ab-angle->ABCF A

Percentage Accurate: 79.3% → 79.4%

Time: 4.4s

Alternatives: 11

Speedup: 1.0×

• 35.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\frac{angle}{180}, \pi, \pi \cdot 0.5\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* angle (* 0.005555555555555556 PI)))) 2.0)
  (pow (* b (sin (fma (/ angle 180.0) PI (* PI 0.5)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((angle * (0.005555555555555556 * ((double) M_PI))))), 2.0) + pow((b * sin(fma((angle / 180.0), ((double) M_PI), (((double) M_PI) * 0.5)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(angle * Float64(0.005555555555555556 * pi)))) ^ 2.0) + (Float64(b * sin(fma(Float64(angle / 180.0), pi, Float64(pi * 0.5)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

3. Applied rewrites 79.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

5. Applied rewrites 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. mult-flip N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-sin.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. mult-flip N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 79.4

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

7. Applied rewrites 79.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. mult-flip N/A

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

   5. lower-/.f64 79.4

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

9. Applied rewrites 79.4%

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

Alternative 2: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{fma}\left(0.005555555555555556, angle, 0.5\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* angle (* 0.005555555555555556 PI)))) 2.0)
  (pow (* b (sin (* PI (fma 0.005555555555555556 angle 0.5)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((angle * (0.005555555555555556 * ((double) M_PI))))), 2.0) + pow((b * sin((((double) M_PI) * fma(0.005555555555555556, angle, 0.5)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(angle * Float64(0.005555555555555556 * pi)))) ^ 2.0) + (Float64(b * sin(Float64(pi * fma(0.005555555555555556, angle, 0.5)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(0.005555555555555556 * angle + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

3. Applied rewrites 79.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

5. Applied rewrites 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. mult-flip N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-sin.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. mult-flip N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 79.4

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

7. Applied rewrites 79.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 79.4

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

9. Applied rewrites 79.4%

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

Alternative 3: 79.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 PI))))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

3. Applied rewrites 79.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

5. Applied rewrites 79.4%

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

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2} + {\left(\cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b\right)}^{2} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

3. Applied rewrites 79.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

5. Applied rewrites 79.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. mult-flip N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-sin.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. mult-flip N/A

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. mult-flip N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 79.4

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

7. Applied rewrites 79.4%

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

9. Applied rewrites 79.3%

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

Alternative 5: 79.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (+ (pow (* (sin t_0) a) 2.0) (pow (* (cos t_0) b) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 79.3

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 79.3

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

   7. lift-/.f64 N/A

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

   8. div-flip N/A

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

   9. associate-/r/ N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 79.4

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

3. Applied rewrites 79.4%

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

Alternative 6: 79.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

3. Applied rewrites 79.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval 79.4

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

5. Applied rewrites 79.4%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 79.4%

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

Alternative 7: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot angle, angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.26e+32)
   (fma
    (*
     (*
      (* PI PI)
      (fma (* b b) -3.08641975308642e-5 (* (* a a) 3.08641975308642e-5)))
     angle)
    angle
    (* b b))
   (* (pow b 2.0) (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.26e+32) {
		tmp = fma((((((double) M_PI) * ((double) M_PI)) * fma((b * b), -3.08641975308642e-5, ((a * a) * 3.08641975308642e-5))) * angle), angle, (b * b));
	} else {
		tmp = pow(b, 2.0) * pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.26e+32)
		tmp = fma(Float64(Float64(Float64(pi * pi) * fma(Float64(b * b), -3.08641975308642e-5, Float64(Float64(a * a) * 3.08641975308642e-5))) * angle), angle, Float64(b * b));
	else
		tmp = Float64((b ^ 2.0) * (cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.26e+32], N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5 + N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * angle + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.26e32

   1. Initial program 79.3%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({angle}^{2}, \color{blue}{\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, {b}^{2}\right) \]

   4. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, {b}^{2} \cdot {\pi}^{2}, 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right), {b}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto {angle}^{2} \cdot \mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) + \color{blue}{{b}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{b}}^{2} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot {angle}^{2} + {b}^{2} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(angle \cdot angle\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot angle\right) \cdot angle + {\color{blue}{b}}^{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot angle, \color{blue}{angle}, {b}^{2}\right) \]

   6. Applied rewrites 43.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot angle, \color{blue}{angle}, b \cdot b\right) \]

   if 1.26e32 < b

   1. Initial program 79.3%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 57.0

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

   4. Applied rewrites 57.0%

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

Alternative 8: 57.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot angle, angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.26e+32)
   (fma
    (*
     (*
      (* PI PI)
      (fma (* b b) -3.08641975308642e-5 (* (* a a) 3.08641975308642e-5)))
     angle)
    angle
    (* b b))
   (* b b)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.26e+32) {
		tmp = fma((((((double) M_PI) * ((double) M_PI)) * fma((b * b), -3.08641975308642e-5, ((a * a) * 3.08641975308642e-5))) * angle), angle, (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.26e+32)
		tmp = fma(Float64(Float64(Float64(pi * pi) * fma(Float64(b * b), -3.08641975308642e-5, Float64(Float64(a * a) * 3.08641975308642e-5))) * angle), angle, Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.26e+32], N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5 + N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision] * angle + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.26e32

   1. Initial program 79.3%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({angle}^{2}, \color{blue}{\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, {b}^{2}\right) \]

   4. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, {b}^{2} \cdot {\pi}^{2}, 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right), {b}^{2}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto {angle}^{2} \cdot \mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) + \color{blue}{{b}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot {angle}^{2} + {\color{blue}{b}}^{2} \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot {angle}^{2} + {b}^{2} \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(angle \cdot angle\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot angle\right) \cdot angle + {\color{blue}{b}}^{2} \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{32400}, {b}^{2} \cdot {\pi}^{2}, \frac{1}{32400} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) \cdot angle, \color{blue}{angle}, {b}^{2}\right) \]

