ab-angle->ABCF C

Percentage Accurate: 80.6% → 80.6%

Time: 7.3s

Alternatives: 12

Speedup: 1.0×

• 36.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 80.6% 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: 80.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. lift-cos.f64 N/A

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

   2. cos-fabs-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. associate-*l* N/A

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

   10. fabs-mul N/A

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

   11. fabs-mul N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

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

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

   16. sqrt-prod N/A

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

   17. lift-PI.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. add-sqr-sqrt N/A

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

   20. lift-PI.f64 N/A

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

3. Applied rewrites 80.6%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-log-exp N/A

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

   4. log-pow-rev N/A

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

   5. lower-log.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. pow-exp N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-exp.f64 80.6

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 80.6

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

5. Applied rewrites 80.6%

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

Alternative 2: 80.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. lift-cos.f64 N/A

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

   2. cos-fabs-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. associate-*l* N/A

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

   10. fabs-mul N/A

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

   11. fabs-mul N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

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

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

   16. sqrt-prod N/A

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

   17. lift-PI.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. add-sqr-sqrt N/A

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

   20. lift-PI.f64 N/A

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

3. Applied rewrites 80.6%

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

Alternative 3: 80.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. distribute-neg-frac2 N/A

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

   9. associate-/l* N/A

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

   10. distribute-neg-frac2 N/A

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

   11. distribute-neg-frac N/A

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

   12. lift-PI.f64 N/A

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

   13. mult-flip N/A

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

   14. distribute-lft-out N/A

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

   15. lower-*.f64 N/A

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

   16. distribute-neg-frac N/A

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

   17. mult-flip N/A

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

   18. *-commutative N/A

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

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

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

   20. distribute-frac-neg2 N/A

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

   21. lower-fma.f64 N/A

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

3. Applied rewrites 80.6%

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

Alternative 4: 80.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (/ (* angle PI) 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 = (angle * ((double) M_PI)) / 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 = (angle * Math.PI) / 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 = (angle * math.pi) / 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(Float64(angle * pi) / 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 = (angle * pi) / 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[(N[(angle * Pi), $MachinePrecision] / 180.0), $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]]

Derivation

1. Initial program 80.6%

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

   1. remove-double-neg N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. distribute-neg-frac N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

3. Applied rewrites 80.5%

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

   1. remove-double-neg N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. distribute-neg-frac N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 80.5%

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

Alternative 5: 80.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI)))
   (+ (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 = (0.005555555555555556 * angle) * ((double) M_PI);
	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 = (0.005555555555555556 * angle) * Math.PI;
	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 = (0.005555555555555556 * angle) * math.pi
	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(Float64(0.005555555555555556 * angle) * pi)
	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 = (0.005555555555555556 * angle) * pi;
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $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]]

Derivation

1. Initial program 80.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 80.6

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.6

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

3. Applied rewrites 80.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 80.6

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.6

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

5. Applied rewrites 80.6%

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

Alternative 6: 80.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. pow-neg N/A

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

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

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

   6. lower-special-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. mult-flip N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval 80.6

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

3. Applied rewrites 80.6%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\frac{1}{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{-2}}} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. mult-flip N/A

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

   12. lift-/.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. add-cube-cbrt N/A

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

   15. lift-PI.f64 N/A

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

   16. lift-cbrt.f64 N/A

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

   17. associate-*r* N/A

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

   18. lift-cbrt.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. add-cube-cbrt N/A

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

   21. lift-PI.f64 N/A

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

   22. lift-cbrt.f64 N/A

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

   23. cbrt-prod N/A

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

5. Applied rewrites 80.5%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left(\left(\left({\pi}^{0.6666666666666666} \cdot angle\right) \cdot 0.005555555555555556\right) \cdot {\pi}^{0.2222222222222222}\right) \cdot {\pi}^{0.1111111111111111}\right)}\right)}^{2} + \frac{1}{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{-2}} \]

6. Taylor expanded in angle around 0

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

8. Applied rewrites 80.5%

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

Alternative 7: 80.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 80.6

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.6

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

3. Applied rewrites 80.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 80.6

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.6

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

5. Applied rewrites 80.6%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 80.5%

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

Alternative 8: 68.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-143}:\\ \;\;\;\;{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.3e-143)
   (* (pow b 2.0) (pow (sin (* 0.005555555555555556 (* angle PI))) 2.0))
   (fma
    1.0
    (* a a)
    (* (* (- 0.5 (* 0.5 (cos (* (* PI angle) 0.011111111111111112)))) b) b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.3e-143) {
		tmp = pow(b, 2.0) * pow(sin((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0);
	} else {
		tmp = fma(1.0, (a * a), (((0.5 - (0.5 * cos(((((double) M_PI) * angle) * 0.011111111111111112)))) * b) * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.3e-143)
		tmp = Float64((b ^ 2.0) * (sin(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0));
	else
		tmp = fma(1.0, Float64(a * a), Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(Float64(pi * angle) * 0.011111111111111112)))) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.3e-143], N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(a * a), $MachinePrecision] + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.29999999999999994e-143

