ab-angle->ABCF A

Percentage Accurate: 79.7% → 79.7%

Time: 5.0s

Alternatives: 14

Speedup: 1.2×

• 36.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 1.2× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (-
   0.5
   (* 0.5 (sin (fma (fma PI (/ angle_m 180.0) (/ PI 2.0)) 2.0 (/ PI 2.0)))))
  (* b b)
  (pow (* (sin (* PI (/ angle_m 180.0))) a) 2.0)))

C

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

Julia

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(0.5 - N[(0.5 * N[Sin[N[(N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[Power[N[(N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. 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. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

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

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

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

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

   9. sqr-sin-a N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-/.f64 N/A

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

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

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

   9. lower-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-fma.f64 N/A

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

7. Applied rewrites 79.7%

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

Alternative 2: 79.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. 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. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

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

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

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

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

   9. sqr-sin-a N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-/.f64 N/A

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

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

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

   9. lower-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-fma.f64 N/A

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

7. Applied rewrites 79.7%

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

8. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.7

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

10. Applied rewrites 79.7%

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

Alternative 3: 79.8% accurate, 1.2× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (fma
  (- 0.5 (* 0.5 (cos (* 2.0 (fma PI (/ angle_m 180.0) (/ PI 2.0))))))
  (* b b)
  (pow (* (sin (* PI (/ angle_m 180.0))) a) 2.0)))

C

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * fma(pi, Float64(angle_m / 180.0), Float64(pi / 2.0)))))), Float64(b * b), (Float64(sin(Float64(pi * Float64(angle_m / 180.0))) * a) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[Power[N[(N[Sin[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. 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. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

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

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

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

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

   9. sqr-sin-a N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

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

Alternative 4: 79.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[Power[N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. 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. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

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

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

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

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

   9. sqr-sin-a N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

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

6. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.8

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

8. Applied rewrites 79.8%

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

Alternative 5: 79.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. 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. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-cos-a N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-PI.f64 79.8

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

5. Applied rewrites 79.8%

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

Alternative 6: 79.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. 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. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

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

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

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

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

   9. sqr-sin-a N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-/.f64 N/A

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

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

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

   9. lower-sin.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-/.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lower-fma.f64 N/A

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

7. Applied rewrites 79.7%

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

9. Applied rewrites 79.8%

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

Alternative 7: 79.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 79.7

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

4. Applied rewrites 79.7%

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

Alternative 8: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle\_m\\ \mathbf{if}\;b \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;{\left(\sin \left(\left(\pi \cdot angle\_m\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(\pi, \frac{angle\_m}{180}, \frac{\pi}{2}\right)\right), b \cdot b, t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* (* PI a) 0.005555555555555556) angle_m)))
   (if (<= b 3.2e-156)
     (pow (* (sin (* (* PI angle_m) 0.005555555555555556)) a) 2.0)
     (fma
      (- 0.5 (* 0.5 (cos (* 2.0 (fma PI (/ angle_m 180.0) (/ PI 2.0))))))
      (* b b)
      (* t_0 t_0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((((double) M_PI) * a) * 0.005555555555555556) * angle_m;
	double tmp;
	if (b <= 3.2e-156) {
		tmp = pow((sin(((((double) M_PI) * angle_m) * 0.005555555555555556)) * a), 2.0);
	} else {
		tmp = fma((0.5 - (0.5 * cos((2.0 * fma(((double) M_PI), (angle_m / 180.0), (((double) M_PI) / 2.0)))))), (b * b), (t_0 * t_0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(Float64(pi * a) * 0.005555555555555556) * angle_m)
	tmp = 0.0
	if (b <= 3.2e-156)
		tmp = Float64(sin(Float64(Float64(pi * angle_m) * 0.005555555555555556)) * a) ^ 2.0;
	else
		tmp = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * fma(pi, Float64(angle_m / 180.0), Float64(pi / 2.0)))))), Float64(b * b), Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(Pi * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]}, If[LessEqual[b, 3.2e-156], N[Power[N[(N[Sin[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.19999999999999982e-156

   1. Initial program 79.7%

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

   2. Taylor expanded in a around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 46.5

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

   4. Applied rewrites 46.5%

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

   if 3.19999999999999982e-156 < b

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. 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. Applied rewrites 79.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

