ab-angle->ABCF A

Percentage Accurate: 80.0% → 80.0%

Time: 5.8s

Alternatives: 11

Speedup: 1.0×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 78.6

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

4. Applied rewrites 78.6%

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

Alternative 2: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 78.6

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

4. Applied rewrites 78.6%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 78.5

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

7. Applied rewrites 78.5%

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

Alternative 3: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 78.6

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

4. Applied rewrites 78.6%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 78.5

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

7. Applied rewrites 78.5%

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

8. Taylor expanded in angle around 0

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

   1. lift-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 78.5

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

10. Applied rewrites 78.5%

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

11. Final simplification 78.5%

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

Alternative 4: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

   1. lower-*.f64 78.5

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

5. Applied rewrites 78.5%

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

Alternative 5: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-PI.f64 78.6

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

4. Applied rewrites 78.6%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 78.5

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

7. Applied rewrites 78.5%

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

8. Taylor expanded in angle around 0

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

   1. lift-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 78.5

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

10. Applied rewrites 78.5%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-pow.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. *-commutative N/A

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

    6. unpow-prod-down N/A

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

    7. lower-fma.f64 N/A

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

12. Applied rewrites 78.4%

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

Alternative 6: 80.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.4

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

5. Applied rewrites 78.4%

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

6. Final simplification 78.4%

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

Alternative 7: 58.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-162}:\\ \;\;\;\;{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot a\right) \cdot \pi\right) \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8e-162)
   (pow (* (sin (* (* PI angle) 0.005555555555555556)) a) 2.0)
   (+
    (* (* (* (* angle a) PI) (* angle (* PI a))) 3.08641975308642e-5)
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8e-162) {
		tmp = pow((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * a), 2.0);
	} else {
		tmp = ((((angle * a) * ((double) M_PI)) * (angle * (((double) M_PI) * a))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8e-162) {
		tmp = Math.pow((Math.sin(((Math.PI * angle) * 0.005555555555555556)) * a), 2.0);
	} else {
		tmp = ((((angle * a) * Math.PI) * (angle * (Math.PI * a))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 8e-162:
		tmp = math.pow((math.sin(((math.pi * angle) * 0.005555555555555556)) * a), 2.0)
	else:
		tmp = ((((angle * a) * math.pi) * (angle * (math.pi * a))) * 3.08641975308642e-5) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8e-162)
		tmp = Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * a) ^ 2.0;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(angle * a) * pi) * Float64(angle * Float64(pi * a))) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 8e-162)
		tmp = (sin(((pi * angle) * 0.005555555555555556)) * a) ^ 2.0;
	else
		tmp = ((((angle * a) * pi) * (angle * (pi * a))) * 3.08641975308642e-5) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8e-162], N[Power[N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(N[(N[(angle * a), $MachinePrecision] * Pi), $MachinePrecision] * N[(angle * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.99999999999999963e-162

   1. Initial program 79.7%

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

   3. Taylor expanded in a around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 44.5

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

   5. Applied rewrites 44.5%

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

   if 7.99999999999999963e-162 < b

   1. Initial program 76.2%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in angle around 0

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

      1. unpow-prod-down N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 73.9

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

   8. Applied rewrites 73.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 74.0

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

   10. Applied rewrites 74.0%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 74.0

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

   12. Applied rewrites 74.0%

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

Alternative 8: 67.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{-126}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot a\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.6e-126)
   (* b b)
   (+
    (* (* (* (* angle PI) a) (* (* angle a) PI)) 3.08641975308642e-5)
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.6e-126) {
		tmp = b * b;
	} else {
		tmp = ((((angle * ((double) M_PI)) * a) * ((angle * a) * ((double) M_PI))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.6e-126) {
		tmp = b * b;
	} else {
		tmp = ((((angle * Math.PI) * a) * ((angle * a) * Math.PI)) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 4.6e-126:
		tmp = b * b
	else:
		tmp = ((((angle * math.pi) * a) * ((angle * a) * math.pi)) * 3.08641975308642e-5) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.6e-126)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(angle * pi) * a) * Float64(Float64(angle * a) * pi)) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.6e-126)
		tmp = b * b;
	else
		tmp = ((((angle * pi) * a) * ((angle * a) * pi)) * 3.08641975308642e-5) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 4.6e-126], N[(b * b), $MachinePrecision], N[(N[(N[(N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision] * N[(N[(angle * a), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.60000000000000021e-126

