ab-angle->ABCF B

Percentage Accurate: 54.1% → 68.0%

Time: 15.9s

Alternatives: 15

Speedup: 6.6×

• 32.5% 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: 54.1% 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: 68.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.2 \cdot 10^{+197}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\pi}{180}, angle\_m, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(\frac{180 \cdot \left(\left(\pi + \pi\right) \cdot angle\_m\right)}{\left(180 \cdot 180\right) \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \frac{\sin \left(\frac{angle\_m}{180} \cdot \left(\pi + \pi\right)\right) + \left(\cos \left(0 - \frac{\pi}{2}\right) - \cos \left(\mathsf{fma}\left(\pi + \pi, \frac{angle\_m}{180}, \frac{\pi}{2}\right)\right)\right)}{2}\right)\\ \end{array} \end{array} \]

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 1.2e+197)
    (*
     (+ b a)
     (*
      (* (* (- b a) 2.0) (sin (fma (/ PI 180.0) angle_m (/ PI 2.0))))
      (sin (/ (* 180.0 (* (+ PI PI) angle_m)) (* (* 180.0 180.0) 2.0)))))
    (*
     (+ b a)
     (*
      (- b a)
      (/
       (+
        (sin (* (/ angle_m 180.0) (+ PI PI)))
        (-
         (cos (- 0.0 (/ PI 2.0)))
         (cos (fma (+ PI PI) (/ angle_m 180.0) (/ PI 2.0)))))
       2.0))))))

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 (angle_m <= 1.2e+197) {
		tmp = (b + a) * ((((b - a) * 2.0) * sin(fma((((double) M_PI) / 180.0), angle_m, (((double) M_PI) / 2.0)))) * sin(((180.0 * ((((double) M_PI) + ((double) M_PI)) * angle_m)) / ((180.0 * 180.0) * 2.0))));
	} else {
		tmp = (b + a) * ((b - a) * ((sin(((angle_m / 180.0) * (((double) M_PI) + ((double) M_PI)))) + (cos((0.0 - (((double) M_PI) / 2.0))) - cos(fma((((double) M_PI) + ((double) M_PI)), (angle_m / 180.0), (((double) M_PI) / 2.0))))) / 2.0));
	}
	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 (angle_m <= 1.2e+197)
		tmp = Float64(Float64(b + a) * Float64(Float64(Float64(Float64(b - a) * 2.0) * sin(fma(Float64(pi / 180.0), angle_m, Float64(pi / 2.0)))) * sin(Float64(Float64(180.0 * Float64(Float64(pi + pi) * angle_m)) / Float64(Float64(180.0 * 180.0) * 2.0)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(Float64(sin(Float64(Float64(angle_m / 180.0) * Float64(pi + pi))) + Float64(cos(Float64(0.0 - Float64(pi / 2.0))) - cos(fma(Float64(pi + pi), Float64(angle_m / 180.0), Float64(pi / 2.0))))) / 2.0)));
	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[angle$95$m, 1.2e+197], N[(N[(b + a), $MachinePrecision] * N[(N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] * angle$95$m + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(180.0 * N[(N[(Pi + Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(180.0 * 180.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(N[(N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.0 - N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(N[(Pi + Pi), $MachinePrecision] * N[(angle$95$m / 180.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.1999999999999999e197

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 67.4%

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

   9. Applied rewrites 66.9%

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

   if 1.1999999999999999e197 < angle

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   5. Applied rewrites 42.9%

      \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\frac{\sin \left(\frac{angle}{180} \cdot \left(\pi + \pi\right)\right) + \left(\cos \left(0 - \frac{\pi}{2}\right) - \cos \left(\mathsf{fma}\left(\pi + \pi, \frac{angle}{180}, \frac{\pi}{2}\right)\right)\right)}{2}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 1.05 \cdot 10^{+269}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(\left(\left(b - a\right) \cdot 2\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\pi}{180}, angle\_m, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(\frac{180 \cdot \left(\left(\pi + \pi\right) \cdot angle\_m\right)}{\left(180 \cdot 180\right) \cdot 2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(-1 \cdot \mathsf{fma}\left(\frac{1}{b}, a \cdot b, -b\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \left(\pi + \pi\right)\right)\right)\\ \end{array} \end{array} \]