   1. Initial program 80.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 34.5

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

   4. Applied rewrites 34.5%

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

   if 1.29999999999999994e-143 < a

   1. Initial program 80.6%

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

      1. lift-cos.f64 N/A

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

      2. cos-fabs-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*l* N/A

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

      10. fabs-mul N/A

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

      11. fabs-mul N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

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

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

      16. sqrt-prod N/A

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

      17. lift-PI.f64 N/A

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

      18. lift-PI.f64 N/A

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

      19. add-sqr-sqrt N/A

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

      20. lift-PI.f64 N/A

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

   3. Applied rewrites 80.6%

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

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right)} \]

   6. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, a \cdot a, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 68.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  1.0
  (* a a)
  (* (* (- 0.5 (* 0.5 (cos (* (* PI angle) 0.011111111111111112)))) b) b)))

C

double code(double a, double b, double angle) {
	return fma(1.0, (a * a), (((0.5 - (0.5 * cos(((((double) M_PI) * angle) * 0.011111111111111112)))) * b) * b));
}

Julia

function code(a, b, angle)
	return fma(1.0, Float64(a * a), Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(Float64(pi * angle) * 0.011111111111111112)))) * b) * b))
end

Wolfram

code[a_, b_, angle_] := N[(1.0 * N[(a * a), $MachinePrecision] + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.6%

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

   1. lift-cos.f64 N/A

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

   2. cos-fabs-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. associate-*l* N/A

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

   10. fabs-mul N/A

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

   11. fabs-mul N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

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

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

   16. sqrt-prod N/A

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

   17. lift-PI.f64 N/A

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

   18. lift-PI.f64 N/A

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

   19. add-sqr-sqrt N/A

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

   20. lift-PI.f64 N/A

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

3. Applied rewrites 80.6%

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

5. Applied rewrites 68.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right)} \]

6. Taylor expanded in angle around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, a \cdot a, \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b\right) \]
7. Step-by-step derivation

8. Applied rewrites 68.4%

   \[\leadsto \mathsf{fma}\left(\color{blue}{1}, a \cdot a, \left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right) \]
9. Add Preprocessing

Alternative 10: 58.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;{\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \leq 10^{+305}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{{a}^{8}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)) 1e+305)
     (* a a)
     (sqrt (sqrt (pow a 8.0))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0)) <= 1e+305) {
		tmp = a * a;
	} else {
		tmp = sqrt(sqrt(pow(a, 8.0)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0)) <= 1e+305) {
		tmp = a * a;
	} else {
		tmp = Math.sqrt(Math.sqrt(Math.pow(a, 8.0)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)) <= 1e+305:
		tmp = a * a
	else:
		tmp = math.sqrt(math.sqrt(math.pow(a, 8.0)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0)) <= 1e+305)
		tmp = Float64(a * a);
	else
		tmp = sqrt(sqrt((a ^ 8.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0)) <= 1e+305)
		tmp = a * a;
	else
		tmp = sqrt(sqrt((a ^ 8.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 1e+305], N[(a * a), $MachinePrecision], N[Sqrt[N[Sqrt[N[Power[a, 8.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 9.9999999999999994e304

   1. Initial program 80.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 9.9999999999999994e304 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 80.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 49.4

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

   8. Applied rewrites 49.4%

      \[\leadsto \sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-prod-up N/A

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

      9. lift-*.f64 N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-up N/A

          \[\leadsto \sqrt{\sqrt{{a}^{\left(\left(2 + 2\right) + \left(2 + 2\right)\right)}}} \]

      12. lower-pow.f64 N/A

          \[\leadsto \sqrt{\sqrt{{a}^{\left(\left(2 + 2\right) + \left(2 + 2\right)\right)}}} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\sqrt{{a}^{\left(4 + \left(2 + 2\right)\right)}}} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\sqrt{{a}^{\left(4 + 4\right)}}} \]

      15. metadata-eval 45.0

          \[\leadsto \sqrt{\sqrt{{a}^{8}}} \]

   10. Applied rewrites 45.0%

       \[\leadsto \sqrt{\sqrt{{a}^{8}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;{\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \leq 10^{+305}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0)) 1e+305)
     (* a a)
     (sqrt (* (* a a) (* a a))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0)) <= 1e+305) {
		tmp = a * a;
	} else {
		tmp = sqrt(((a * a) * (a * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0)) <= 1e+305) {
		tmp = a * a;
	} else {
		tmp = Math.sqrt(((a * a) * (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (math.pow((a * math.cos(t_0)), 2.0) + math.pow((b * math.sin(t_0)), 2.0)) <= 1e+305:
		tmp = a * a
	else:
		tmp = math.sqrt(((a * a) * (a * a)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0)) <= 1e+305)
		tmp = Float64(a * a);
	else
		tmp = sqrt(Float64(Float64(a * a) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0)) <= 1e+305)
		tmp = a * a;
	else
		tmp = sqrt(((a * a) * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[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], 1e+305], N[(a * a), $MachinePrecision], N[Sqrt[N[(N[(a * a), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 9.9999999999999994e304

   1. Initial program 80.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 9.9999999999999994e304 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 80.6%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 49.4

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

   8. Applied rewrites 49.4%

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

Alternative 12: 52.8% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

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 = a * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 57.1

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

4. Applied rewrites 57.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 57.1

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

6. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025151 
(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)))