ab-angle->ABCF A

Percentage Accurate: 78.9% → 78.8%

Time: 18.9s

Alternatives: 20

Speedup: N/A×

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\\ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{t\_0 \cdot t\_0}{180 \cdot \frac{1}{angle}}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (pow (cbrt (sqrt PI)) 3.0)))
   (+
    (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
    (pow (* b (cos (/ (* t_0 t_0) (* 180.0 (/ 1.0 angle))))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = pow(cbrt(sqrt(((double) M_PI))), 3.0);
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((t_0 * t_0) / (180.0 * (1.0 / angle))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0);
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((t_0 * t_0) / (180.0 * (1.0 / angle))))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(sqrt(pi)) ^ 3.0
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(t_0 * t_0) / Float64(180.0 * Float64(1.0 / angle))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(180.0 * N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 76.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. lift-PI.f64 N/A

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

   21. lower-cbrt.f64 77.0

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

4. Applied rewrites 77.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

6. Applied rewrites 76.8%

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

   1. add-cube-cbrt N/A

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

   2. pow3 N/A

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

   3. lift-PI.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. cbrt-prod N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-cbrt.f64 77.1

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

8. Applied rewrites 77.1%

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

9. Final simplification 77.1%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}{180 \cdot \frac{1}{angle}}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 78.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. lift-PI.f64 N/A

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

   21. lower-cbrt.f64 77.0

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

4. Applied rewrites 77.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

6. Applied rewrites 77.0%

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

7. Final simplification 77.0%

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

Alternative 3: 78.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
  (pow (* b (cos (/ 1.0 (/ (/ 180.0 angle) PI)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(1.0 / N[(N[(180.0 / angle), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-PI.f64 N/A

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

   5. add-cube-cbrt N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

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

   20. lift-PI.f64 N/A

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

   21. lower-cbrt.f64 77.0

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

4. Applied rewrites 77.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

6. Applied rewrites 77.0%

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

Alternative 4: 78.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 76.9%

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

6. Final simplification 76.9%

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

Alternative 5: 76.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \pi \cdot \pi, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot -0.5\right) \cdot t\_0, a, \mathsf{fma}\left(a, a \cdot 0.5, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (* PI (* angle 0.011111111111111112)))))
   (if (<= (/ angle 180.0) 5e-6)
     (fma (pow (* angle (* a 0.005555555555555556)) 2.0) (* PI PI) (* b b))
     (fma
      (* (* a -0.5) t_0)
      a
      (fma a (* a 0.5) (* (* b b) (fma 0.5 t_0 0.5)))))))

C

double code(double a, double b, double angle) {
	double t_0 = cos((((double) M_PI) * (angle * 0.011111111111111112)));
	double tmp;
	if ((angle / 180.0) <= 5e-6) {
		tmp = fma(pow((angle * (a * 0.005555555555555556)), 2.0), (((double) M_PI) * ((double) M_PI)), (b * b));
	} else {
		tmp = fma(((a * -0.5) * t_0), a, fma(a, (a * 0.5), ((b * b) * fma(0.5, t_0, 0.5))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cos(Float64(pi * Float64(angle * 0.011111111111111112)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e-6)
		tmp = fma((Float64(angle * Float64(a * 0.005555555555555556)) ^ 2.0), Float64(pi * pi), Float64(b * b));
	else
		tmp = fma(Float64(Float64(a * -0.5) * t_0), a, fma(a, Float64(a * 0.5), Float64(Float64(b * b) * fma(0.5, t_0, 0.5))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-6], N[(N[Power[N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * -0.5), $MachinePrecision] * t$95$0), $MachinePrecision] * a + N[(a * N[(a * 0.5), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000041e-6

   1. Initial program 82.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 82.9

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 17.2%

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

      1. lift-log.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. rem-log-exp N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. exp-prod N/A

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

      10. lift-*.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 79.2%

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

   if 5.00000000000000041e-6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. add-cube-cbrt N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. div-inv N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 61.0

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 61.0%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

   6. Applied rewrites 60.9%

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

   8. Applied rewrites 60.8%

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

Alternative 6: 76.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \pi \cdot \pi, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(t\_0, -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (* PI (* angle 0.011111111111111112)))))
   (if (<= (/ angle 180.0) 5e-6)
     (fma (pow (* angle (* a 0.005555555555555556)) 2.0) (* PI PI) (* b b))
     (fma (* a (fma t_0 -0.5 0.5)) a (* (* b b) (fma 0.5 t_0 0.5))))))

