ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.7%

Time: 16.6s

Alternatives: 12

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "9.084064656945769583893630989888699868062e204")
  2. reg1 = (bf "6.178679066829250277149315602072379400236e-171")
  3. reg2 = (bf #e1.302330821487094256203443247225510774518e70)

• 36.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \mathsf{fma}\left(\cos \left(1 + 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right), \cos 1, \sin \left(\mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle\_m, 1\right)\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (*
    a
    (fma
     (cos (+ 1.0 (* 0.005555555555555556 (* angle_m PI))))
     (cos 1.0)
     (* (sin (fma PI (* 0.005555555555555556 angle_m) 1.0)) (sin 1.0))))
   2.0)
  (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * fma(cos((1.0 + (0.005555555555555556 * (angle_m * ((double) M_PI))))), cos(1.0), (sin(fma(((double) M_PI), (0.005555555555555556 * angle_m), 1.0)) * sin(1.0)))), 2.0) + pow((b * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * fma(cos(Float64(1.0 + Float64(0.005555555555555556 * Float64(angle_m * pi)))), cos(1.0), Float64(sin(fma(pi, Float64(0.005555555555555556 * angle_m), 1.0)) * sin(1.0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[(N[Cos[N[(1.0 + N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[1.0], $MachinePrecision] + N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. div-inv 80.2%

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

   3. expm1-log1p-u 66.0%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. expm1-undefine 66.0%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. cos-diff 66.0%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. div-inv 66.0%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. metadata-eval 66.0%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. associate-*r* 66.0%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot angle\right) \cdot 0.005555555555555556}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. *-commutative 66.0%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 66.0%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. fma-define 66.0%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{fma}\left(\cos \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right), \cos 1, \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. log1p-undefine 66.0%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \left(e^{\color{blue}{\log \left(1 + 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}}\right), \cos 1, \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. rem-exp-log 66.0%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \color{blue}{\left(1 + 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}, \cos 1, \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. *-commutative 66.0%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \left(1 + 0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right), \cos 1, \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. log1p-undefine 66.0%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos 1, \sin \left(e^{\color{blue}{\log \left(1 + 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. rem-exp-log 80.2%

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

   7. *-commutative 80.2%

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

8. Simplified 80.2%

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

   1. *-un-lft-identity 80.2%

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

   2. +-commutative 80.2%

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

   3. associate-*r* 80.3%

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

   4. *-commutative 80.3%

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

   5. *-commutative 80.3%

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

   6. fma-define 80.3%

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

   7. *-commutative 80.3%

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

10. Applied egg-rr 80.3%

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

    1. *-lft-identity 80.3%

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

12. Simplified 80.3%

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

13. Final simplification 80.3%

    \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos 1, \sin \left(\mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle, 1\right)\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} \]
14. Add Preprocessing

Alternative 2: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 1 + 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\cos t\_0, \cos 1, \sin 1 \cdot \sin t\_0\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.005555555555555556 (* angle_m PI)))))
   (+
    (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0)
    (pow (* a (fma (cos t_0) (cos 1.0) (* (sin 1.0) (sin t_0)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = 1.0 + (0.005555555555555556 * (angle_m * ((double) M_PI)));
	return pow((b * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0) + pow((a * fma(cos(t_0), cos(1.0), (sin(1.0) * sin(t_0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(1.0 + Float64(0.005555555555555556 * Float64(angle_m * pi)))
	return Float64((Float64(b * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0) + (Float64(a * fma(cos(t_0), cos(1.0), Float64(sin(1.0) * sin(t_0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(1.0 + N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[1.0], $MachinePrecision] + N[(N[Sin[1.0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. div-inv 80.2%