   6. Applied rewrites 43.1%

      \[\leadsto \mathsf{fma}\left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot angle, \color{blue}{angle}, b \cdot b\right) \]

   if 1.26e32 < b

   1. Initial program 79.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.2

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

   6. Applied rewrites 57.2%

      \[\leadsto b \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.26 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.26e+32)
   (fma
    (* angle angle)
    (*
     (* PI PI)
     (fma (* b b) -3.08641975308642e-5 (* (* a a) 3.08641975308642e-5)))
    (* b b))
   (* b b)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.26e+32) {
		tmp = fma((angle * angle), ((((double) M_PI) * ((double) M_PI)) * fma((b * b), -3.08641975308642e-5, ((a * a) * 3.08641975308642e-5))), (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.26e+32)
		tmp = fma(Float64(angle * angle), Float64(Float64(pi * pi) * fma(Float64(b * b), -3.08641975308642e-5, Float64(Float64(a * a) * 3.08641975308642e-5))), Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.26e+32], N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * -3.08641975308642e-5 + N[(N[(a * a), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.26e32

   1. Initial program 79.3%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({angle}^{2}, \color{blue}{\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, {b}^{2}\right) \]

   4. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, {b}^{2} \cdot {\pi}^{2}, 3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right), {b}^{2}\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 40.7%

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \color{blue}{\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}, b \cdot b\right) \]

   if 1.26e32 < b

   1. Initial program 79.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.2

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

   6. Applied rewrites 57.2%

      \[\leadsto b \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;{\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \leq 10^{+308}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(b \cdot b\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (if (<= (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0)) 1e+308)
     (* b b)
     (sqrt (* (* (* b b) b) b)))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double tmp;
	if ((pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0)) <= 1e+308) {
		tmp = b * b;
	} else {
		tmp = sqrt((((b * b) * b) * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	double tmp;
	if ((Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0)) <= 1e+308) {
		tmp = b * b;
	} else {
		tmp = Math.sqrt((((b * b) * b) * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	tmp = 0
	if (math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)) <= 1e+308:
		tmp = b * b
	else:
		tmp = math.sqrt((((b * b) * b) * b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	tmp = 0.0
	if (Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0)) <= 1e+308)
		tmp = Float64(b * b);
	else
		tmp = sqrt(Float64(Float64(Float64(b * b) * b) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = 0.0;
	if ((((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0)) <= 1e+308)
		tmp = b * b;
	else
		tmp = sqrt((((b * b) * b) * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1e+308], N[(b * b), $MachinePrecision], N[Sqrt[N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) < 1e308

   1. Initial program 79.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.2

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

   6. Applied rewrites 57.2%

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

   if 1e308 < (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)))

   1. Initial program 79.3%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.2

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

   4. Applied rewrites 57.2%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

         \[\leadsto e^{\log b \cdot 2} \]

      3. lower-exp.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      5. lower-log.f64 27.8

         \[\leadsto e^{\log b \cdot 2} \]

   6. Applied rewrites 27.8%

      \[\leadsto e^{\log b \cdot 2} \]
   7. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      2. exp-fabs N/A

         \[\leadsto \left|e^{\log b \cdot 2}\right| \]

      3. lift-*.f64 N/A

         \[\leadsto \left|e^{\log b \cdot 2}\right| \]

      4. lift-log.f64 N/A

         \[\leadsto \left|e^{\log b \cdot 2}\right| \]

      5. pow-to-exp N/A

         \[\leadsto \left|{b}^{2}\right| \]

      6. lift-pow.f64 N/A

         \[\leadsto \left|{b}^{2}\right| \]

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

         \[\leadsto \sqrt{{b}^{2} \cdot {b}^{2}} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{{b}^{2} \cdot {b}^{2}} \]

      9. lift-pow.f64 N/A

         \[\leadsto \sqrt{{b}^{2} \cdot {b}^{2}} \]

      10. unpow2 N/A

          \[\leadsto \sqrt{{b}^{2} \cdot \left(b \cdot b\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sqrt{\left({b}^{2} \cdot b\right) \cdot b} \]

      12. lift-pow.f64 N/A

          \[\leadsto \sqrt{\left({b}^{2} \cdot b\right) \cdot b} \]

      13. pow-plus-rev N/A

          \[\leadsto \sqrt{{b}^{\left(2 + 1\right)} \cdot b} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{{b}^{3} \cdot b} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{{b}^{3} \cdot b} \]

      16. metadata-eval N/A

          \[\leadsto \sqrt{{b}^{\left(2 + 1\right)} \cdot b} \]

      17. pow-plus-rev N/A

          \[\leadsto \sqrt{\left({b}^{2} \cdot b\right) \cdot b} \]

      18. lift-pow.f64 N/A

          \[\leadsto \sqrt{\left({b}^{2} \cdot b\right) \cdot b} \]

      19. lower-*.f64 49.2

          \[\leadsto \sqrt{\left({b}^{2} \cdot b\right) \cdot b} \]

      20. lift-pow.f64 N/A

          \[\leadsto \sqrt{\left({b}^{2} \cdot b\right) \cdot b} \]

      21. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \]

      22. lower-*.f64 49.2

          \[\leadsto \sqrt{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \]

   8. Applied rewrites 49.2%

      \[\leadsto \sqrt{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 53.2% accurate, 29.7× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

public static double code(double a, double b, double angle) {
	return b * b;
}

Python

def code(a, b, angle):
	return b * b

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 57.2

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

4. Applied rewrites 57.2%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 57.2

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

6. Applied rewrites 57.2%

   \[\leadsto b \cdot \color{blue}{b} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))