C

double code(double a, double b, double angle) {
	double t_0 = cos((((double) M_PI) * (angle * 0.011111111111111112)));
	double tmp;
	if ((angle / 180.0) <= 5e-6) {
		tmp = fma(pow((angle * (a * 0.005555555555555556)), 2.0), (((double) M_PI) * ((double) M_PI)), (b * b));
	} else {
		tmp = fma((a * fma(t_0, -0.5, 0.5)), a, ((b * b) * fma(0.5, t_0, 0.5)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cos(Float64(pi * Float64(angle * 0.011111111111111112)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e-6)
		tmp = fma((Float64(angle * Float64(a * 0.005555555555555556)) ^ 2.0), Float64(pi * pi), Float64(b * b));
	else
		tmp = fma(Float64(a * fma(t_0, -0.5, 0.5)), a, Float64(Float64(b * b) * fma(0.5, t_0, 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-6], N[(N[Power[N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(t$95$0 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000041e-6

   1. Initial program 82.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 82.9

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 17.2%

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

      1. lift-log.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. rem-log-exp N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. exp-prod N/A

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

      10. lift-*.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 79.2%

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

   if 5.00000000000000041e-6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. add-cube-cbrt N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. div-inv N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 61.0

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 61.0%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]

   6. Applied rewrites 60.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \pi \cdot \pi, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \pi \cdot \pi, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right), \mathsf{fma}\left(t\_0, -0.5, 0.5\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (* PI (* angle 0.011111111111111112)))))
   (if (<= (/ angle 180.0) 5e-6)
     (fma (pow (* angle (* a 0.005555555555555556)) 2.0) (* PI PI) (* b b))
     (fma b (* b (fma 0.5 t_0 0.5)) (* (fma t_0 -0.5 0.5) (* a a))))))

C

double code(double a, double b, double angle) {
	double t_0 = cos((((double) M_PI) * (angle * 0.011111111111111112)));
	double tmp;
	if ((angle / 180.0) <= 5e-6) {
		tmp = fma(pow((angle * (a * 0.005555555555555556)), 2.0), (((double) M_PI) * ((double) M_PI)), (b * b));
	} else {
		tmp = fma(b, (b * fma(0.5, t_0, 0.5)), (fma(t_0, -0.5, 0.5) * (a * a)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cos(Float64(pi * Float64(angle * 0.011111111111111112)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e-6)
		tmp = fma((Float64(angle * Float64(a * 0.005555555555555556)) ^ 2.0), Float64(pi * pi), Float64(b * b));
	else
		tmp = fma(b, Float64(b * fma(0.5, t_0, 0.5)), Float64(fma(t_0, -0.5, 0.5) * Float64(a * a)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-6], N[(N[Power[N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * -0.5 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000041e-6

   1. Initial program 82.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 82.9

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 17.2%

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

      1. lift-log.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. rem-log-exp N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. exp-prod N/A

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

      10. lift-*.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 79.2%

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

   if 5.00000000000000041e-6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. add-cube-cbrt N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. div-inv N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 61.0

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 61.0%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]

   6. Applied rewrites 60.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \pi \cdot \pi, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right), \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right) \cdot \left(a \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{+84}:\\ \;\;\;\;{b}^{2} + \mathsf{fma}\left(a \cdot a, 0.5, \left(a \cdot a\right) \cdot \left(-0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \pi \cdot \pi, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 7.2e+84)
   (+
    (pow b 2.0)
    (fma
     (* a a)
     0.5
     (* (* a a) (* -0.5 (cos (* PI (* angle 0.011111111111111112)))))))
   (fma (pow (* angle (* a 0.005555555555555556)) 2.0) (* PI PI) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.2e+84) {
		tmp = pow(b, 2.0) + fma((a * a), 0.5, ((a * a) * (-0.5 * cos((((double) M_PI) * (angle * 0.011111111111111112))))));
	} else {
		tmp = fma(pow((angle * (a * 0.005555555555555556)), 2.0), (((double) M_PI) * ((double) M_PI)), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 7.2e+84)
		tmp = Float64((b ^ 2.0) + fma(Float64(a * a), 0.5, Float64(Float64(a * a) * Float64(-0.5 * cos(Float64(pi * Float64(angle * 0.011111111111111112)))))));
	else
		tmp = fma((Float64(angle * Float64(a * 0.005555555555555556)) ^ 2.0), Float64(pi * pi), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 7.2e+84], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * 0.5 + N[(N[(a * a), $MachinePrecision] * N[(-0.5 * N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.1999999999999999e84