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

   3. expm1-log1p-u 66.0%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. expm1-undefine 66.0%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. cos-diff 66.0%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. div-inv 66.0%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. metadata-eval 66.0%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. associate-*r* 66.0%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot angle\right) \cdot 0.005555555555555556}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. *-commutative 66.0%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{0.005555555555555556 \cdot \left(\pi \cdot angle\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 66.0%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. fma-define 66.0%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{fma}\left(\cos \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right), \cos 1, \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. log1p-undefine 66.0%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \left(e^{\color{blue}{\log \left(1 + 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}}\right), \cos 1, \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. rem-exp-log 66.0%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \color{blue}{\left(1 + 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}, \cos 1, \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. *-commutative 66.0%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \left(1 + 0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \pi\right)}\right), \cos 1, \sin \left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. log1p-undefine 66.0%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\cos \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos 1, \sin \left(e^{\color{blue}{\log \left(1 + 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. rem-exp-log 80.2%

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

   7. *-commutative 80.2%

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

8. Simplified 80.2%

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

9. Final simplification 80.2%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\cos \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos 1, \sin 1 \cdot \sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle\_m \cdot \pi}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (* PI (* 0.005555555555555556 angle_m)))) 2.0)
  (pow (* b (sin (/ (* angle_m PI) 180.0))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0) + pow((b * sin(((angle_m * ((double) M_PI)) / 180.0))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.cos((Math.PI * (0.005555555555555556 * angle_m)))), 2.0) + Math.pow((b * Math.sin(((angle_m * Math.PI) / 180.0))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.cos((math.pi * (0.005555555555555556 * angle_m)))), 2.0) + math.pow((b * math.sin(((angle_m * math.pi) / 180.0))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0) + (Float64(b * sin(Float64(Float64(angle_m * pi) / 180.0))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * cos((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + ((b * sin(((angle_m * pi) / 180.0))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   2. div-inv 80.2%

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

   3. associate-*r/ 80.3%

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

6. Applied egg-rr 80.3%

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

7. Final simplification 80.3%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (* PI (* 0.005555555555555556 angle_m)))) 2.0)
  (pow (* b (sin (/ PI (/ 180.0 angle_m)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0) + pow((b * sin((((double) M_PI) / (180.0 / angle_m)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.cos((Math.PI * (0.005555555555555556 * angle_m)))), 2.0) + Math.pow((b * Math.sin((Math.PI / (180.0 / angle_m)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.cos((math.pi * (0.005555555555555556 * angle_m)))), 2.0) + math.pow((b * math.sin((math.pi / (180.0 / angle_m)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0) + (Float64(b * sin(Float64(pi / Float64(180.0 / angle_m)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * cos((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + ((b * sin((pi / (180.0 / angle_m)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} \]

   2. div-inv 80.2%

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

   3. clear-num 80.2%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} \]

   4. un-div-inv 80.2%

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

6. Applied egg-rr 80.2%

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

7. Final simplification 80.2%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 5: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\\ {\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle_m))))
   (+ (pow (* b (sin t_0)) 2.0) (pow (* a (cos t_0)) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle_m);
	return pow((b * sin(t_0)), 2.0) + pow((a * cos(t_0)), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle_m);
	return Math.pow((b * Math.sin(t_0)), 2.0) + Math.pow((a * Math.cos(t_0)), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (0.005555555555555556 * angle_m)
	return math.pow((b * math.sin(t_0)), 2.0) + math.pow((a * math.cos(t_0)), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle_m))
	return Float64((Float64(b * sin(t_0)) ^ 2.0) + (Float64(a * cos(t_0)) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = pi * (0.005555555555555556 * angle_m);
	tmp = ((b * sin(t_0)) ^ 2.0) + ((a * cos(t_0)) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