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 1.05e+269)
    (*
     (+ b a)
     (*
      (* (* (- b a) 2.0) (sin (fma (/ PI 180.0) angle_m (/ PI 2.0))))
      (sin (/ (* 180.0 (* (+ PI PI) angle_m)) (* (* 180.0 180.0) 2.0)))))
    (*
     (+ b a)
     (*
      (* -1.0 (fma (/ 1.0 b) (* a b) (- b)))
      (sin (* (* 0.005555555555555556 angle_m) (+ PI 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 <= 1.05e+269) {
		tmp = (b + a) * ((((b - a) * 2.0) * sin(fma((((double) M_PI) / 180.0), angle_m, (((double) M_PI) / 2.0)))) * sin(((180.0 * ((((double) M_PI) + ((double) M_PI)) * angle_m)) / ((180.0 * 180.0) * 2.0))));
	} else {
		tmp = (b + a) * ((-1.0 * fma((1.0 / b), (a * b), -b)) * sin(((0.005555555555555556 * angle_m) * (((double) M_PI) + ((double) M_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 (angle_m <= 1.05e+269)
		tmp = Float64(Float64(b + a) * Float64(Float64(Float64(Float64(b - a) * 2.0) * sin(fma(Float64(pi / 180.0), angle_m, Float64(pi / 2.0)))) * sin(Float64(Float64(180.0 * Float64(Float64(pi + pi) * angle_m)) / Float64(Float64(180.0 * 180.0) * 2.0)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(-1.0 * fma(Float64(1.0 / b), Float64(a * b), Float64(-b))) * sin(Float64(Float64(0.005555555555555556 * angle_m) * Float64(pi + 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[angle$95$m, 1.05e+269], N[(N[(b + a), $MachinePrecision] * N[(N[(N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision] * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] * angle$95$m + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(180.0 * N[(N[(Pi + Pi), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(180.0 * 180.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(-1.0 * N[(N[(1.0 / b), $MachinePrecision] * N[(a * b), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 1.05e269

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 67.4%

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

   9. Applied rewrites 66.9%

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

   if 1.05e269 < angle

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 67.6%

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

   7. Taylor expanded in b around -inf

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

   9. Applied rewrites 64.1%

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

   11. Applied rewrites 63.9%

       \[\leadsto \left(b + a\right) \cdot \left(\left(-1 \cdot \mathsf{fma}\left(\frac{1}{b}, \color{blue}{a \cdot b}, -b\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 67.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 8 \cdot 10^{+108}:\\ \;\;\;\;\left(\left(b - a\right) \cdot \left(\left(b + a\right) \cdot \left(\sin \left(\frac{angle\_m \cdot \pi}{180}\right) \cdot 2\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{\pi}{180}, angle\_m, \frac{\pi}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m}{180} \cdot \left(\pi + \pi\right)\right)\right)\\ \end{array} \end{array} \]

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 8e+108)
    (*
     (* (- b a) (* (+ b a) (* (sin (/ (* angle_m PI) 180.0)) 2.0)))
     (sin (fma (/ PI 180.0) angle_m (/ PI 2.0))))
    (* (+ b a) (* (- b a) (sin (* (/ angle_m 180.0) (+ PI 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 <= 8e+108) {
		tmp = ((b - a) * ((b + a) * (sin(((angle_m * ((double) M_PI)) / 180.0)) * 2.0))) * sin(fma((((double) M_PI) / 180.0), angle_m, (((double) M_PI) / 2.0)));
	} else {
		tmp = (b + a) * ((b - a) * sin(((angle_m / 180.0) * (((double) M_PI) + ((double) M_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 (angle_m <= 8e+108)
		tmp = Float64(Float64(Float64(b - a) * Float64(Float64(b + a) * Float64(sin(Float64(Float64(angle_m * pi) / 180.0)) * 2.0))) * sin(fma(Float64(pi / 180.0), angle_m, Float64(pi / 2.0))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m / 180.0) * Float64(pi + 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[angle$95$m, 8e+108], N[(N[(N[(b - a), $MachinePrecision] * N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(Pi / 180.0), $MachinePrecision] * angle$95$m + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 8.0000000000000003e108

