ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.6%

Time: 14.7s

Alternatives: 8

Speedup: 1.0×

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (* PI (* angle_m 0.005555555555555556)))) 2.0)
  (pow
   (* b (sin (* PI (expm1 (log1p (* angle_m 0.005555555555555556))))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos((((double) M_PI) * (angle_m * 0.005555555555555556)))), 2.0) + pow((b * sin((((double) M_PI) * expm1(log1p((angle_m * 0.005555555555555556)))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.cos((Math.PI * (angle_m * 0.005555555555555556)))), 2.0) + Math.pow((b * Math.sin((Math.PI * Math.expm1(Math.log1p((angle_m * 0.005555555555555556)))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.cos((math.pi * (angle_m * 0.005555555555555556)))), 2.0) + math.pow((b * math.sin((math.pi * math.expm1(math.log1p((angle_m * 0.005555555555555556)))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0) + (Float64(b * sin(Float64(pi * expm1(log1p(Float64(angle_m * 0.005555555555555556)))))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(Exp[N[Log[1 + N[(angle$95$m * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. associate-*r/ 79.8%

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

   2. metadata-eval 79.8%

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

   3. metadata-eval 79.8%

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

   4. distribute-neg-frac2 79.8%

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

   5. distribute-frac-neg 79.8%

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

   6. distribute-rgt-neg-out 79.8%

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

   7. associate-/l* 79.8%

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

   8. neg-mul-1 79.8%

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

   9. *-commutative 79.8%

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

   10. associate-/l* 79.9%

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

   11. metadata-eval 79.9%

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

   12. metadata-eval 79.9%

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

3. Simplified 80.0%

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

   1. expm1-log1p-u 65.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} \]

6. Applied egg-rr 65.6%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 2: 79.6% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* PI angle_m)))))
   (+
    (* t_0 (* a (* a t_0)))
    (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = cos((0.005555555555555556 * (((double) M_PI) * angle_m)));
	return (t_0 * (a * (a * t_0))) + pow((b * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.cos((0.005555555555555556 * (Math.PI * angle_m)));
	return (t_0 * (a * (a * t_0))) + Math.pow((b * Math.sin((Math.PI * (angle_m * 0.005555555555555556)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.cos((0.005555555555555556 * (math.pi * angle_m)))
	return (t_0 * (a * (a * t_0))) + math.pow((b * math.sin((math.pi * (angle_m * 0.005555555555555556)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = cos(Float64(0.005555555555555556 * Float64(pi * angle_m)))
	return Float64(Float64(t_0 * Float64(a * Float64(a * t_0))) + (Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = cos((0.005555555555555556 * (pi * angle_m)));
	tmp = (t_0 * (a * (a * t_0))) + ((b * sin((pi * (angle_m * 0.005555555555555556)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[(a * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.8%

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

   1. associate-*r/ 79.8%

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

   2. metadata-eval 79.8%

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

   3. metadata-eval 79.8%

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

   4. distribute-neg-frac2 79.8%

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

   5. distribute-frac-neg 79.8%

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

   6. distribute-rgt-neg-out 79.8%

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

   7. associate-/l* 79.8%

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

   8. neg-mul-1 79.8%

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

   9. *-commutative 79.8%

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

   10. associate-/l* 79.9%

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

   11. metadata-eval 79.9%

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

   12. metadata-eval 79.9%

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

3. Simplified 80.0%

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

   1. metadata-eval 80.0%

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

   2. div-inv 80.0%

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

   3. unpow2 80.0%

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

   4. associate-*r* 80.0%

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

   5. div-inv 77.9%

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

   6. metadata-eval 77.9%

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

   7. associate-*r* 77.4%

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

   8. *-commutative 77.4%

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

   9. div-inv 78.6%

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

   10. metadata-eval 78.6%

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

   11. associate-*r* 80.0%

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

   12. *-commutative 80.0%

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

6. Applied egg-rr 80.0%

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

7. Final simplification 80.0%

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

Alternative 3: 79.6% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (pow (hypot (* b (sin t_0)) (* a (cos t_0))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	return pow(hypot((b * sin(t_0)), (a * cos(t_0))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	return Math.pow(Math.hypot((b * Math.sin(t_0)), (a * Math.cos(t_0))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	return math.pow(math.hypot((b * math.sin(t_0)), (a * math.cos(t_0))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	return hypot(Float64(b * sin(t_0)), Float64(a * cos(t_0))) ^ 2.0
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = pi * (angle_m * 0.005555555555555556);
	tmp = hypot((b * sin(t_0)), (a * cos(t_0))) ^ 2.0;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[Power[N[Sqrt[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Initial program 79.8%

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

   1. associate-*r/ 79.8%

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

   2. metadata-eval 79.8%

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

   3. metadata-eval 79.8%

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

   4. distribute-neg-frac2 79.8%

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

   5. distribute-frac-neg 79.8%

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

   6. distribute-rgt-neg-out 79.8%

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

   7. associate-/l* 79.8%

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

   8. neg-mul-1 79.8%

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

   9. *-commutative 79.8%

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

   10. associate-/l* 79.9%

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

   11. metadata-eval 79.9%

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

   12. metadata-eval 79.9%

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

3. Simplified 80.0%

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

   1. expm1-log1p-u 65.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} \]

6. Applied egg-rr 65.6%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. *-commutative 65.6%

      \[\leadsto {\color{blue}{\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot a\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

   2. associate-*r* 65.6%

      \[\leadsto {\left(\cos \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot a\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

   3. *-commutative 65.6%

      \[\leadsto {\left(\cos \color{blue}{\left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)} \cdot a\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