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

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 77.4

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

   8. Applied rewrites 77.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 77.4

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

   10. Applied rewrites 77.4%

       \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(\pi, \frac{angle}{180}, \frac{\pi}{2}\right)\right), b \cdot b, \color{blue}{\left(\left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle\_m\\ \mathbf{if}\;a \leq 3.95 \cdot 10^{-140}:\\ \;\;\;\;\left(0.5 - \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot angle\_m, 0.005555555555555556, \pi \cdot 0.5\right), 2, \pi \cdot 0.5\right)\right) \cdot 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(\pi, \frac{angle\_m}{180}, \frac{\pi}{2}\right)\right), b \cdot b, t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* (* PI a) 0.005555555555555556) angle_m)))
   (if (<= a 3.95e-140)
     (*
      (-
       0.5
       (*
        (sin
         (fma
          (fma (* PI angle_m) 0.005555555555555556 (* PI 0.5))
          2.0
          (* PI 0.5)))
        0.5))
      (* b b))
     (fma
      (- 0.5 (* 0.5 (cos (* 2.0 (fma PI (/ angle_m 180.0) (/ PI 2.0))))))
      (* b b)
      (* t_0 t_0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((((double) M_PI) * a) * 0.005555555555555556) * angle_m;
	double tmp;
	if (a <= 3.95e-140) {
		tmp = (0.5 - (sin(fma(fma((((double) M_PI) * angle_m), 0.005555555555555556, (((double) M_PI) * 0.5)), 2.0, (((double) M_PI) * 0.5))) * 0.5)) * (b * b);
	} else {
		tmp = fma((0.5 - (0.5 * cos((2.0 * fma(((double) M_PI), (angle_m / 180.0), (((double) M_PI) / 2.0)))))), (b * b), (t_0 * t_0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(Float64(pi * a) * 0.005555555555555556) * angle_m)
	tmp = 0.0
	if (a <= 3.95e-140)
		tmp = Float64(Float64(0.5 - Float64(sin(fma(fma(Float64(pi * angle_m), 0.005555555555555556, Float64(pi * 0.5)), 2.0, Float64(pi * 0.5))) * 0.5)) * Float64(b * b));
	else
		tmp = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * fma(pi, Float64(angle_m / 180.0), Float64(pi / 2.0)))))), Float64(b * b), Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(Pi * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]}, If[LessEqual[a, 3.95e-140], N[(N[(0.5 - N[(N[Sin[N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.94999999999999983e-140

   1. Initial program 78.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. 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. Applied rewrites 78.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

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

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

   5. Applied rewrites 78.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

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

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.7%

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

   if 3.94999999999999983e-140 < a

   1. Initial program 81.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. 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. Applied rewrites 81.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

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

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 78.1

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

   8. Applied rewrites 78.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 78.1

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

   10. Applied rewrites 78.1%

       \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(\pi, \frac{angle}{180}, \frac{\pi}{2}\right)\right), b \cdot b, \color{blue}{\left(\left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle\right) \cdot \left(\left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle\_m\\ \mathbf{if}\;a \leq 3.95 \cdot 10^{-140}:\\ \;\;\;\;\left(0.5 - \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot angle\_m, 0.005555555555555556, \pi \cdot 0.5\right), 2, \pi \cdot 0.5\right)\right) \cdot 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \frac{angle\_m}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* (* PI a) 0.005555555555555556) angle_m)))
   (if (<= a 3.95e-140)
     (*
      (-
       0.5
       (*
        (sin
         (fma
          (fma (* PI angle_m) 0.005555555555555556 (* PI 0.5))
          2.0
          (* PI 0.5)))
        0.5))
      (* b b))
     (fma
      t_0
      t_0
      (* (+ 0.5 (* 0.5 (cos (* 2.0 (* PI (/ angle_m 180.0)))))) (* b b))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((((double) M_PI) * a) * 0.005555555555555556) * angle_m;
	double tmp;
	if (a <= 3.95e-140) {
		tmp = (0.5 - (sin(fma(fma((((double) M_PI) * angle_m), 0.005555555555555556, (((double) M_PI) * 0.5)), 2.0, (((double) M_PI) * 0.5))) * 0.5)) * (b * b);
	} else {
		tmp = fma(t_0, t_0, ((0.5 + (0.5 * cos((2.0 * (((double) M_PI) * (angle_m / 180.0)))))) * (b * b)));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(Float64(pi * a) * 0.005555555555555556) * angle_m)
	tmp = 0.0
	if (a <= 3.95e-140)
		tmp = Float64(Float64(0.5 - Float64(sin(fma(fma(Float64(pi * angle_m), 0.005555555555555556, Float64(pi * 0.5)), 2.0, Float64(pi * 0.5))) * 0.5)) * Float64(b * b));
	else
		tmp = fma(t_0, t_0, Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(angle_m / 180.0)))))) * Float64(b * b)));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(Pi * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]}, If[LessEqual[a, 3.95e-140], N[(N[(0.5 - N[(N[Sin[N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.94999999999999983e-140