5. Final simplification 80.2%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 6: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+144}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.6e+144)
   (pow (* a (cos (* angle_m (* 0.005555555555555556 PI)))) 2.0)
   (pow (* b (sin (* 0.005555555555555556 (* angle_m PI)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.6e+144) {
		tmp = pow((a * cos((angle_m * (0.005555555555555556 * ((double) M_PI))))), 2.0);
	} else {
		tmp = pow((b * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.6e+144) {
		tmp = Math.pow((a * Math.cos((angle_m * (0.005555555555555556 * Math.PI)))), 2.0);
	} else {
		tmp = Math.pow((b * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 5.6e+144:
		tmp = math.pow((a * math.cos((angle_m * (0.005555555555555556 * math.pi)))), 2.0)
	else:
		tmp = math.pow((b * math.sin((0.005555555555555556 * (angle_m * math.pi)))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5.6e+144)
		tmp = Float64(a * cos(Float64(angle_m * Float64(0.005555555555555556 * pi)))) ^ 2.0;
	else
		tmp = Float64(b * sin(Float64(0.005555555555555556 * Float64(angle_m * pi)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 5.6e+144)
		tmp = (a * cos((angle_m * (0.005555555555555556 * pi)))) ^ 2.0;
	else
		tmp = (b * sin((0.005555555555555556 * (angle_m * pi)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5.6e+144], N[Power[N[(a * N[Cos[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.60000000000000013e144

   1. Initial program 77.5%

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

   3. Simplified 77.5%

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

      1. metadata-eval 77.5%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      2. div-inv 77.5%

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

      3. unpow2 77.5%

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

      4. associate-*r* 77.5%

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

      5. div-inv 77.4%

         \[\leadsto \left(\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot a\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      6. metadata-eval 77.4%

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

      7. associate-*r* 74.2%

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

      8. *-commutative 74.2%

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

      9. div-inv 74.2%

         \[\leadsto \left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot a\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      10. metadata-eval 74.2%

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

      11. associate-*r* 77.5%

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

      12. *-commutative 77.5%

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

   6. Applied egg-rr 77.5%

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

   7. Taylor expanded in a around inf 65.0%

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

      1. unpow2 65.0%

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

      2. associate-*r* 65.0%

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

      3. *-commutative 65.0%

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

      4. unpow2 65.0%

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} \]

      5. swap-sqr 65.0%

         \[\leadsto \color{blue}{\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} \]

      6. unpow2 65.0%

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

      7. associate-*r* 65.1%

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

      8. *-commutative 65.1%

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

   9. Simplified 65.1%

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

   if 5.60000000000000013e144 < b

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around 0 71.4%

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

      1. *-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. unpow2 71.4%

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

      4. swap-sqr 78.2%

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

      5. unpow2 78.2%

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

      6. *-commutative 78.2%

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

   7. Simplified 78.2%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{+144}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{a}^{6}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.5e+153)
   (pow (* a (cos (* angle_m (* 0.005555555555555556 PI)))) 2.0)
   (cbrt (pow a 6.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.5e+153) {
		tmp = pow((a * cos((angle_m * (0.005555555555555556 * ((double) M_PI))))), 2.0);
	} else {
		tmp = cbrt(pow(a, 6.0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.5e+153) {
		tmp = Math.pow((a * Math.cos((angle_m * (0.005555555555555556 * Math.PI)))), 2.0);
	} else {
		tmp = Math.cbrt(Math.pow(a, 6.0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5.5e+153)
		tmp = Float64(a * cos(Float64(angle_m * Float64(0.005555555555555556 * pi)))) ^ 2.0;
	else
		tmp = cbrt((a ^ 6.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5.5e+153], N[Power[N[(a * N[Cos[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[Power[a, 6.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5000000000000003e153

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

      1. metadata-eval 77.7%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      2. div-inv 77.7%

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

      3. unpow2 77.7%

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

      4. associate-*r* 77.7%

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

      5. div-inv 77.6%

         \[\leadsto \left(\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right) \cdot a\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      6. metadata-eval 77.6%

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

      7. associate-*r* 74.4%

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

      8. *-commutative 74.4%

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

      9. div-inv 74.4%

         \[\leadsto \left(\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot a\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

      10. metadata-eval 74.4%

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

      11. associate-*r* 77.7%

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

      12. *-commutative 77.7%

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

   6. Applied egg-rr 77.7%

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

   7. Taylor expanded in a around inf 64.9%

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

      1. unpow2 64.9%

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

      2. associate-*r* 64.9%

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

      3. *-commutative 64.9%

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

      4. unpow2 64.9%

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} \]

      5. swap-sqr 64.9%

         \[\leadsto \color{blue}{\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \cdot \left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)} \]

