ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.8%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{0.005555555555555556}{angle\_m}} \cdot angle\_m\\ {\left(a \cdot \sin \left(\mathsf{fma}\left(t\_0 \cdot t\_0, \pi, \frac{\pi}{2}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 0.005555555555555556 angle_m)) angle_m)))
   (+
    (pow (* a (sin (fma (* t_0 t_0) PI (/ PI 2.0)))) 2.0)
    (pow (* b (sin (* PI (/ angle_m 180.0)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = sqrt((0.005555555555555556 / angle_m)) * angle_m;
	return pow((a * sin(fma((t_0 * t_0), ((double) M_PI), (((double) M_PI) / 2.0)))), 2.0) + pow((b * sin((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(sqrt(Float64(0.005555555555555556 / angle_m)) * angle_m)
	return Float64((Float64(a * sin(fma(Float64(t_0 * t_0), pi, Float64(pi / 2.0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[Sqrt[N[(0.005555555555555556 / angle$95$m), $MachinePrecision]], $MachinePrecision] * angle$95$m), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(N[(t$95$0 * t$95$0), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.8%

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

   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-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. fabs-mul N/A

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

   9. add-exp-log N/A

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

   10. exp-fabs N/A

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

   11. add-exp-log N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. lift-PI.f64 79.8

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

3. Applied rewrites 79.8%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 79.8

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 79.8

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in angle around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 79.8

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

8. Applied rewrites 79.8%

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

9. Taylor expanded in angle around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-sqrt.f64 N/A

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

    4. lower-/.f64 79.8

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

11. Applied rewrites 79.8%

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

Alternative 2: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sqrt{0.005555555555555556 \cdot angle\_m}\\ {\left(a \cdot \sin \left(\mathsf{fma}\left(t\_0, t\_0 \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sqrt (* 0.005555555555555556 angle_m))))
   (+
    (pow (* a (sin (fma t_0 (* t_0 PI) (* 0.5 PI)))) 2.0)
    (pow (* b (sin (* PI (/ angle_m 180.0)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = sqrt((0.005555555555555556 * angle_m));
	return pow((a * sin(fma(t_0, (t_0 * ((double) M_PI)), (0.5 * ((double) M_PI))))), 2.0) + pow((b * sin((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = sqrt(Float64(0.005555555555555556 * angle_m))
	return Float64((Float64(a * sin(fma(t_0, Float64(t_0 * pi), Float64(0.5 * pi)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Sqrt[N[(0.005555555555555556 * angle$95$m), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(t$95$0 * N[(t$95$0 * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.8%

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

   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-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. fabs-mul N/A

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

   9. add-exp-log N/A

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

   10. exp-fabs N/A

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

   11. add-exp-log N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. lift-PI.f64 79.8

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

3. Applied rewrites 79.8%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 79.8

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 79.8

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

5. Applied rewrites 79.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. mult-flip N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

7. Applied rewrites 79.8%

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

Alternative 3: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   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-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. fabs-mul N/A

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

   9. add-exp-log N/A

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

   10. exp-fabs N/A

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

   11. add-exp-log N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. lift-PI.f64 79.8

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

3. Applied rewrites 79.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 79.8

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

5. Applied rewrites 79.8%

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

Alternative 4: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot \left|angle\_m\right|\right) \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\_m\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (*
   (+ 0.5 (* 0.5 (cos (* 2.0 (* (* 0.005555555555555556 (fabs angle_m)) PI)))))
   (* a a))
  (pow (* b (sin (* (* PI 0.005555555555555556) angle_m))) 2.0)))

C

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

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return ((0.5 + (0.5 * Math.cos((2.0 * ((0.005555555555555556 * Math.abs(angle_m)) * Math.PI))))) * (a * a)) + Math.pow((b * Math.sin(((Math.PI * 0.005555555555555556) * angle_m))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return ((0.5 + (0.5 * math.cos((2.0 * ((0.005555555555555556 * math.fabs(angle_m)) * math.pi))))) * (a * a)) + math.pow((b * math.sin(((math.pi * 0.005555555555555556) * angle_m))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(0.005555555555555556 * abs(angle_m)) * pi))))) * Float64(a * a)) + (Float64(b * sin(Float64(Float64(pi * 0.005555555555555556) * angle_m))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((0.5 + (0.5 * cos((2.0 * ((0.005555555555555556 * abs(angle_m)) * pi))))) * (a * a)) + ((b * sin(((pi * 0.005555555555555556) * angle_m))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * N[Abs[angle$95$m], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   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-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. fabs-mul N/A