   1. Initial program 54.1%

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

   3. Applied rewrites 57.9%

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 67.4%

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

   if 8.0000000000000003e108 < angle

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

Alternative 4: 67.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+157}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m}{180} \cdot \left(\pi + \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(-1 \cdot \left(b \cdot \left(\frac{a}{b} - \mathsf{fma}\left(\frac{1}{180}, \mathsf{fma}\left(angle\_m, \pi, -angle\_m \cdot \pi\right), 1\right)\right)\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \left(\pi + \pi\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= b 3e+157)
    (* (+ b a) (* (- b a) (sin (* (/ angle_m 180.0) (+ PI PI)))))
    (*
     (+ b a)
     (*
      (*
       -1.0
       (*
        b
        (-
         (/ a b)
         (fma (/ 1.0 180.0) (fma angle_m PI (- (* angle_m PI))) 1.0))))
      (sin (* (* 0.005555555555555556 angle_m) (+ PI PI))))))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 3e+157) {
		tmp = (b + a) * ((b - a) * sin(((angle_m / 180.0) * (((double) M_PI) + ((double) M_PI)))));
	} else {
		tmp = (b + a) * ((-1.0 * (b * ((a / b) - fma((1.0 / 180.0), fma(angle_m, ((double) M_PI), -(angle_m * ((double) M_PI))), 1.0)))) * sin(((0.005555555555555556 * angle_m) * (((double) M_PI) + ((double) M_PI)))));
	}
	return angle_s * tmp;
}

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 3e+157)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m / 180.0) * Float64(pi + pi)))));
	else
		tmp = Float64(Float64(b + a) * Float64(Float64(-1.0 * Float64(b * Float64(Float64(a / b) - fma(Float64(1.0 / 180.0), fma(angle_m, pi, Float64(-Float64(angle_m * pi))), 1.0)))) * sin(Float64(Float64(0.005555555555555556 * angle_m) * Float64(pi + 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[b, 3e+157], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + a), $MachinePrecision] * N[(N[(-1.0 * N[(b * N[(N[(a / b), $MachinePrecision] - N[(N[(1.0 / 180.0), $MachinePrecision] * N[(angle$95$m * Pi + (-N[(angle$95$m * Pi), $MachinePrecision])), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 3.0000000000000001e157

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   if 3.0000000000000001e157 < b

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 67.6%

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

   7. Taylor expanded in b around -inf

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

   9. Applied rewrites 64.1%

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

   11. Applied rewrites 61.4%

       \[\leadsto \left(b + a\right) \cdot \left(\left(-1 \cdot \left(b \cdot \left(\frac{a}{b} - \mathsf{fma}\left(\frac{1}{180}, \color{blue}{\mathsf{fma}\left(angle, \pi, -angle \cdot \pi\right)}, 1\right)\right)\right)\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi + \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 67.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 3.9 \cdot 10^{+157}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\frac{angle\_m}{180} \cdot \left(\pi + \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \pi\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 3.9e+157)
    (* (+ b a) (* (- b a) (sin (* (/ angle_m 180.0) (+ PI PI)))))
    (* (* (* (* angle_m 0.011111111111111112) (+ b a)) (- b a)) PI))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 3.9e+157) {
		tmp = (b + a) * ((b - a) * sin(((angle_m / 180.0) * (((double) M_PI) + ((double) M_PI)))));
	} else {
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * ((double) M_PI);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 3.9e+157) {
		tmp = (b + a) * ((b - a) * Math.sin(((angle_m / 180.0) * (Math.PI + Math.PI))));
	} else {
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * Math.PI;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 3.9e+157:
		tmp = (b + a) * ((b - a) * math.sin(((angle_m / 180.0) * (math.pi + math.pi))))
	else:
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * math.pi
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 3.9e+157)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(Float64(angle_m / 180.0) * Float64(pi + pi)))));
	else
		tmp = Float64(Float64(Float64(Float64(angle_m * 0.011111111111111112) * Float64(b + a)) * Float64(b - a)) * pi);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 3.9e+157)
		tmp = (b + a) * ((b - a) * sin(((angle_m / 180.0) * (pi + pi))));
	else
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * pi;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 3.9e+157], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 3.89999999999999971e157