      6. unpow2 64.9%

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

      7. associate-*r* 64.9%

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

      8. *-commutative 64.9%

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

   9. Simplified 64.9%

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

   if 5.5000000000000003e153 < b

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in angle around 0 43.1%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 43.1%

         \[\leadsto \color{blue}{a \cdot a} \]

   7. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{a \cdot a} \]
   8. Step-by-step derivation

      1. pow2 43.1%

         \[\leadsto \color{blue}{{a}^{2}} \]

      2. add-cbrt-cube 49.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left({a}^{2} \cdot {a}^{2}\right) \cdot {a}^{2}}} \]

      3. pow1/3 49.5%

         \[\leadsto \color{blue}{{\left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {a}^{2}\right)}^{0.3333333333333333}} \]

      4. pow2 49.5%

         \[\leadsto {\left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {a}^{2}\right) \cdot {a}^{2}\right)}^{0.3333333333333333} \]

      5. pow2 49.5%

         \[\leadsto {\left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot {a}^{2}\right)}^{0.3333333333333333} \]

      6. pow2 49.5%

         \[\leadsto {\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)}^{0.3333333333333333} \]

      7. pow3 49.5%

         \[\leadsto {\color{blue}{\left({\left(a \cdot a\right)}^{3}\right)}}^{0.3333333333333333} \]

      8. pow2 49.5%

         \[\leadsto {\left({\color{blue}{\left({a}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      9. pow-pow 49.5%

         \[\leadsto {\color{blue}{\left({a}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      10. metadata-eval 49.5%

          \[\leadsto {\left({a}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{{\left({a}^{6}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 49.5%

          \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]

   11. Simplified 49.5%

       \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{a}^{6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+153}:\\ \;\;\;\;{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{a}^{6}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.8e+153)
   (pow (* a (cos (* 0.005555555555555556 (* angle_m PI)))) 2.0)
   (cbrt (pow a 6.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.8e+153) {
		tmp = pow((a * cos((0.005555555555555556 * (angle_m * ((double) M_PI))))), 2.0);
	} else {
		tmp = cbrt(pow(a, 6.0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.8e+153) {
		tmp = Math.pow((a * Math.cos((0.005555555555555556 * (angle_m * Math.PI)))), 2.0);
	} else {
		tmp = Math.cbrt(Math.pow(a, 6.0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5.8e+153)
		tmp = Float64(a * cos(Float64(0.005555555555555556 * Float64(angle_m * pi)))) ^ 2.0;
	else
		tmp = cbrt((a ^ 6.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5.8e+153], N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[Power[a, 6.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.80000000000000004e153

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in a around inf 64.9%

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

      1. unpow2 64.9%

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

      2. *-commutative 64.9%

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

      3. unpow2 64.9%

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      4. swap-sqr 64.9%

         \[\leadsto \color{blue}{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      5. unpow2 64.9%

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

      6. *-commutative 64.9%

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

   7. Simplified 64.9%

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

   if 5.80000000000000004e153 < b

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in angle around 0 43.1%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 43.1%

         \[\leadsto \color{blue}{a \cdot a} \]

   7. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{a \cdot a} \]
   8. Step-by-step derivation

      1. pow2 43.1%

         \[\leadsto \color{blue}{{a}^{2}} \]

      2. add-cbrt-cube 49.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left({a}^{2} \cdot {a}^{2}\right) \cdot {a}^{2}}} \]

      3. pow1/3 49.5%

         \[\leadsto \color{blue}{{\left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {a}^{2}\right)}^{0.3333333333333333}} \]

      4. pow2 49.5%

         \[\leadsto {\left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {a}^{2}\right) \cdot {a}^{2}\right)}^{0.3333333333333333} \]

      5. pow2 49.5%

         \[\leadsto {\left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot {a}^{2}\right)}^{0.3333333333333333} \]

      6. pow2 49.5%

         \[\leadsto {\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)}^{0.3333333333333333} \]

      7. pow3 49.5%

         \[\leadsto {\color{blue}{\left({\left(a \cdot a\right)}^{3}\right)}}^{0.3333333333333333} \]