   1. Initial program 73.1%

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

      1. clear-num N/A

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

      2. lift-PI.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 73.1

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

   4. Applied rewrites 73.1%

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

   6. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, 0.5, \left(a \cdot a\right) \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 61.3%

      \[\leadsto \mathsf{fma}\left(a \cdot a, 0.5, \left(a \cdot a\right) \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right) + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

   if 7.1999999999999999e84 < a

   1. Initial program 90.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 90.6

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 45.5%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 42.5%

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

      1. lift-log.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. rem-log-exp N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. exp-prod N/A

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

      10. lift-*.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 89.6%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{+84}:\\ \;\;\;\;{b}^{2} + \mathsf{fma}\left(a \cdot a, 0.5, \left(a \cdot a\right) \cdot \left(-0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \pi \cdot \pi, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.005555555555555556 \cdot \left(a \cdot \pi\right)}{\frac{-1}{angle}}\\ \mathbf{if}\;b \leq 2.25 \cdot 10^{-160}:\\ \;\;\;\;{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (/ (* 0.005555555555555556 (* a PI)) (/ -1.0 angle))))
   (if (<= b 2.25e-160)
     (pow (* a (sin (* 0.005555555555555556 (* angle PI)))) 2.0)
     (fma b b (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * (a * ((double) M_PI))) / (-1.0 / angle);
	double tmp;
	if (b <= 2.25e-160) {
		tmp = pow((a * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
	} else {
		tmp = fma(b, b, (t_0 * t_0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * Float64(a * pi)) / Float64(-1.0 / angle))
	tmp = 0.0
	if (b <= 2.25e-160)
		tmp = Float64(a * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0;
	else
		tmp = fma(b, b, Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * N[(a * Pi), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.25e-160], N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(b * b + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.25000000000000013e-160

   1. Initial program 76.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. add-cube-cbrt N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. div-inv N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 76.6

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 76.6%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]

   5. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. rem-exp-log N/A

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

      5. *-lft-identity N/A

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

      6. log-E N/A

         \[\leadsto e^{\color{blue}{\log \mathsf{E}\left(\right)} \cdot \log \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. rem-exp-log N/A

         \[\leadsto e^{\log \mathsf{E}\left(\right) \cdot \log \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \color{blue}{e^{\log \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

      8. *-lft-identity N/A

         \[\leadsto e^{\log \mathsf{E}\left(\right) \cdot \log \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot e^{\color{blue}{1 \cdot \log \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

      9. log-E N/A

         \[\leadsto e^{\log \mathsf{E}\left(\right) \cdot \log \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot e^{\color{blue}{\log \mathsf{E}\left(\right)} \cdot \log \left(a \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      10. unpow2 N/A

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

   7. Applied rewrites 44.2%

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

   if 2.25000000000000013e-160 < b

   1. Initial program 77.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 77.6

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 44.4%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in angle around -inf

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

      1. distribute-rgt-in N/A

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

      2. exp-sum N/A

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

      3. exp-to-pow N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. exp-lft-sqr N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 74.4%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.25 \cdot 10^{-160}:\\ \;\;\;\;{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, \frac{0.005555555555555556 \cdot \left(a \cdot \pi\right)}{\frac{-1}{angle}} \cdot \frac{0.005555555555555556 \cdot \left(a \cdot \pi\right)}{\frac{-1}{angle}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right)}^{2}, \pi \cdot \pi, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 7.2e-9)
   (* b b)
   (fma (pow (* angle (* a 0.005555555555555556)) 2.0) (* PI PI) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 7.2e-9) {
		tmp = b * b;
	} else {
		tmp = fma(pow((angle * (a * 0.005555555555555556)), 2.0), (((double) M_PI) * ((double) M_PI)), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 7.2e-9)
		tmp = Float64(b * b);
	else
		tmp = fma((Float64(angle * Float64(a * 0.005555555555555556)) ^ 2.0), Float64(pi * pi), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 7.2e-9], N[(b * b), $MachinePrecision], N[(N[Power[N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.2e-9

