ab-angle->ABCF A

Percentage Accurate: 79.2% → 79.2%

Time: 4.8s

Alternatives: 9

Speedup: 1.0×

• 64.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.2% accurate, 1.0× speedup

Math

\[{\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{fma}\left(-0.005555555555555556, angle, 0.5\right)\right)\right)}^{2} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.2%

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

5. Applied rewrites 79.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. mult-flip N/A

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

   12. lift-/.f64 N/A

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

   13. distribute-rgt-neg-out N/A

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

   14. lift-/.f64 N/A

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

   15. distribute-neg-frac N/A

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

   16. lift-PI.f64 N/A

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

   17. mult-flip N/A

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

   18. metadata-eval N/A

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

   19. distribute-lft-out N/A

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

   20. lower-*.f64 N/A

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

7. Applied rewrites 79.2%

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

Alternative 2: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(0.005555555555555556 \cdot \pi\right) \cdot angle\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 PI) angle)))
   (+ (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 = (0.005555555555555556 * ((double) M_PI)) * angle;
	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 = (0.005555555555555556 * Math.PI) * angle;
	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 = (0.005555555555555556 * math.pi) * angle
	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(0.005555555555555556 * pi) * angle)
	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 = (0.005555555555555556 * pi) * angle;
	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[(0.005555555555555556 * Pi), $MachinePrecision] * angle), $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]]

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.2%

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

5. Applied rewrites 79.2%

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

Alternative 3: 79.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.2%

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

3. Applied rewrites 79.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 79.2%

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in angle around 0

   \[\leadsto {\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
7. Step-by-step derivation

8. Applied rewrites 79.1%

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

Alternative 4: 67.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 75000:\\ \;\;\;\;{b}^{2} \cdot {\sin \left(\mathsf{fma}\left(0.005555555555555556, \left|angle\right| \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, \frac{2}{2} \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs angle) 75000.0)
   (*
    (pow b 2.0)
    (pow (sin (fma 0.005555555555555556 (* (fabs angle) PI) (* 0.5 PI))) 2.0))
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* (fabs angle) (* PI 0.005555555555555556))))))
    (* a a)
    (* (/ 2.0 2.0) (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 75000.0) {
		tmp = pow(b, 2.0) * pow(sin(fma(0.005555555555555556, (fabs(angle) * ((double) M_PI)), (0.5 * ((double) M_PI)))), 2.0);
	} else {
		tmp = fma((0.5 - (0.5 * cos((2.0 * (fabs(angle) * (((double) M_PI) * 0.005555555555555556)))))), (a * a), ((2.0 / 2.0) * (b * b)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 75000.0)
		tmp = Float64((b ^ 2.0) * (sin(fma(0.005555555555555556, Float64(abs(angle) * pi), Float64(0.5 * pi))) ^ 2.0));
	else
		tmp = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(abs(angle) * Float64(pi * 0.005555555555555556)))))), Float64(a * a), Float64(Float64(2.0 / 2.0) * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[angle], $MachinePrecision], 75000.0], N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[N[Sin[N[(0.005555555555555556 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(2.0 / 2.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 75000

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   3. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   5. Applied rewrites 79.2%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. mult-flip N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. lift-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-PI.f64 N/A

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

      17. mult-flip N/A

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

      18. metadata-eval N/A

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

      19. distribute-lft-out N/A

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

      20. lower-*.f64 N/A

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

   7. Applied rewrites 79.2%

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

   9. Applied rewrites 79.2%

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

   10. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-PI.f64 56.7%

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

   12. Applied rewrites 56.7%

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

   if 75000 < angle

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   3. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, \color{blue}{\frac{\cos \left(\mathsf{fma}\left(-0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \cos \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{2}} \cdot \left(b \cdot b\right)\right) \]

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 63.2%

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

Alternative 5: 67.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 75000:\\ \;\;\;\;{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(\left|angle\right| \cdot \pi\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, \frac{2}{2} \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs angle) 75000.0)
   (* (pow b 2.0) (pow (cos (* 0.005555555555555556 (* (fabs angle) PI))) 2.0))
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* (fabs angle) (* PI 0.005555555555555556))))))
    (* a a)
    (* (/ 2.0 2.0) (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 75000.0) {
		tmp = pow(b, 2.0) * pow(cos((0.005555555555555556 * (fabs(angle) * ((double) M_PI)))), 2.0);
	} else {
		tmp = fma((0.5 - (0.5 * cos((2.0 * (fabs(angle) * (((double) M_PI) * 0.005555555555555556)))))), (a * a), ((2.0 / 2.0) * (b * b)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 75000.0)
		tmp = Float64((b ^ 2.0) * (cos(Float64(0.005555555555555556 * Float64(abs(angle) * pi))) ^ 2.0));
	else
		tmp = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(abs(angle) * Float64(pi * 0.005555555555555556)))))), Float64(a * a), Float64(Float64(2.0 / 2.0) * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[angle], $MachinePrecision], 75000.0], N[(N[Power[b, 2.0], $MachinePrecision] * N[Power[N[Cos[N[(0.005555555555555556 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(2.0 / 2.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 75000