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

   9. add-exp-log N/A

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

   10. exp-fabs N/A

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

   11. add-exp-log N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lift-PI.f64 N/A

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

   20. lower-/.f64 N/A

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

   21. lift-PI.f64 79.8

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

3. Applied rewrites 79.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 79.8

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

5. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow-prod-down N/A

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

7. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot \left|angle\right|\right) \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 5: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a 1.0) 2.0) (pow (* b (sin (* PI (/ angle_m 180.0)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * 1.0), 2.0) + pow((b * sin((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * 1.0), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle_m / 180.0)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * 1.0), 2.0) + math.pow((b * math.sin((math.pi * (angle_m / 180.0)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * 1.0) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * 1.0) ^ 2.0) + ((b * sin((pi * (angle_m / 180.0)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 79.8%

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

Alternative 6: 75.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2}\\ \mathbf{if}\;t\_1 \leq 10^{-312}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), \left(\pi \cdot \pi\right) \cdot \left(angle\_m \cdot angle\_m\right), a \cdot a\right) + {\left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot b\right)\right) \cdot angle\_m\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\_m\right) \cdot angle\_m\right) \cdot \left(\pi \cdot b\right), \pi \cdot b, \left(\left(0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right) \cdot 0.5\right) \cdot a\right) \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))
   (if (<= t_1 1e-312)
     (* a a)
     (if (<= t_1 5e+125)
       (+
        (fma
         (* -3.08641975308642e-5 (* a a))
         (* (* PI PI) (* angle_m angle_m))
         (* a a))
        (pow
         (*
          (fma
           (* 0.005555555555555556 b)
           PI
           (*
            (* -2.8577960676726107e-8 (* angle_m angle_m))
            (* (* (* PI PI) PI) b)))
          angle_m)
         2.0))
       (fma
        (* (* (* 3.08641975308642e-5 angle_m) angle_m) (* PI b))
        (* PI b)
        (*
         (*
          (+ 0.5 (* (cos (* (* 2.0 PI) (* 0.005555555555555556 angle_m))) 0.5))
          a)
         a))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0);
	double tmp;
	if (t_1 <= 1e-312) {
		tmp = a * a;
	} else if (t_1 <= 5e+125) {
		tmp = fma((-3.08641975308642e-5 * (a * a)), ((((double) M_PI) * ((double) M_PI)) * (angle_m * angle_m)), (a * a)) + pow((fma((0.005555555555555556 * b), ((double) M_PI), ((-2.8577960676726107e-8 * (angle_m * angle_m)) * (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * b))) * angle_m), 2.0);
	} else {
		tmp = fma((((3.08641975308642e-5 * angle_m) * angle_m) * (((double) M_PI) * b)), (((double) M_PI) * b), (((0.5 + (cos(((2.0 * ((double) M_PI)) * (0.005555555555555556 * angle_m))) * 0.5)) * a) * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
	tmp = 0.0
	if (t_1 <= 1e-312)
		tmp = Float64(a * a);
	elseif (t_1 <= 5e+125)
		tmp = Float64(fma(Float64(-3.08641975308642e-5 * Float64(a * a)), Float64(Float64(pi * pi) * Float64(angle_m * angle_m)), Float64(a * a)) + (Float64(fma(Float64(0.005555555555555556 * b), pi, Float64(Float64(-2.8577960676726107e-8 * Float64(angle_m * angle_m)) * Float64(Float64(Float64(pi * pi) * pi) * b))) * angle_m) ^ 2.0));
	else
		tmp = fma(Float64(Float64(Float64(3.08641975308642e-5 * angle_m) * angle_m) * Float64(pi * b)), Float64(pi * b), Float64(Float64(Float64(0.5 + Float64(cos(Float64(Float64(2.0 * pi) * Float64(0.005555555555555556 * angle_m))) * 0.5)) * a) * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 1e-312], N[(a * a), $MachinePrecision], If[LessEqual[t$95$1, 5e+125], N[(N[(N[(-3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[(0.005555555555555556 * b), $MachinePrecision] * Pi + N[(N[(-2.8577960676726107e-8 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(3.08641975308642e-5 * angle$95$m), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * b), $MachinePrecision] + N[(N[(N[(0.5 + N[(N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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))) < 9.9999999999847e-313

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

   if 9.9999999999847e-313 < (+.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))) < 4.99999999999999962e125

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{angle}\right)}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{angle}\right)}^{2} \]