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 7.2e-9 < a

   1. Initial program 80.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 80.3

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 33.8%

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

      1. lift-log.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. rem-log-exp N/A

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

      7. lift-+.f64 N/A

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

      8. pow-to-exp N/A

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

      9. exp-prod N/A

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

      10. lift-*.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 75.6%

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

Alternative 11: 66.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.005555555555555556 \cdot \left(a \cdot \pi\right)}{\frac{-1}{angle}}\\ \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (/ (* 0.005555555555555556 (* a PI)) (/ -1.0 angle))))
   (if (<= a 7.2e-9) (* b b) (fma b b (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * (a * ((double) M_PI))) / (-1.0 / angle);
	double tmp;
	if (a <= 7.2e-9) {
		tmp = b * b;
	} else {
		tmp = fma(b, b, (t_0 * t_0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * Float64(a * pi)) / Float64(-1.0 / angle))
	tmp = 0.0
	if (a <= 7.2e-9)
		tmp = Float64(b * b);
	else
		tmp = fma(b, b, Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * N[(a * Pi), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 7.2e-9], N[(b * b), $MachinePrecision], N[(b * b + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.2e-9

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 7.2e-9 < a

   1. Initial program 80.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 80.3

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 33.8%

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

   8. Taylor expanded in angle around -inf

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

      1. distribute-rgt-in N/A

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

      2. exp-sum N/A

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

      3. exp-to-pow N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. exp-lft-sqr N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 75.6%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, \frac{0.005555555555555556 \cdot \left(a \cdot \pi\right)}{\frac{-1}{angle}} \cdot \frac{0.005555555555555556 \cdot \left(a \cdot \pi\right)}{\frac{-1}{angle}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := a \cdot t\_0\\ \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, t\_0 \cdot t\_0, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI 0.005555555555555556))) (t_1 (* a t_0)))
   (if (<= a 7.2e-9)
     (* b b)
     (if (<= a 1.35e+154) (fma (* a a) (* t_0 t_0) (* b b)) (* t_1 t_1)))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_1 = a * t_0;
	double tmp;
	if (a <= 7.2e-9) {
		tmp = b * b;
	} else if (a <= 1.35e+154) {
		tmp = fma((a * a), (t_0 * t_0), (b * b));
	} else {
		tmp = t_1 * t_1;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * 0.005555555555555556))
	t_1 = Float64(a * t_0)
	tmp = 0.0
	if (a <= 7.2e-9)
		tmp = Float64(b * b);
	elseif (a <= 1.35e+154)
		tmp = fma(Float64(a * a), Float64(t_0 * t_0), Float64(b * b));
	else
		tmp = Float64(t_1 * t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * t$95$0), $MachinePrecision]}, If[LessEqual[a, 7.2e-9], N[(b * b), $MachinePrecision], If[LessEqual[a, 1.35e+154], N[(N[(a * a), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.2e-9

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 7.2e-9 < a < 1.35000000000000003e154

   1. Initial program 60.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 60.9

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 26.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 23.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 48.9%

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

   if 1.35000000000000003e154 < a

   1. Initial program 99.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 99.8

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 43.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 83.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot a, \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \left(a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (* angle (* PI 0.005555555555555556)))))
   (if (<= a 7.2e-9)
     (* b b)
     (if (<= a 4.8e+154)
       (fma
        (* angle angle)
        (* (* PI PI) (* a (* a 3.08641975308642e-5)))
        (* b b))
       (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = a * (angle * (((double) M_PI) * 0.005555555555555556));
	double tmp;
	if (a <= 7.2e-9) {
		tmp = b * b;
	} else if (a <= 4.8e+154) {
		tmp = fma((angle * angle), ((((double) M_PI) * ((double) M_PI)) * (a * (a * 3.08641975308642e-5))), (b * b));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(a * Float64(angle * Float64(pi * 0.005555555555555556)))
	tmp = 0.0
	if (a <= 7.2e-9)
		tmp = Float64(b * b);
	elseif (a <= 4.8e+154)
		tmp = fma(Float64(angle * angle), Float64(Float64(pi * pi) * Float64(a * Float64(a * 3.08641975308642e-5))), Float64(b * b));
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 7.2e-9], N[(b * b), $MachinePrecision], If[LessEqual[a, 4.8e+154], N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.2e-9