   1. Initial program 78.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. 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. Applied rewrites 78.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

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

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

   5. Applied rewrites 78.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

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

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.7%

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

   if 3.94999999999999983e-140 < a

   1. Initial program 81.3%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. 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. Applied rewrites 81.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

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

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 78.1

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

   8. Applied rewrites 78.1%

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

   10. Applied rewrites 78.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle, \left(\left(\pi \cdot a\right) \cdot 0.005555555555555556\right) \cdot angle, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(b \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.95 \cdot 10^{-140}:\\ \;\;\;\;\left(0.5 - \sin \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot angle\_m, 0.005555555555555556, \pi \cdot 0.5\right), 2, \pi \cdot 0.5\right)\right) \cdot 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot angle\_m\right) \cdot \left(a \cdot angle\_m\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.95e-140)
   (*
    (-
     0.5
     (*
      (sin
       (fma
        (fma (* PI angle_m) 0.005555555555555556 (* PI 0.5))
        2.0
        (* PI 0.5)))
      0.5))
    (* b b))
   (+
    (* (* (* (* a angle_m) (* a angle_m)) (* PI PI)) 3.08641975308642e-5)
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.95e-140) {
		tmp = (0.5 - (sin(fma(fma((((double) M_PI) * angle_m), 0.005555555555555556, (((double) M_PI) * 0.5)), 2.0, (((double) M_PI) * 0.5))) * 0.5)) * (b * b);
	} else {
		tmp = ((((a * angle_m) * (a * angle_m)) * (((double) M_PI) * ((double) M_PI))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.95e-140)
		tmp = Float64(Float64(0.5 - Float64(sin(fma(fma(Float64(pi * angle_m), 0.005555555555555556, Float64(pi * 0.5)), 2.0, Float64(pi * 0.5))) * 0.5)) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * angle_m) * Float64(a * angle_m)) * Float64(pi * pi)) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3.95e-140], N[(N[(0.5 - N[(N[Sin[N[(N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.005555555555555556 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] * 2.0 + N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * angle$95$m), $MachinePrecision] * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.94999999999999983e-140

   1. Initial program 78.9%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. 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. Applied rewrites 78.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

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

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

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

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

   5. Applied rewrites 78.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-/.f64 N/A

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

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

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

      9. lower-sin.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lower-fma.f64 N/A

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

   7. Applied rewrites 78.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 61.7%

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

   if 3.94999999999999983e-140 < a

   1. Initial program 81.3%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

         \[\leadsto \left({a}^{2} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. pow-prod-down N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. lift-PI.f64 78.0

          \[\leadsto {\left(\left(\pi \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{{\left(\left(\pi \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

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

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

      5. pow2 N/A

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

      6. lift-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      3. lower-*.f64 77.9

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      8. *-commutative N/A

         \[\leadsto \left(\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      12. lift-PI.f64 78.0

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      16. *-commutative N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      17. *-commutative N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      18. associate-*r* N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      19. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      21. lift-PI.f64 77.9

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   9. Applied rewrites 77.9%

      \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. swap-sqr N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lift-PI.f64 N/A

           \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       17. lift-PI.f64 78.0

           \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   11. Applied rewrites 78.0%

