ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.9%

Time: 5.5s

Alternatives: 10

Speedup: 1.0×

• 37.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-fabs-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. fabs-mul N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-fabs.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lift-PI.f64 N/A

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

   21. lower-/.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 79.8

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

5. Applied rewrites 79.8%

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

Alternative 2: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-fabs-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. fabs-mul N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-fabs.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lift-PI.f64 N/A

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

   21. lower-/.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 79.8

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

5. Applied rewrites 79.8%

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

7. Applied rewrites 79.8%

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

Alternative 3: 79.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-cos.f64 N/A

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

   2. cos-fabs-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. fabs-mul N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-fabs.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lift-PI.f64 N/A

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

   21. lower-/.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 79.8

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

5. Applied rewrites 79.8%

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in angle around 0

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

10. Applied rewrites 79.9%

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

Alternative 4: 67.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.5e-62)
   (*
    (fma (cos (* 2.0 (* PI (* angle 0.005555555555555556)))) 0.5 0.5)
    (* a a))
   (+
    (pow (* a 1.0) 2.0)
    (pow
     (*
      b
      (*
       (fma
        0.005555555555555556
        PI
        (* (* -2.8577960676726107e-8 (* angle angle)) (* (* PI PI) PI)))
       angle))
     2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.5e-62) {
		tmp = fma(cos((2.0 * (((double) M_PI) * (angle * 0.005555555555555556)))), 0.5, 0.5) * (a * a);
	} else {
		tmp = pow((a * 1.0), 2.0) + pow((b * (fma(0.005555555555555556, ((double) M_PI), ((-2.8577960676726107e-8 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))) * angle)), 2.0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.5e-62)
		tmp = Float64(fma(cos(Float64(2.0 * Float64(pi * Float64(angle * 0.005555555555555556)))), 0.5, 0.5) * Float64(a * a));
	else
		tmp = Float64((Float64(a * 1.0) ^ 2.0) + (Float64(b * Float64(fma(0.005555555555555556, pi, Float64(Float64(-2.8577960676726107e-8 * Float64(angle * angle)) * Float64(Float64(pi * pi) * pi))) * angle)) ^ 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.5e-62], N[(N[(N[Cos[N[(2.0 * N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(a * 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[(N[(0.005555555555555556 * Pi + N[(N[(-2.8577960676726107e-8 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5000000000000001e-62

   1. Initial program 79.8%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 46.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-cos-a-rev N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. cos-fabs-rev N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. cos-fabs-rev N/A

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

   6. Applied rewrites 46.1%

      \[\leadsto \mathsf{fma}\left(b \cdot b, \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{a \cdot a}, 1 - \sin \left(\left|\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right|\right) \cdot \sin \left(\left|\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right|\right)\right) \cdot \left(a \cdot a\right) \]

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 57.2%

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

   if 1.5000000000000001e-62 < b

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 73.8%

      \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)\right)}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 63.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 9.6 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(angle, angle \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 9.6e+78)
   (fma
    angle
    (*
     angle
     (*
      (* PI PI)
      (fma (* b b) 3.08641975308642e-5 (* (* a a) -3.08641975308642e-5))))
    (* a a))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 9.6e+78) {
		tmp = fma(angle, (angle * ((((double) M_PI) * ((double) M_PI)) * fma((b * b), 3.08641975308642e-5, ((a * a) * -3.08641975308642e-5)))), (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 9.6e+78)
		tmp = fma(angle, Float64(angle * Float64(Float64(pi * pi) * fma(Float64(b * b), 3.08641975308642e-5, Float64(Float64(a * a) * -3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 9.6e+78], N[(angle * N[(angle * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5 + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.5999999999999994e78

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   7. Applied rewrites 40.8%

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

   9. Applied rewrites 43.2%

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

   if 9.5999999999999994e78 < a

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

Alternative 6: 61.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{-62}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot b\right) \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right) \cdot \left(\pi \cdot b\right)\right) \cdot \left(\pi \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.5e-62)
   (* a a)
   (if (<= b 1.8e+157)
     (fma
      a
      a
      (* (* angle angle) (* (* PI PI) (* (* 3.08641975308642e-5 b) b))))
     (* (* (* (* 3.08641975308642e-5 angle) angle) (* PI b)) (* PI b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.5e-62) {
		tmp = a * a;
	} else if (b <= 1.8e+157) {
		tmp = fma(a, a, ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((3.08641975308642e-5 * b) * b))));
	} else {
		tmp = (((3.08641975308642e-5 * angle) * angle) * (((double) M_PI) * b)) * (((double) M_PI) * b);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.5e-62)
		tmp = Float64(a * a);
	elseif (b <= 1.8e+157)
		tmp = fma(a, a, Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * Float64(Float64(3.08641975308642e-5 * b) * b))));
	else
		tmp = Float64(Float64(Float64(Float64(3.08641975308642e-5 * angle) * angle) * Float64(pi * b)) * Float64(pi * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.5e-62], N[(a * a), $MachinePrecision], If[LessEqual[b, 1.8e+157], N[(a * a + N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(3.08641975308642e-5 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(3.08641975308642e-5 * angle), $MachinePrecision] * angle), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.5000000000000001e-62