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 7.2e-9 < a < 4.8000000000000003e154

   1. Initial program 60.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. add-cube-cbrt N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. div-inv N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 60.9

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 60.9%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 28.6%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 48.8

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

   10. Applied rewrites 48.8%

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

   if 4.8000000000000003e154 < a

   1. Initial program 99.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 99.8

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 43.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 83.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \left(a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(a \cdot a\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (* angle (* PI 0.005555555555555556)))))
   (if (<= a 7.2e-9)
     (* b b)
     (if (<= a 1.35e+154)
       (fma
        (* angle angle)
        (* (* a a) (* PI (* PI 3.08641975308642e-5)))
        (* b b))
       (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = a * (angle * (((double) M_PI) * 0.005555555555555556));
	double tmp;
	if (a <= 7.2e-9) {
		tmp = b * b;
	} else if (a <= 1.35e+154) {
		tmp = fma((angle * angle), ((a * a) * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))), (b * b));
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(a * Float64(angle * Float64(pi * 0.005555555555555556)))
	tmp = 0.0
	if (a <= 7.2e-9)
		tmp = Float64(b * b);
	elseif (a <= 1.35e+154)
		tmp = fma(Float64(angle * angle), Float64(Float64(a * a) * Float64(pi * Float64(pi * 3.08641975308642e-5))), Float64(b * b));
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 7.2e-9], N[(b * b), $MachinePrecision], If[LessEqual[a, 1.35e+154], N[(N[(angle * angle), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 7.2e-9

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 7.2e-9 < a < 1.35000000000000003e154

   1. Initial program 60.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. add-cube-cbrt N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. div-inv N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval N/A

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

      20. lift-PI.f64 N/A

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

      21. lower-cbrt.f64 60.9

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 60.9%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Applied rewrites 28.6%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 48.8

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

   10. Applied rewrites 48.8%

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

   if 1.35000000000000003e154 < a

   1. Initial program 99.8%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 99.8

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 43.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 83.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(a \cdot a\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 66.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;a \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI (* a 0.005555555555555556)))))
   (if (<= a 7.2e-9) (* b b) (fma t_0 t_0 (* b b)))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * (a * 0.005555555555555556));
	double tmp;
	if (a <= 7.2e-9) {
		tmp = b * b;
	} else {
		tmp = fma(t_0, t_0, (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * Float64(a * 0.005555555555555556)))
	tmp = 0.0
	if (a <= 7.2e-9)
		tmp = Float64(b * b);
	else
		tmp = fma(t_0, t_0, Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 7.2e-9], N[(b * b), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 7.2e-9

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   if 7.2e-9 < a

   1. Initial program 80.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 80.3

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

      7. rem-exp-log N/A

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

      8. unpow1 N/A

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

      9. log-pow N/A

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

      10. lift-pow.f64 N/A

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

      11. log-pow N/A

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

      12. *-commutative N/A

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

      13. exp-prod N/A

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

      14. lower-pow.f64 N/A

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. log-E N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-log.f64 N/A

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

   7. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, e^{2 \cdot \left(\log angle + \log \left(a \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}\right)} \]
   8. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto b \cdot b + e^{2 \cdot \left(\color{blue}{\log angle} + \log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lift-PI.f64 N/A

         \[\leadsto b \cdot b + e^{2 \cdot \left(\log angle + \log \left(a \cdot \left(\frac{1}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto b \cdot b + e^{2 \cdot \left(\log angle + \log \left(a \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto b \cdot b + e^{2 \cdot \left(\log angle + \log \color{blue}{\left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)} \]

      5. lift-log.f64 N/A

         \[\leadsto b \cdot b + e^{2 \cdot \left(\log angle + \color{blue}{\log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)} \]

      6. rem-log-exp N/A

         \[\leadsto b \cdot b + e^{\color{blue}{\log \left(e^{2}\right)} \cdot \left(\log angle + \log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      7. lift-+.f64 N/A

         \[\leadsto b \cdot b + e^{\log \left(e^{2}\right) \cdot \color{blue}{\left(\log angle + \log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