   4. Applied rewrites 73.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot b\right)\right) \cdot angle\right)}}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}\right)} + {\left(\mathsf{fma}\left(\frac{1}{180} \cdot b, \pi, \left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot b\right)\right) \cdot angle\right)}^{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), \left(\pi \cdot \pi\right) \cdot \left(angle \cdot angle\right), a \cdot a\right)} + {\left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot b\right)\right) \cdot angle\right)}^{2} \]

   if 4.99999999999999962e125 < (+.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 79.8%

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

      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-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 79.8

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

   3. Applied rewrites 79.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 79.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \mathsf{PI}\left(\right) \cdot {\left(\sqrt{\frac{1}{180} \cdot angle}\right)}^{2}\right)}^{2}} \]

   8. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right) \cdot \left(\pi \cdot b\right), \pi \cdot b, \left(\left(0.5 + \cos \left(\left(2 \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot 0.5\right) \cdot a\right) \cdot a\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle\_m}{180}\\ t_1 := {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2}\\ \mathbf{if}\;t\_1 \leq 10^{-312}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;t\_1 \leq 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), \left(\pi \cdot \pi\right) \cdot \left(angle\_m \cdot angle\_m\right), a \cdot a\right) + {\left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot b\right)\right) \cdot angle\_m\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right) \cdot 2\right), 0.5, 0.5\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0)))
        (t_1 (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))
   (if (<= t_1 1e-312)
     (* a a)
     (if (<= t_1 1e+246)
       (+
        (fma
         (* -3.08641975308642e-5 (* a a))
         (* (* PI PI) (* angle_m angle_m))
         (* a a))
        (pow
         (*
          (fma
           (* 0.005555555555555556 b)
           PI
           (*
            (* -2.8577960676726107e-8 (* angle_m angle_m))
            (* (* (* PI PI) PI) b)))
          angle_m)
         2.0))
       (fma
        (*
         (fma (cos (* (* PI (* angle_m 0.005555555555555556)) 2.0)) 0.5 0.5)
         a)
        a
        (*
         (* 3.08641975308642e-5 (* angle_m angle_m))
         (* (* PI b) (* PI b))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m / 180.0);
	double t_1 = pow((a * cos(t_0)), 2.0) + pow((b * sin(t_0)), 2.0);
	double tmp;
	if (t_1 <= 1e-312) {
		tmp = a * a;
	} else if (t_1 <= 1e+246) {
		tmp = fma((-3.08641975308642e-5 * (a * a)), ((((double) M_PI) * ((double) M_PI)) * (angle_m * angle_m)), (a * a)) + pow((fma((0.005555555555555556 * b), ((double) M_PI), ((-2.8577960676726107e-8 * (angle_m * angle_m)) * (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * b))) * angle_m), 2.0);
	} else {
		tmp = fma((fma(cos(((((double) M_PI) * (angle_m * 0.005555555555555556)) * 2.0)), 0.5, 0.5) * a), a, ((3.08641975308642e-5 * (angle_m * angle_m)) * ((((double) M_PI) * b) * (((double) M_PI) * b))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m / 180.0))
	t_1 = Float64((Float64(a * cos(t_0)) ^ 2.0) + (Float64(b * sin(t_0)) ^ 2.0))
	tmp = 0.0
	if (t_1 <= 1e-312)
		tmp = Float64(a * a);
	elseif (t_1 <= 1e+246)
		tmp = Float64(fma(Float64(-3.08641975308642e-5 * Float64(a * a)), Float64(Float64(pi * pi) * Float64(angle_m * angle_m)), Float64(a * a)) + (Float64(fma(Float64(0.005555555555555556 * b), pi, Float64(Float64(-2.8577960676726107e-8 * Float64(angle_m * angle_m)) * Float64(Float64(Float64(pi * pi) * pi) * b))) * angle_m) ^ 2.0));
	else
		tmp = fma(Float64(fma(cos(Float64(Float64(pi * Float64(angle_m * 0.005555555555555556)) * 2.0)), 0.5, 0.5) * a), a, Float64(Float64(3.08641975308642e-5 * Float64(angle_m * angle_m)) * Float64(Float64(pi * b) * Float64(pi * b))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 1e-312], N[(a * a), $MachinePrecision], If[LessEqual[t$95$1, 1e+246], N[(N[(N[(-3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[(0.005555555555555556 * b), $MachinePrecision] * Pi + N[(N[(-2.8577960676726107e-8 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[(N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(3.08641975308642e-5 * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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))) < 9.9999999999847e-313

