)))));
end

Wolfram

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

Derivation

1. Initial program 52.6%

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

   1. associate-*l* 52.6%

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

   2. *-commutative 52.6%

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

   3. associate-*l* 52.6%

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

3. Simplified 52.6%

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

   1. unpow2 52.6%

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

   2. unpow2 52.6%

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

   3. difference-of-squares 59.7%

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

6. Applied egg-rr 59.7%

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

7. Taylor expanded in angle around 0 55.6%

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

8. Taylor expanded in angle around 0 55.9%

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

Reproduce

herbie shell --seed 2024089 
(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)))))