      8. pow-to-exp N/A

         \[\leadsto b \cdot b + \color{blue}{{\left(e^{2}\right)}^{\left(\log angle + \log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

      9. exp-prod N/A

         \[\leadsto b \cdot b + \color{blue}{e^{2 \cdot \left(\log angle + \log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto b \cdot b + e^{\color{blue}{2 \cdot \left(\log angle + \log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

      11. lift-exp.f64 N/A

          \[\leadsto b \cdot b + \color{blue}{e^{2 \cdot \left(\log angle + \log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \color{blue}{b \cdot b} + e^{2 \cdot \left(\log angle + \log \left(a \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   9. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right), angle \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right), b \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathbf{if}\;a \leq 2.2 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (* angle (* PI 0.005555555555555556)))))
   (if (<= a 2.2e+137) (* b b) (* t_0 t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = a * (angle * (((double) M_PI) * 0.005555555555555556));
	double tmp;
	if (a <= 2.2e+137) {
		tmp = b * b;
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = a * (angle * (Math.PI * 0.005555555555555556));
	double tmp;
	if (a <= 2.2e+137) {
		tmp = b * b;
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = a * (angle * (math.pi * 0.005555555555555556))
	tmp = 0
	if a <= 2.2e+137:
		tmp = b * b
	else:
		tmp = t_0 * t_0
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(a * Float64(angle * Float64(pi * 0.005555555555555556)))
	tmp = 0.0
	if (a <= 2.2e+137)
		tmp = Float64(b * b);
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = a * (angle * (pi * 0.005555555555555556));
	tmp = 0.0;
	if (a <= 2.2e+137)
		tmp = b * b;
	else
		tmp = t_0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.2e+137], N[(b * b), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.20000000000000015e137

   1. Initial program 73.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 58.4

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 2.20000000000000015e137 < a

   1. Initial program 95.8%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. lift-sin.f64 N/A

         \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lift-*.f64 N/A

         \[\leadsto {\color{blue}{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. lift-pow.f64 95.8

         \[\leadsto \color{blue}{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. rem-exp-log N/A

         \[\leadsto \color{blue}{e^{\log \left({\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. unpow1 N/A

         \[\leadsto e^{\log \color{blue}{\left({\left({\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{1}\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. log-pow N/A

         \[\leadsto e^{\color{blue}{1 \cdot \log \left({\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      10. lift-pow.f64 N/A

          \[\leadsto e^{1 \cdot \log \color{blue}{\left({\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      11. log-pow N/A

          \[\leadsto e^{1 \cdot \color{blue}{\left(2 \cdot \log \left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      12. *-commutative N/A

          \[\leadsto e^{1 \cdot \color{blue}{\left(\log \left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      13. exp-prod N/A

          \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      14. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. Applied rewrites 43.0%

      \[\leadsto \color{blue}{{e}^{\left(2 \cdot \log \left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + {b}^{2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{b}^{2} + e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} + e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right)} \]

      4. lower-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, b, \color{blue}{e^{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, b, e^{\color{blue}{2 \cdot \left(\log \mathsf{E}\left(\right) \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}\right) \]

      6. log-E N/A

         \[\leadsto \mathsf{fma}\left(b, b, e^{2 \cdot \left(\color{blue}{1} \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(b, b, e^{2 \cdot \color{blue}{\left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right) \]

      8. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, b, e^{2 \cdot \color{blue}{\left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right) \]

      9. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, b, e^{2 \cdot \left(\color{blue}{\log angle} + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      10. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, b, e^{2 \cdot \left(\log angle + \color{blue}{\log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right) \]

   7. Applied rewrites 40.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, e^{2 \cdot \left(\log angle + \log \left(a \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{e^{2 \cdot \left(\log angle + \log \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}} \]