      8. pow2 49.5%

         \[\leadsto {\left({\color{blue}{\left({a}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      9. pow-pow 49.5%

         \[\leadsto {\color{blue}{\left({a}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      10. metadata-eval 49.5%

          \[\leadsto {\left({a}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{{\left({a}^{6}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 49.5%

          \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]

   11. Simplified 49.5%

       \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 79.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} + a \cdot a \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0) (* a a)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0) + (a * a);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((b * Math.sin((Math.PI * (0.005555555555555556 * angle_m)))), 2.0) + (a * a);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((b * math.sin((math.pi * (0.005555555555555556 * angle_m)))), 2.0) + (a * a)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0) + Float64(a * a))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + (a * a);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

   1. metadata-eval 80.2%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot \color{blue}{\frac{1}{180}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. div-inv 80.2%

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

   3. log1p-expm1-u 71.7%

      \[\leadsto {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. log1p-undefine 61.8%

      \[\leadsto {\color{blue}{\log \left(1 + \mathsf{expm1}\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. pow1 61.8%

      \[\leadsto {\log \left(1 + \mathsf{expm1}\left(\color{blue}{{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. pow1 61.8%

      \[\leadsto {\log \left(1 + \mathsf{expm1}\left(\color{blue}{a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. div-inv 61.8%

      \[\leadsto {\log \left(1 + \mathsf{expm1}\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. metadata-eval 61.8%

      \[\leadsto {\log \left(1 + \mathsf{expm1}\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. associate-*r* 61.8%

      \[\leadsto {\log \left(1 + \mathsf{expm1}\left(a \cdot \cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. *-commutative 61.8%

       \[\leadsto {\log \left(1 + \mathsf{expm1}\left(a \cdot \cos \color{blue}{\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 61.8%

   \[\leadsto {\color{blue}{\log \left(1 + \mathsf{expm1}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

7. Taylor expanded in angle around 0 61.8%

   \[\leadsto {\log \left(1 + \mathsf{expm1}\left(\color{blue}{a}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
8. Step-by-step derivation

   1. log1p-define 71.8%

      \[\leadsto {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(a\right)\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. log1p-expm1-u 80.2%

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

   3. pow2 80.2%

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

9. Applied egg-rr 80.2%

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

10. Final simplification 80.2%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + a \cdot a \]
11. Add Preprocessing

Alternative 10: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.1 \cdot 10^{+158}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left({a}^{6}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.1e+158) (* a a) (pow (pow a 6.0) 0.3333333333333333)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.1e+158) {
		tmp = a * a;
	} else {
		tmp = pow(pow(a, 6.0), 0.3333333333333333);
	}
	return tmp;
}

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    real(8) :: tmp
    if (b <= 5.1d+158) then
        tmp = a * a
    else
        tmp = (a ** 6.0d0) ** 0.3333333333333333d0
    end if
    code = tmp
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.1e+158) {
		tmp = a * a;
	} else {
		tmp = Math.pow(Math.pow(a, 6.0), 0.3333333333333333);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 5.1e+158:
		tmp = a * a
	else:
		tmp = math.pow(math.pow(a, 6.0), 0.3333333333333333)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5.1e+158)
		tmp = Float64(a * a);
	else
		tmp = (a ^ 6.0) ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 5.1e+158)
		tmp = a * a;
	else
		tmp = (a ^ 6.0) ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5.1e+158], N[(a * a), $MachinePrecision], N[Power[N[Power[a, 6.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.09999999999999987e158

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in angle around 0 64.8%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 64.8%

         \[\leadsto \color{blue}{a \cdot a} \]

   7. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 5.09999999999999987e158 < b

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in angle around 0 43.1%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 43.1%

         \[\leadsto \color{blue}{a \cdot a} \]

   7. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{a \cdot a} \]
   8. Step-by-step derivation

      1. pow2 43.1%

         \[\leadsto \color{blue}{{a}^{2}} \]

      2. add-cbrt-cube 49.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left({a}^{2} \cdot {a}^{2}\right) \cdot {a}^{2}}} \]