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

   if 9.9999999999847e-313 < (+.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))) < 1.00000000000000007e246

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{angle}\right)}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{angle}\right)}^{2} \]

   4. Applied rewrites 73.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot b\right)\right) \cdot angle\right)}}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}\right)} + {\left(\mathsf{fma}\left(\frac{1}{180} \cdot b, \pi, \left(\frac{-1}{34992000} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot b\right)\right) \cdot angle\right)}^{2} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), \left(\pi \cdot \pi\right) \cdot \left(angle \cdot angle\right), a \cdot a\right)} + {\left(\mathsf{fma}\left(0.005555555555555556 \cdot b, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot b\right)\right) \cdot angle\right)}^{2} \]

   if 1.00000000000000007e246 < (+.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 79.8%

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

      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-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 79.8

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

   3. Applied rewrites 79.8%

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

   4. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + \mathsf{PI}\left(\right) \cdot \left|\frac{1}{180} \cdot angle\right|\right)}^{2}} \]

   6. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right), 0.5, 0.5\right) \cdot a, a, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(\pi \cdot b\right)\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right), \pi \cdot \pi, \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1, 0\right) \cdot \left(a \cdot a\right)\right), angle\_m, \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot 0\right), angle\_m, \left(a \cdot a\right) \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.45e+152)
   (fma
    (fma
     (fma
      (* 3.08641975308642e-5 (* b b))
      (* PI PI)
      (* (fma (* -3.08641975308642e-5 (* PI PI)) 1.0 0.0) (* a a)))
     angle_m
     (* (* 0.011111111111111112 (* a a)) 0.0))
    angle_m
    (* (* a a) 1.0))
   (* a a)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.45e+152) {
		tmp = fma(fma(fma((3.08641975308642e-5 * (b * b)), (((double) M_PI) * ((double) M_PI)), (fma((-3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI))), 1.0, 0.0) * (a * a))), angle_m, ((0.011111111111111112 * (a * a)) * 0.0)), angle_m, ((a * a) * 1.0));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.45e+152)
		tmp = fma(fma(fma(Float64(3.08641975308642e-5 * Float64(b * b)), Float64(pi * pi), Float64(fma(Float64(-3.08641975308642e-5 * Float64(pi * pi)), 1.0, 0.0) * Float64(a * a))), angle_m, Float64(Float64(0.011111111111111112 * Float64(a * a)) * 0.0)), angle_m, Float64(Float64(a * a) * 1.0));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.45e+152], N[(N[(N[(N[(3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(N[(-3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 1.0 + 0.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m + N[(N[(0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] * angle$95$m + N[(N[(a * a), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4499999999999999e152

   1. Initial program 79.8%

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

      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-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. fabs-mul N/A

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

      9. add-exp-log N/A

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

      10. exp-fabs N/A

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

      11. add-exp-log N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-PI.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-PI.f64 79.8

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

   3. Applied rewrites 79.8%

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{angle \cdot \left(\frac{1}{90} \cdot \left({a}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + angle \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {a}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)\right) + {a}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} \]

   8. Applied rewrites 43.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right), \pi \cdot \pi, \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1, 0\right) \cdot \left(a \cdot a\right)\right), angle, \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot 0\right), angle, \left(a \cdot a\right) \cdot 1\right)} \]

   if 1.4499999999999999e152 < a

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

Alternative 9: 55.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(angle\_m \cdot angle\_m\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), \pi \cdot \pi, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.45e+152)
   (fma
    a
    a
    (*
     (* angle_m angle_m)
     (fma
      (* -3.08641975308642e-5 (* a a))
      (* PI PI)
      (* (* 3.08641975308642e-5 (* b b)) (* PI PI)))))
   (* a a)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.45e+152) {
		tmp = fma(a, a, ((angle_m * angle_m) * fma((-3.08641975308642e-5 * (a * a)), (((double) M_PI) * ((double) M_PI)), ((3.08641975308642e-5 * (b * b)) * (((double) M_PI) * ((double) M_PI))))));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.45e+152)
		tmp = fma(a, a, Float64(Float64(angle_m * angle_m) * fma(Float64(-3.08641975308642e-5 * Float64(a * a)), Float64(pi * pi), Float64(Float64(3.08641975308642e-5 * Float64(b * b)) * Float64(pi * pi)))));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.45e+152], N[(a * a + N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4499999999999999e152

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 40.9%

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

   if 1.4499999999999999e152 < a

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

Alternative 10: 53.3% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.1

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

4. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025142 
(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)))