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   if 3.89999999999999971e157 < b

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 62.2%

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

Alternative 6: 67.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{+157}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \left(\pi + \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \pi\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 2.25e+157)
    (*
     (+ b a)
     (* (- b a) (sin (* (* 0.005555555555555556 angle_m) (+ PI PI)))))
    (* (* (* (* angle_m 0.011111111111111112) (+ b a)) (- b a)) PI))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.25e+157) {
		tmp = (b + a) * ((b - a) * sin(((0.005555555555555556 * angle_m) * (((double) M_PI) + ((double) M_PI)))));
	} else {
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * ((double) M_PI);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.25e+157) {
		tmp = (b + a) * ((b - a) * Math.sin(((0.005555555555555556 * angle_m) * (Math.PI + Math.PI))));
	} else {
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * Math.PI;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 2.25e+157:
		tmp = (b + a) * ((b - a) * math.sin(((0.005555555555555556 * angle_m) * (math.pi + math.pi))))
	else:
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * math.pi
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 2.25e+157)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(Float64(0.005555555555555556 * angle_m) * Float64(pi + pi)))));
	else
		tmp = Float64(Float64(Float64(Float64(angle_m * 0.011111111111111112) * Float64(b + a)) * Float64(b - a)) * pi);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.25e+157)
		tmp = (b + a) * ((b - a) * sin(((0.005555555555555556 * angle_m) * (pi + pi))));
	else
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * pi;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 2.25e+157], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 2.24999999999999992e157

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 67.6%

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

   if 2.24999999999999992e157 < b

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 62.2%

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

Alternative 7: 67.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{+155}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\sin \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot \left(b - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right) \cdot \pi\\ \end{array} \end{array} \]

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (*
  angle_s
  (if (<= b 2.2e+155)
    (* (+ b a) (* (sin (* 0.011111111111111112 (* angle_m PI))) (- b a)))
    (* (* (* (* angle_m 0.011111111111111112) (+ b a)) (- b a)) PI))))

C

angle\_m = fabs(angle);
angle\_s = copysign(1.0, angle);
double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.2e+155) {
		tmp = (b + a) * (sin((0.011111111111111112 * (angle_m * ((double) M_PI)))) * (b - a));
	} else {
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * ((double) M_PI);
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.2e+155) {
		tmp = (b + a) * (Math.sin((0.011111111111111112 * (angle_m * Math.PI))) * (b - a));
	} else {
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * Math.PI;
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if b <= 2.2e+155:
		tmp = (b + a) * (math.sin((0.011111111111111112 * (angle_m * math.pi))) * (b - a))
	else:
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * math.pi
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (b <= 2.2e+155)
		tmp = Float64(Float64(b + a) * Float64(sin(Float64(0.011111111111111112 * Float64(angle_m * pi))) * Float64(b - a)));
	else
		tmp = Float64(Float64(Float64(Float64(angle_m * 0.011111111111111112) * Float64(b + a)) * Float64(b - a)) * pi);
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.2e+155)
		tmp = (b + a) * (sin((0.011111111111111112 * (angle_m * pi))) * (b - a));
	else
		tmp = (((angle_m * 0.011111111111111112) * (b + a)) * (b - a)) * pi;
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[b, 2.2e+155], N[(N[(b + a), $MachinePrecision] * N[(N[Sin[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if b < 2.2000000000000002e155

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   4. Taylor expanded in angle around inf

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

   6. Applied rewrites 67.6%

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

   if 2.2000000000000002e155 < b

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 62.2%

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

Alternative 8: 65.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;angle\_m \leq 9.8 \cdot 10^{+96}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(angle\_m \cdot 0.011111111111111112\right) \cdot \pi\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\\ \end{array} \end{array} \]