   1. Initial program 79.2%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 56.7%

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

   4. Applied rewrites 56.7%

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

   if 75000 < angle

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   3. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, \color{blue}{\frac{\cos \left(\mathsf{fma}\left(-0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \cos \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{2}} \cdot \left(b \cdot b\right)\right) \]

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 63.2%

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

Alternative 6: 67.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 75000:\\ \;\;\;\;{b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(\left|angle\right| \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left|angle\right| \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, \frac{2}{2} \cdot \left(b \cdot b\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs angle) 75000.0)
   (*
    (pow b 2.0)
    (+ 0.5 (* 0.5 (cos (* -0.011111111111111112 (* (fabs angle) PI))))))
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* (fabs angle) (* PI 0.005555555555555556))))))
    (* a a)
    (* (/ 2.0 2.0) (* b b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 75000.0) {
		tmp = pow(b, 2.0) * (0.5 + (0.5 * cos((-0.011111111111111112 * (fabs(angle) * ((double) M_PI))))));
	} else {
		tmp = fma((0.5 - (0.5 * cos((2.0 * (fabs(angle) * (((double) M_PI) * 0.005555555555555556)))))), (a * a), ((2.0 / 2.0) * (b * b)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 75000.0)
		tmp = Float64((b ^ 2.0) * Float64(0.5 + Float64(0.5 * cos(Float64(-0.011111111111111112 * Float64(abs(angle) * pi))))));
	else
		tmp = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(abs(angle) * Float64(pi * 0.005555555555555556)))))), Float64(a * a), Float64(Float64(2.0 / 2.0) * Float64(b * b)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[angle], $MachinePrecision], 75000.0], N[(N[Power[b, 2.0], $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(-0.011111111111111112 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[Abs[angle], $MachinePrecision] * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(2.0 / 2.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 75000

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   3. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 56.7%

         \[\leadsto {b}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 56.7%

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

   if 75000 < angle

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   3. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 63.2%

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

   9. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, \color{blue}{\frac{\cos \left(\mathsf{fma}\left(-0.005555555555555556 \cdot angle, \pi, \left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right) + \cos \left(\left(-0.011111111111111112 \cdot angle\right) \cdot \pi\right)}{2}} \cdot \left(b \cdot b\right)\right) \]

   10. Taylor expanded in angle around 0

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

   12. Applied rewrites 63.2%

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

Alternative 7: 60.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 1.4 \cdot 10^{-114}:\\ \;\;\;\;{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left|b\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (fabs b) 1.4e-114)
   (* (pow a 2.0) (- 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
   (* (fabs b) (fabs b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 1.4e-114) {
		tmp = pow(a, 2.0) * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))));
	} else {
		tmp = fabs(b) * fabs(b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 1.4e-114) {
		tmp = Math.pow(a, 2.0) * (0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)))));
	} else {
		tmp = Math.abs(b) * Math.abs(b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 1.4e-114:
		tmp = math.pow(a, 2.0) * (0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))))
	else:
		tmp = math.fabs(b) * math.fabs(b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 1.4e-114)
		tmp = Float64((a ^ 2.0) * Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))));
	else
		tmp = Float64(abs(b) * abs(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 1.4e-114)
		tmp = (a ^ 2.0) * (0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi)))));
	else
		tmp = abs(b) * abs(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 1.4e-114], N[(N[Power[a, 2.0], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4000000000000001e-114

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   3. Applied rewrites 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 79.2%

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-PI.f64 26.0%

         \[\leadsto {a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 26.0%

       \[\leadsto \color{blue}{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   if 1.4000000000000001e-114 < b

   1. Initial program 79.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 56.8%

         \[\leadsto {b}^{\color{blue}{2}} \]

   4. Applied rewrites 56.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto {b}^{\color{blue}{2}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 56.8%