   10. Applied rewrites 76.9%

       \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.3% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.2e+137)
   (* b b)
   (* (* angle (* a (* a (* PI PI)))) (* angle 3.08641975308642e-5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.2e+137) {
		tmp = b * b;
	} else {
		tmp = (angle * (a * (a * (((double) M_PI) * ((double) M_PI))))) * (angle * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.2e+137) {
		tmp = b * b;
	} else {
		tmp = (angle * (a * (a * (Math.PI * Math.PI)))) * (angle * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.2e+137:
		tmp = b * b
	else:
		tmp = (angle * (a * (a * (math.pi * math.pi)))) * (angle * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.2e+137)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(angle * Float64(a * Float64(a * Float64(pi * pi)))) * Float64(angle * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.2e+137)
		tmp = b * b;
	else
		tmp = (angle * (a * (a * (pi * pi)))) * (angle * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.2e+137], N[(b * b), $MachinePrecision], N[(N[(angle * N[(a * N[(a * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.20000000000000015e137

   1. Initial program 73.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 58.4

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 2.20000000000000015e137 < a

   1. Initial program 95.8%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

      2. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)}^{2} \]

      4. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}^{2} \]

      5. add-cube-cbrt N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{angle}{180}\right)\right)}^{2} \]

      6. associate-*l* N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)}^{2} \]

      7. associate-*l* N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

      9. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      10. lower-cbrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)\right)}^{2} \]

      12. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      13. lower-cbrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      14. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)\right)}^{2} \]

      16. lift-/.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{angle}{180}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      17. div-inv N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      18. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      19. metadata-eval N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      20. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}^{2} \]

      21. lower-cbrt.f64 95.8

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 95.8%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {b}^{2}\right)} \]

   7. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      12. lower-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right)\right) \]

      17. lower-*.f64 68.0

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \]

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right) \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right)\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)}\right) \]

       7. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

       10. associate-*l* N/A

           \[\leadsto \left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \frac{1}{32400}\right)\right)} \]

       11. associate-*r* N/A

           \[\leadsto \color{blue}{\left(\left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right) \cdot \left(angle \cdot \frac{1}{32400}\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(\left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right) \cdot \left(angle \cdot \frac{1}{32400}\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(\left(\left(a \cdot a\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)} \cdot \left(angle \cdot \frac{1}{32400}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\left(\color{blue}{\left(a \cdot a\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right) \cdot \left(angle \cdot \frac{1}{32400}\right) \]

       15. associate-*l* N/A

           \[\leadsto \left(\color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot angle\right) \cdot \left(angle \cdot \frac{1}{32400}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\color{blue}{\left(a \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot angle\right) \cdot \left(angle \cdot \frac{1}{32400}\right) \]

       17. lower-*.f64 N/A

           \[\leadsto \left(\left(a \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot angle\right) \cdot \left(angle \cdot \frac{1}{32400}\right) \]

       18. lower-*.f64 72.9

           \[\leadsto \left(\left(a \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot angle\right) \cdot \color{blue}{\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]

   12. Applied rewrites 72.9%

       \[\leadsto \color{blue}{\left(\left(a \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot angle\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.2 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 60.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{+140}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \pi\right) \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.45e+140)
   (* b b)
   (* a (* (* a PI) (* (* angle 3.08641975308642e-5) (* angle PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.45e+140) {
		tmp = b * b;
	} else {
		tmp = a * ((a * ((double) M_PI)) * ((angle * 3.08641975308642e-5) * (angle * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.45e+140) {
		tmp = b * b;
	} else {
		tmp = a * ((a * Math.PI) * ((angle * 3.08641975308642e-5) * (angle * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.45e+140:
		tmp = b * b
	else:
		tmp = a * ((a * math.pi) * ((angle * 3.08641975308642e-5) * (angle * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.45e+140)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(Float64(a * pi) * Float64(Float64(angle * 3.08641975308642e-5) * Float64(angle * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.45e+140)
		tmp = b * b;
	else
		tmp = a * ((a * pi) * ((angle * 3.08641975308642e-5) * (angle * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.45e+140], N[(b * b), $MachinePrecision], N[(a * N[(N[(a * Pi), $MachinePrecision] * N[(N[(angle * 3.08641975308642e-5), $MachinePrecision] * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4499999999999999e140

   1. Initial program 73.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 58.4

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 1.4499999999999999e140 < a

   1. Initial program 95.8%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

      2. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)}^{2} \]

      4. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}^{2} \]

      5. add-cube-cbrt N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{angle}{180}\right)\right)}^{2} \]

      6. associate-*l* N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)}^{2} \]