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 9.8e+96)
    (* (+ b a) (* (- b a) (* 0.011111111111111112 (* angle_m PI))))
    (* (sin (* (* angle_m 0.011111111111111112) 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) {
	double tmp;
	if (angle_m <= 9.8e+96) {
		tmp = (b + a) * ((b - a) * (0.011111111111111112 * (angle_m * ((double) M_PI))));
	} else {
		tmp = sin(((angle_m * 0.011111111111111112) * ((double) M_PI))) * ((a + b) * (b - a));
	}
	return angle_s * tmp;
}

Java

angle\_m = Math.abs(angle);
angle\_s = Math.copySign(1.0, angle);
public static double code(double angle_s, double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 9.8e+96) {
		tmp = (b + a) * ((b - a) * (0.011111111111111112 * (angle_m * Math.PI)));
	} else {
		tmp = Math.sin(((angle_m * 0.011111111111111112) * Math.PI)) * ((a + b) * (b - a));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if angle_m <= 9.8e+96:
		tmp = (b + a) * ((b - a) * (0.011111111111111112 * (angle_m * math.pi)))
	else:
		tmp = math.sin(((angle_m * 0.011111111111111112) * math.pi)) * ((a + b) * (b - a))
	return angle_s * tmp

Julia

angle\_m = abs(angle)
angle\_s = copysign(1.0, angle)
function code(angle_s, a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 9.8e+96)
		tmp = Float64(Float64(b + a) * Float64(Float64(b - a) * Float64(0.011111111111111112 * Float64(angle_m * pi))));
	else
		tmp = Float64(sin(Float64(Float64(angle_m * 0.011111111111111112) * pi)) * Float64(Float64(a + b) * Float64(b - a)));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 9.8e+96)
		tmp = (b + a) * ((b - a) * (0.011111111111111112 * (angle_m * pi)));
	else
		tmp = sin(((angle_m * 0.011111111111111112) * pi)) * ((a + b) * (b - a));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[angle$95$m, 9.8e+96], N[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[(angle$95$m * 0.011111111111111112), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if angle < 9.7999999999999993e96

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   4. Taylor expanded in angle around 0

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

   6. Applied rewrites 62.3%

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

   if 9.7999999999999993e96 < angle

   1. Initial program 54.1%

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

   3. Applied rewrites 67.7%

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

   4. Taylor expanded in angle around inf

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

   6. Applied rewrites 57.7%

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

   8. Applied rewrites 57.7%

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

Alternative 9: 62.3% accurate, 6.6× speedup

Math

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

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 (* (+ b a) (* (- b a) (* 0.011111111111111112 (* angle_m PI))))))

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.1%

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

3. Applied rewrites 67.7%

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

4. Taylor expanded in angle around 0

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

6. Applied rewrites 62.3%

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

Alternative 10: 62.3% accurate, 6.6× speedup

Math

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

FPCore

angle\_m = (fabs.f64 angle)
angle\_s = (copysign.f64 #s(literal 1 binary64) angle)
(FPCore (angle_s a b angle_m)
 :precision binary64
 (* angle_s (* (* (* (* 0.011111111111111112 PI) angle_m) (- b a)) (+ 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 * ((double) M_PI)) * angle_m) * (b - a)) * (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 * Math.PI) * angle_m) * (b - a)) * (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 * math.pi) * angle_m) * (b - a)) * (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(Float64(Float64(Float64(0.011111111111111112 * pi) * angle_m) * Float64(b - a)) * 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 * pi) * angle_m) * (b - a)) * (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[(N[(N[(N[(0.011111111111111112 * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 50.6%

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

6. Applied rewrites 62.3%

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

Alternative 11: 54.2% accurate, 6.6× speedup

Math

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

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 (* PI (* (* (* (+ b a) (- b a)) 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 * (((double) M_PI) * ((((b + a) * (b - a)) * 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 * (Math.PI * ((((b + a) * (b - a)) * 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 * (math.pi * ((((b + a) * (b - a)) * 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(pi * Float64(Float64(Float64(Float64(b + a) * Float64(b - a)) * 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 * (pi * ((((b + a) * (b - a)) * 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[(Pi * N[(N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 50.6%