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

   6. Applied rewrites 56.8%

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

Alternative 8: 59.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(b \cdot b\right) \cdot b\right) \cdot b\\ t_1 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;{\left(a \cdot \sin t\_1\right)}^{2} + {\left(b \cdot \cos t\_1\right)}^{2} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{t\_0 \cdot t\_0}}\\ \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* (* b b) b) b)) (t_1 (* (/ angle 180.0) PI)))
   (if (<= (+ (pow (* a (sin t_1)) 2.0) (pow (* b (cos t_1)) 2.0)) 2e+303)
     (* b b)
     (sqrt (sqrt (* t_0 t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((b * b) * b) * b;
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double tmp;
	if ((pow((a * sin(t_1)), 2.0) + pow((b * cos(t_1)), 2.0)) <= 2e+303) {
		tmp = b * b;
	} else {
		tmp = sqrt(sqrt((t_0 * t_0)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = ((b * b) * b) * b;
	double t_1 = (angle / 180.0) * Math.PI;
	double tmp;
	if ((Math.pow((a * Math.sin(t_1)), 2.0) + Math.pow((b * Math.cos(t_1)), 2.0)) <= 2e+303) {
		tmp = b * b;
	} else {
		tmp = Math.sqrt(Math.sqrt((t_0 * t_0)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = ((b * b) * b) * b
	t_1 = (angle / 180.0) * math.pi
	tmp = 0
	if (math.pow((a * math.sin(t_1)), 2.0) + math.pow((b * math.cos(t_1)), 2.0)) <= 2e+303:
		tmp = b * b
	else:
		tmp = math.sqrt(math.sqrt((t_0 * t_0)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(Float64(b * b) * b) * b)
	t_1 = Float64(Float64(angle / 180.0) * pi)
	tmp = 0.0
	if (Float64((Float64(a * sin(t_1)) ^ 2.0) + (Float64(b * cos(t_1)) ^ 2.0)) <= 2e+303)
		tmp = Float64(b * b);
	else
		tmp = sqrt(sqrt(Float64(t_0 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = ((b * b) * b) * b;
	t_1 = (angle / 180.0) * pi;
	tmp = 0.0;
	if ((((a * sin(t_1)) ^ 2.0) + ((b * cos(t_1)) ^ 2.0)) <= 2e+303)
		tmp = b * b;
	else
		tmp = sqrt(sqrt((t_0 * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(a * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e+303], N[(b * b), $MachinePrecision], N[Sqrt[N[Sqrt[N[(t$95$0 * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) < 2e303

   1. Initial program 79.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 56.8%

         \[\leadsto {b}^{\color{blue}{2}} \]

   4. Applied rewrites 56.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto {b}^{\color{blue}{2}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 56.8%

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

   6. Applied rewrites 56.8%

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

   if 2e303 < (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)))

   1. Initial program 79.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 56.8%

         \[\leadsto {b}^{\color{blue}{2}} \]

   4. Applied rewrites 56.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto {b}^{\color{blue}{2}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 56.8%

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

   6. Applied rewrites 56.8%

      \[\leadsto \color{blue}{b \cdot b} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]

      4. lower-*.f64 48.8%

         \[\leadsto \sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]

   8. Applied rewrites 48.8%

      \[\leadsto \sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \]
   9. Step-by-step derivation

      1. rem-square-sqrt N/A

         \[\leadsto \sqrt{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \cdot \sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      4. lower-*.f64 44.4%

         \[\leadsto \sqrt{\sqrt{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\sqrt{\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{\sqrt{\left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      10. *-commutative N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      11. lower-*.f64 44.4%

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      12. lift-*.f64 N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}} \]

      14. associate-*l* N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}} \]

      15. *-commutative N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b\right)}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot \left(b \cdot b\right)\right) \cdot b\right)}} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)}} \]

      18. lower-*.f64 44.4%

          \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)}} \]

   10. Applied rewrites 44.4%

       \[\leadsto \sqrt{\sqrt{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right) \cdot \left(\left(\left(b \cdot b\right) \cdot b\right) \cdot b\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.8% accurate, 29.7× speedup

Math

\[b \cdot b \]

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 79.2%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 56.8%

      \[\leadsto {b}^{\color{blue}{2}} \]

4. Applied rewrites 56.8%

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

   1. lift-pow.f64 N/A

      \[\leadsto {b}^{\color{blue}{2}} \]

   2. unpow2 N/A

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

   3. lower-*.f64 56.8%

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

6. Applied rewrites 56.8%

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

Reproduce

herbie shell --seed 2025189 
(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)))