      7. associate-*l* N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

      9. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      10. lower-cbrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)\right)}^{2} \]

      12. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      13. lower-cbrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      14. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)\right)}^{2} \]

      16. lift-/.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{angle}{180}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      17. div-inv N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      18. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      19. metadata-eval N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      20. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}^{2} \]

      21. lower-cbrt.f64 95.8

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 95.8%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {b}^{2}\right)} \]

   7. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      12. lower-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right)\right) \]

      17. lower-*.f64 68.0

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \]

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right)\right) \]

       5. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right)} \]

       7. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right)\right)} \]

       8. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right)\right) \cdot a} \]

       9. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right)\right)\right) \cdot a} \]

   12. Applied rewrites 72.9%

       \[\leadsto \color{blue}{\left(\left(a \cdot \pi\right) \cdot \left(\left(\pi \cdot angle\right) \cdot \left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{+140}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \pi\right) \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 60.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.8e+137)
   (* b b)
   (* (* a a) (* 3.08641975308642e-5 (* angle (* angle (* PI PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.8e+137) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (3.08641975308642e-5 * (angle * (angle * (((double) M_PI) * ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.8e+137) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (3.08641975308642e-5 * (angle * (angle * (Math.PI * Math.PI))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.8e+137:
		tmp = b * b
	else:
		tmp = (a * a) * (3.08641975308642e-5 * (angle * (angle * (math.pi * math.pi))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.8e+137)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(a * a) * Float64(3.08641975308642e-5 * Float64(angle * Float64(angle * Float64(pi * pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.8e+137)
		tmp = b * b;
	else
		tmp = (a * a) * (3.08641975308642e-5 * (angle * (angle * (pi * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.8e+137], N[(b * b), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(3.08641975308642e-5 * N[(angle * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.79999999999999969e137

   1. Initial program 73.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 58.4

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 5.79999999999999969e137 < a

   1. Initial program 95.8%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

      2. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. *-commutative N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)}\right)}^{2} \]

      4. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}^{2} \]

      5. add-cube-cbrt N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot \frac{angle}{180}\right)\right)}^{2} \]

      6. associate-*l* N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)}^{2} \]

      7. associate-*l* N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} \]

      9. lift-PI.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      10. lower-cbrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)}\right)\right)}^{2} \]

      12. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      13. lower-cbrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)\right)\right)\right)}^{2} \]

      14. *-commutative N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\frac{angle}{180} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)\right)}^{2} \]

      16. lift-/.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{angle}{180}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      17. div-inv N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      18. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      19. metadata-eval N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)}^{2} \]

      20. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)}^{2} \]

      21. lower-cbrt.f64 95.8

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \color{blue}{\sqrt[3]{\pi}}\right)\right)\right)\right)}^{2} \]

   4. Applied rewrites 95.8%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {b}^{2}\right)} \]

   7. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)} \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      12. lower-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right)\right) \]

      17. lower-*.f64 68.0

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \]

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]

   11. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. metadata-eval N/A

          \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       7. distribute-lft-neg-in N/A

          \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)} \]

       9. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right) \]

       11. distribute-lft-neg-in N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \cdot {angle}^{2}\right) \]

       12. metadata-eval N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\color{blue}{\frac{1}{32400}} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       13. associate-*r* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)\right)} \]

       14. *-commutative N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       16. *-commutative N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

       17. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right) \]

       18. associate-*r* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot angle\right) \cdot angle\right)}\right) \]

       19. lower-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\left({\mathsf{PI}\left(\right)}^{2} \cdot angle\right) \cdot angle\right)}\right) \]

   13. Applied rewrites 67.9%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot angle\right) \cdot angle\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.8 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.2% accurate, 74.7× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* b b))

C

double code(double a, double b, double angle) {
	return b * b;
}

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = b * b
end function

Java

public static double code(double a, double b, double angle) {
	return b * b;
}

Python

def code(a, b, angle):
	return b * b

Julia

function code(a, b, angle)
	return Float64(b * b)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = b * b;
end

Wolfram

code[a_, b_, angle_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{b}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{b \cdot b} \]

   2. lower-*.f64 54.7

      \[\leadsto \color{blue}{b \cdot b} \]

5. Applied rewrites 54.7%

   \[\leadsto \color{blue}{b \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))