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   if 1.5000000000000001e-62 < b < 1.80000000000000012e157

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in a around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 64.4

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 64.5

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

   10. Applied rewrites 64.5%

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

   if 1.80000000000000012e157 < b

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 34.7

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

   10. Applied rewrites 34.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 36.9

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 36.9

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

   12. Applied rewrites 36.9%

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

Alternative 7: 57.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+157}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot angle\right) \cdot \left(\pi \cdot b\right)\right) \cdot \left(\pi \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.8e+157)
   (* a a)
   (* (* (* (* 3.08641975308642e-5 angle) angle) (* PI b)) (* PI b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+157) {
		tmp = a * a;
	} else {
		tmp = (((3.08641975308642e-5 * angle) * angle) * (((double) M_PI) * b)) * (((double) M_PI) * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+157) {
		tmp = a * a;
	} else {
		tmp = (((3.08641975308642e-5 * angle) * angle) * (Math.PI * b)) * (Math.PI * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.8e+157:
		tmp = a * a
	else:
		tmp = (((3.08641975308642e-5 * angle) * angle) * (math.pi * b)) * (math.pi * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.8e+157)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(Float64(Float64(3.08641975308642e-5 * angle) * angle) * Float64(pi * b)) * Float64(pi * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.8e+157)
		tmp = a * a;
	else
		tmp = (((3.08641975308642e-5 * angle) * angle) * (pi * b)) * (pi * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.8e+157], N[(a * a), $MachinePrecision], N[(N[(N[(N[(3.08641975308642e-5 * angle), $MachinePrecision] * angle), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.80000000000000012e157

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   if 1.80000000000000012e157 < b

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 34.7

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

   10. Applied rewrites 34.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 36.9

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 36.9

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

   12. Applied rewrites 36.9%

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

Alternative 8: 56.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{-124}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.8e-124)
   (* (* 3.08641975308642e-5 (* angle angle)) (* (* PI PI) (* b b)))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.8e-124) {
		tmp = (3.08641975308642e-5 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * (b * b));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.8e-124) {
		tmp = (3.08641975308642e-5 * (angle * angle)) * ((Math.PI * Math.PI) * (b * b));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.8e-124:
		tmp = (3.08641975308642e-5 * (angle * angle)) * ((math.pi * math.pi) * (b * b))
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.8e-124)
		tmp = Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(pi * pi) * Float64(b * b)));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.8e-124)
		tmp = (3.08641975308642e-5 * (angle * angle)) * ((pi * pi) * (b * b));
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.8e-124], N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.80000000000000012e-124

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 34.7

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

   10. Applied rewrites 34.7%

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

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. pow-prod-down N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lift-PI.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 34.7

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

   12. Applied rewrites 34.7%

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

   if 3.80000000000000012e-124 < a

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

Alternative 9: 48.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.8 \cdot 10^{-124}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\pi \cdot \left(b \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.8e-124)
   (* (* 3.08641975308642e-5 (* angle angle)) (* PI (* b (* PI b))))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.8e-124) {
		tmp = (3.08641975308642e-5 * (angle * angle)) * (((double) M_PI) * (b * (((double) M_PI) * b)));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.8e-124) {
		tmp = (3.08641975308642e-5 * (angle * angle)) * (Math.PI * (b * (Math.PI * b)));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 3.8e-124:
		tmp = (3.08641975308642e-5 * (angle * angle)) * (math.pi * (b * (math.pi * b)))
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.8e-124)
		tmp = Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(pi * Float64(b * Float64(pi * b))));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 3.8e-124)
		tmp = (3.08641975308642e-5 * (angle * angle)) * (pi * (b * (pi * b)));
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.8e-124], N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.80000000000000012e-124

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-prod-down N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 34.7

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

   10. Applied rewrites 34.7%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lower-*.f64 34.7

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

   12. Applied rewrites 34.7%

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

   if 3.80000000000000012e-124 < a

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

Alternative 10: 48.3% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.4

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

4. Applied rewrites 57.4%

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

Reproduce

herbie shell --seed 2025132 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))