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

Alternative 12: 67.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.95 \cdot 10^{-140}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(a \cdot angle\_m\right) \cdot \left(a \cdot angle\_m\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.95e-140)
   (* b b)
   (+
    (* (* (* (* a angle_m) (* a angle_m)) (* PI PI)) 3.08641975308642e-5)
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.95e-140) {
		tmp = b * b;
	} else {
		tmp = ((((a * angle_m) * (a * angle_m)) * (((double) M_PI) * ((double) M_PI))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.95e-140) {
		tmp = b * b;
	} else {
		tmp = ((((a * angle_m) * (a * angle_m)) * (Math.PI * Math.PI)) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 3.95e-140:
		tmp = b * b
	else:
		tmp = ((((a * angle_m) * (a * angle_m)) * (math.pi * math.pi)) * 3.08641975308642e-5) + (b * b)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.95e-140)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * angle_m) * Float64(a * angle_m)) * Float64(pi * pi)) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 3.95e-140)
		tmp = b * b;
	else
		tmp = ((((a * angle_m) * (a * angle_m)) * (pi * pi)) * 3.08641975308642e-5) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3.95e-140], N[(b * b), $MachinePrecision], N[(N[(N[(N[(N[(a * angle$95$m), $MachinePrecision] * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.94999999999999983e-140

   1. Initial program 78.9%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   if 3.94999999999999983e-140 < a

   1. Initial program 81.3%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

         \[\leadsto \left({a}^{2} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. pow-prod-down N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. lift-PI.f64 78.0

          \[\leadsto {\left(\left(\pi \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{{\left(\left(\pi \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

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

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

      5. pow2 N/A

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

      6. lift-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      3. lower-*.f64 77.9

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      8. *-commutative N/A

         \[\leadsto \left(\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      12. lift-PI.f64 78.0

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      16. *-commutative N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      17. *-commutative N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      18. associate-*r* N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      19. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      21. lift-PI.f64 77.9

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   9. Applied rewrites 77.9%

      \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. swap-sqr N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lift-PI.f64 N/A

           \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       17. lift-PI.f64 78.0

           \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   11. Applied rewrites 78.0%

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

Alternative 13: 67.8% accurate, 10.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.95 \cdot 10^{-140}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\left(a \cdot angle\_m\right) \cdot \pi\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.95e-140)
   (* b b)
   (+
    (* (* (* a angle_m) (* PI (* (* a angle_m) PI))) 3.08641975308642e-5)
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.95e-140) {
		tmp = b * b;
	} else {
		tmp = (((a * angle_m) * (((double) M_PI) * ((a * angle_m) * ((double) M_PI)))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.95e-140) {
		tmp = b * b;
	} else {
		tmp = (((a * angle_m) * (Math.PI * ((a * angle_m) * Math.PI))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 3.95e-140:
		tmp = b * b
	else:
		tmp = (((a * angle_m) * (math.pi * ((a * angle_m) * math.pi))) * 3.08641975308642e-5) + (b * b)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.95e-140)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(a * angle_m) * Float64(pi * Float64(Float64(a * angle_m) * pi))) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 3.95e-140)
		tmp = b * b;
	else
		tmp = (((a * angle_m) * (pi * ((a * angle_m) * pi))) * 3.08641975308642e-5) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3.95e-140], N[(b * b), $MachinePrecision], N[(N[(N[(N[(a * angle$95$m), $MachinePrecision] * N[(Pi * N[(N[(a * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.94999999999999983e-140

   1. Initial program 78.9%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

   if 3.94999999999999983e-140 < a

   1. Initial program 81.3%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

         \[\leadsto \left({a}^{2} \cdot {\left(angle \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. pow-prod-down N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lower-pow.f64 N/A

         \[\leadsto {\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto {\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. *-commutative N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)}^{2} \cdot \frac{1}{32400} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. lift-PI.f64 78.0

          \[\leadsto {\left(\left(\pi \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 78.0%

      \[\leadsto \color{blue}{{\left(\left(\pi \cdot angle\right) \cdot a\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

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

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

      5. pow2 N/A

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

      6. lift-*.f64 77.9

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

   7. Applied rewrites 77.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      3. lower-*.f64 77.9

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      5. lift-PI.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      8. *-commutative N/A

         \[\leadsto \left(\left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      11. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      12. lift-PI.f64 78.0

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

      13. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\pi \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      14. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      16. *-commutative N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      17. *-commutative N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      18. associate-*r* N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      19. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

      21. lift-PI.f64 77.9

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   9. Applied rewrites 77.9%

      \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       3. lift-PI.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       4. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(a \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       5. associate-*l* N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       8. lower-*.f64 N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right)\right) \cdot \frac{1}{32400} + b \cdot b \]

       9. lift-PI.f64 77.9

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\pi \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b \]

   11. Applied rewrites 77.9%

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

Alternative 14: 57.1% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

angle_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.1

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

4. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025093 
(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)))