      3. pow1/3 49.5%

         \[\leadsto \color{blue}{{\left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {a}^{2}\right)}^{0.3333333333333333}} \]

      4. pow2 49.5%

         \[\leadsto {\left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {a}^{2}\right) \cdot {a}^{2}\right)}^{0.3333333333333333} \]

      5. pow2 49.5%

         \[\leadsto {\left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot {a}^{2}\right)}^{0.3333333333333333} \]

      6. pow2 49.5%

         \[\leadsto {\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)}^{0.3333333333333333} \]

      7. pow3 49.5%

         \[\leadsto {\color{blue}{\left({\left(a \cdot a\right)}^{3}\right)}}^{0.3333333333333333} \]

      8. pow2 49.5%

         \[\leadsto {\left({\color{blue}{\left({a}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      9. pow-pow 49.5%

         \[\leadsto {\color{blue}{\left({a}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      10. metadata-eval 49.5%

          \[\leadsto {\left({a}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{{\left({a}^{6}\right)}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 6 \cdot 10^{+153}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{a}^{6}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 6e+153) (* a a) (cbrt (pow a 6.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 6e+153) {
		tmp = a * a;
	} else {
		tmp = cbrt(pow(a, 6.0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 6e+153) {
		tmp = a * a;
	} else {
		tmp = Math.cbrt(Math.pow(a, 6.0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 6e+153)
		tmp = Float64(a * a);
	else
		tmp = cbrt((a ^ 6.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 6e+153], N[(a * a), $MachinePrecision], N[Power[N[Power[a, 6.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.00000000000000037e153

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in angle around 0 64.8%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 64.8%

         \[\leadsto \color{blue}{a \cdot a} \]

   7. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 6.00000000000000037e153 < b

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in angle around 0 43.1%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 43.1%

         \[\leadsto \color{blue}{a \cdot a} \]

   7. Applied egg-rr 43.1%

      \[\leadsto \color{blue}{a \cdot a} \]
   8. Step-by-step derivation

      1. pow2 43.1%

         \[\leadsto \color{blue}{{a}^{2}} \]

      2. add-cbrt-cube 49.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left({a}^{2} \cdot {a}^{2}\right) \cdot {a}^{2}}} \]

      3. pow1/3 49.5%

         \[\leadsto \color{blue}{{\left(\left({a}^{2} \cdot {a}^{2}\right) \cdot {a}^{2}\right)}^{0.3333333333333333}} \]

      4. pow2 49.5%

         \[\leadsto {\left(\left(\color{blue}{\left(a \cdot a\right)} \cdot {a}^{2}\right) \cdot {a}^{2}\right)}^{0.3333333333333333} \]

      5. pow2 49.5%

         \[\leadsto {\left(\left(\left(a \cdot a\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \cdot {a}^{2}\right)}^{0.3333333333333333} \]

      6. pow2 49.5%

         \[\leadsto {\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)}^{0.3333333333333333} \]

      7. pow3 49.5%

         \[\leadsto {\color{blue}{\left({\left(a \cdot a\right)}^{3}\right)}}^{0.3333333333333333} \]

      8. pow2 49.5%

         \[\leadsto {\left({\color{blue}{\left({a}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      9. pow-pow 49.5%

         \[\leadsto {\color{blue}{\left({a}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      10. metadata-eval 49.5%

          \[\leadsto {\left({a}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 49.5%

      \[\leadsto \color{blue}{{\left({a}^{6}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 49.5%

          \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]

   11. Simplified 49.5%

       \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 56.7% accurate, 139.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* a a))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return a * a;
}

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = a * a
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return a * a;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return a * a

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(a * a)
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = a * a;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Simplified 80.2%

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

5. Taylor expanded in angle around 0 62.4%

   \[\leadsto \color{blue}{{a}^{2}} \]
6. Step-by-step derivation

   1. unpow2 62.4%

      \[\leadsto \color{blue}{a \cdot a} \]

7. Applied egg-rr 62.4%

   \[\leadsto \color{blue}{a \cdot a} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))