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

6. Applied rewrites 54.2%

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

Alternative 12: 54.2% accurate, 6.6× speedup

Math

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

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 (* 0.011111111111111112 (* (* (+ b a) (- b a)) (* 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 * (0.011111111111111112 * (((b + a) * (b - a)) * (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 * (0.011111111111111112 * (((b + a) * (b - a)) * (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 * (0.011111111111111112 * (((b + a) * (b - a)) * (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(0.011111111111111112 * Float64(Float64(Float64(b + a) * Float64(b - a)) * 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 * (0.011111111111111112 * (((b + a) * (b - a)) * (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[(0.011111111111111112 * N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 50.6%

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

6. Applied rewrites 54.2%

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

Alternative 13: 54.2% accurate, 6.6× speedup

Math

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

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) (* (+ b a) 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) * ((b + a) * ((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) * ((b + a) * 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) * ((b + a) * 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(b + a) * 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) * ((b + a) * 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[(b + a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 50.6%

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

6. Applied rewrites 54.2%

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

Alternative 14: 53.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} angle\_m = \left|angle\right| \\ angle\_s = \mathsf{copysign}\left(1, angle\right) \\ angle\_s \cdot \begin{array}{l} \mathbf{if}\;2 \cdot \left({b}^{2} - {a}^{2}\right) \leq 10^{-256}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\right) \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle\_m \cdot \left(\left(b - a\right) \cdot \left(b \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

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 (<= (* 2.0 (- (pow b 2.0) (pow a 2.0))) 1e-256)
    (* 0.011111111111111112 (* angle_m (* (- b a) (* a PI))))
    (* 0.011111111111111112 (* angle_m (* (- b 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) {
	double tmp;
	if ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) <= 1e-256) {
		tmp = 0.011111111111111112 * (angle_m * ((b - a) * (a * ((double) M_PI))));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a) * (b * ((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 ((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) <= 1e-256) {
		tmp = 0.011111111111111112 * (angle_m * ((b - a) * (a * Math.PI)));
	} else {
		tmp = 0.011111111111111112 * (angle_m * ((b - a) * (b * Math.PI)));
	}
	return angle_s * tmp;
}

Python

angle\_m = math.fabs(angle)
angle\_s = math.copysign(1.0, angle)
def code(angle_s, a, b, angle_m):
	tmp = 0
	if (2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) <= 1e-256:
		tmp = 0.011111111111111112 * (angle_m * ((b - a) * (a * math.pi)))
	else:
		tmp = 0.011111111111111112 * (angle_m * ((b - a) * (b * 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(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) <= 1e-256)
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a) * Float64(a * pi))));
	else
		tmp = Float64(0.011111111111111112 * Float64(angle_m * Float64(Float64(b - a) * Float64(b * pi))));
	end
	return Float64(angle_s * tmp)
end

MATLAB

angle\_m = abs(angle);
angle\_s = sign(angle) * abs(1.0);
function tmp_2 = code(angle_s, a, b, angle_m)
	tmp = 0.0;
	if ((2.0 * ((b ^ 2.0) - (a ^ 2.0))) <= 1e-256)
		tmp = 0.011111111111111112 * (angle_m * ((b - a) * (a * pi)));
	else
		tmp = 0.011111111111111112 * (angle_m * ((b - a) * (b * pi)));
	end
	tmp_2 = angle_s * tmp;
end

Wolfram

angle\_m = N[Abs[angle], $MachinePrecision]
angle\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[angle]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[angle$95$s_, a_, b_, angle$95$m_] := N[(angle$95$s * If[LessEqual[N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-256], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a), $MachinePrecision] * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.011111111111111112 * N[(angle$95$m * N[(N[(b - a), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) < 9.99999999999999977e-257

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 37.6%

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

   if 9.99999999999999977e-257 < (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64))))

   1. Initial program 54.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 54.2%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 37.9%

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

Alternative 15: 37.6% accurate, 7.8× speedup

Math

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

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 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 * ((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 * 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 * 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(a * 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 * 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[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 50.6%

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

6. Applied rewrites 54.2%

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

7. Taylor expanded in a around inf

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

9. Applied rewrites 37.6%

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

Reproduce

herbie shell --seed 2025153 
(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)))))