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.7

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

   5. Applied rewrites 59.7%

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

   if 4.60000000000000021e-126 < a

   1. Initial program 84.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in angle around 0

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

      1. unpow-prod-down N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 82.9

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

   8. Applied rewrites 82.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 82.9

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

   10. Applied rewrites 82.9%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-*.f64 82.9

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

       9. lift-PI.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. lift-PI.f64 82.9

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

   12. Applied rewrites 82.9%

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

Alternative 9: 67.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{-126}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot a\right) \cdot \pi\right) \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.6e-126)
   (* b b)
   (+
    (* (* (* (* angle a) PI) (* angle (* PI a))) 3.08641975308642e-5)
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.6e-126) {
		tmp = b * b;
	} else {
		tmp = ((((angle * a) * ((double) M_PI)) * (angle * (((double) M_PI) * a))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.6e-126) {
		tmp = b * b;
	} else {
		tmp = ((((angle * a) * Math.PI) * (angle * (Math.PI * a))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 4.6e-126:
		tmp = b * b
	else:
		tmp = ((((angle * a) * math.pi) * (angle * (math.pi * a))) * 3.08641975308642e-5) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.6e-126)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(angle * a) * pi) * Float64(angle * Float64(pi * a))) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.6e-126)
		tmp = b * b;
	else
		tmp = ((((angle * a) * pi) * (angle * (pi * a))) * 3.08641975308642e-5) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 4.6e-126], N[(b * b), $MachinePrecision], N[(N[(N[(N[(N[(angle * a), $MachinePrecision] * Pi), $MachinePrecision] * N[(angle * N[(Pi * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.60000000000000021e-126

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.7

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

   5. Applied rewrites 59.7%

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

   if 4.60000000000000021e-126 < a

   1. Initial program 84.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in angle around 0

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

      1. unpow-prod-down N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 82.9

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

   8. Applied rewrites 82.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 82.9

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

   10. Applied rewrites 82.9%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 82.9

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

   12. Applied rewrites 82.9%

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

Alternative 10: 66.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.6 \cdot 10^{-126}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \pi\right) \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.6e-126)
   (* b b)
   (+
    (* (* (* angle PI) (* a (* (* angle PI) a))) 3.08641975308642e-5)
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.6e-126) {
		tmp = b * b;
	} else {
		tmp = (((angle * ((double) M_PI)) * (a * ((angle * ((double) M_PI)) * a))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.6e-126) {
		tmp = b * b;
	} else {
		tmp = (((angle * Math.PI) * (a * ((angle * Math.PI) * a))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 4.6e-126:
		tmp = b * b
	else:
		tmp = (((angle * math.pi) * (a * ((angle * math.pi) * a))) * 3.08641975308642e-5) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.6e-126)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(Float64(angle * pi) * Float64(a * Float64(Float64(angle * pi) * a))) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.6e-126)
		tmp = b * b;
	else
		tmp = (((angle * pi) * (a * ((angle * pi) * a))) * 3.08641975308642e-5) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 4.6e-126], N[(b * b), $MachinePrecision], N[(N[(N[(N[(angle * Pi), $MachinePrecision] * N[(a * N[(N[(angle * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.60000000000000021e-126

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 59.7

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

   5. Applied rewrites 59.7%

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

   if 4.60000000000000021e-126 < a

   1. Initial program 84.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in angle around 0

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

      1. unpow-prod-down N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. *-commutative N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 82.9

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

   8. Applied rewrites 82.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 82.9

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

   10. Applied rewrites 82.9%

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

       1. lift-*.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. lift-*.f64 N/A

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

       9. lift-PI.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. associate-*l* N/A

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

       14. lower-*.f64 N/A

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

       15. lift-PI.f64 N/A

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lower-*.f64 N/A

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

       19. lift-PI.f64 N/A

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

       20. lift-*.f64 N/A

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

       21. lift-PI.f64 N/A

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

       22. lift-*.f64 N/A

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

       23. associate-*l* N/A

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

       24. *-commutative N/A

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

       25. lift-*.f64 N/A

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

   12. Applied rewrites 80.6%

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

Alternative 11: 57.7% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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 = b * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

   \[\leadsto \color{blue}{b \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025072 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))