   4. *-commutative 65.6%

      \[\leadsto {\color{blue}{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

   5. expm1-log1p-u 45.4%

      \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

   6. unpow2 45.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)} + {\left(b \cdot \sin \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} \]

   7. *-commutative 45.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) + {\color{blue}{\left(\sin \left(\pi \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)\right) \cdot b\right)}}^{2} \]

   8. expm1-log1p-u 56.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) + {\left(\sin \left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right) \cdot b\right)}^{2} \]

   9. *-commutative 56.2%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) + {\color{blue}{\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}}^{2} \]

8. Applied egg-rr 80.0%

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

Alternative 4: 79.4% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0) (pow a 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. associate-*r/ 79.8%

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

   2. metadata-eval 79.8%

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

   3. metadata-eval 79.8%

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

   4. distribute-neg-frac2 79.8%

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

   5. distribute-frac-neg 79.8%

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

   6. distribute-rgt-neg-out 79.8%

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

   7. associate-/l* 79.8%

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

   8. neg-mul-1 79.8%

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

   9. *-commutative 79.8%

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

   10. associate-/l* 79.9%

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

   11. metadata-eval 79.9%

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

   12. metadata-eval 79.9%

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

3. Simplified 80.0%

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

5. Taylor expanded in angle around 0 79.8%

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

6. Final simplification 79.8%

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

Alternative 5: 62.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+130}:\\ \;\;\;\;{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.3e+130)
   (pow (* a (cos (* 0.005555555555555556 (* PI angle_m)))) 2.0)
   (pow (* b (sin (* angle_m (* PI 0.005555555555555556)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.3e+130) {
		tmp = pow((a * cos((0.005555555555555556 * (((double) M_PI) * angle_m)))), 2.0);
	} else {
		tmp = pow((b * sin((angle_m * (((double) M_PI) * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.3e+130) {
		tmp = Math.pow((a * Math.cos((0.005555555555555556 * (Math.PI * angle_m)))), 2.0);
	} else {
		tmp = Math.pow((b * Math.sin((angle_m * (Math.PI * 0.005555555555555556)))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 1.3e+130:
		tmp = math.pow((a * math.cos((0.005555555555555556 * (math.pi * angle_m)))), 2.0)
	else:
		tmp = math.pow((b * math.sin((angle_m * (math.pi * 0.005555555555555556)))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.3e+130)
		tmp = Float64(a * cos(Float64(0.005555555555555556 * Float64(pi * angle_m)))) ^ 2.0;
	else
		tmp = Float64(b * sin(Float64(angle_m * Float64(pi * 0.005555555555555556)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.3e+130)
		tmp = (a * cos((0.005555555555555556 * (pi * angle_m)))) ^ 2.0;
	else
		tmp = (b * sin((angle_m * (pi * 0.005555555555555556)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.3e+130], N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(b * N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2999999999999999e130

   1. Initial program 77.2%

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

      1. associate-*r/ 77.2%

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

      2. metadata-eval 77.2%

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

      3. metadata-eval 77.2%

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

      4. distribute-neg-frac2 77.2%

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

      5. distribute-frac-neg 77.2%

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

      6. distribute-rgt-neg-out 77.2%

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

      7. associate-/l* 77.2%

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

      8. neg-mul-1 77.2%

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

      9. *-commutative 77.2%

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

      10. associate-/l* 77.2%

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

      11. metadata-eval 77.2%

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

      12. metadata-eval 77.2%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in a around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

      3. unpow2 63.5%

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

      4. swap-sqr 63.5%

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

      5. unpow2 63.5%

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

      6. *-commutative 63.5%

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

   7. Simplified 63.5%

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

   if 1.2999999999999999e130 < b

   1. Initial program 94.9%

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

      1. associate-*r/ 94.9%

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

      2. metadata-eval 94.9%

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

      3. metadata-eval 94.9%

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

      4. distribute-neg-frac2 94.9%

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

      5. distribute-frac-neg 94.9%

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

      6. distribute-rgt-neg-out 94.9%

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

      7. associate-/l* 94.9%

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

      8. neg-mul-1 94.9%

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

      9. *-commutative 94.9%

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

      10. associate-/l* 94.9%

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

      11. metadata-eval 94.9%

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

      12. metadata-eval 94.9%

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

   3. Simplified 95.0%

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

      1. expm1-log1p-u 73.3%

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} \]

   6. Applied egg-rr 73.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot 0.005555555555555556\right)\right)}\right)\right)}^{2} \]

   7. Taylor expanded in a around 0 77.2%

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

      1. unpow2 77.2%

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

      2. associate-*r* 77.2%

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

      3. *-commutative 77.2%

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

      4. *-commutative 77.2%

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

      5. unpow2 77.2%

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

      6. swap-sqr 82.4%

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

      7. unpow2 82.4%

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

      8. *-commutative 82.4%

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

      9. associate-*l* 82.4%

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

   9. Simplified 82.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+130}:\\ \;\;\;\;{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\\ \mathbf{if}\;b \leq 3.4 \cdot 10^{+128}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \sin t\_0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle_m))))
   (if (<= b 3.4e+128) (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle_m);
	double tmp;
	if (b <= 3.4e+128) {
		tmp = pow((a * cos(t_0)), 2.0);
	} else {
		tmp = pow((b * sin(t_0)), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle_m);
	double tmp;
	if (b <= 3.4e+128) {
		tmp = Math.pow((a * Math.cos(t_0)), 2.0);
	} else {
		tmp = Math.pow((b * Math.sin(t_0)), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = 0.005555555555555556 * (math.pi * angle_m)
	tmp = 0
	if b <= 3.4e+128:
		tmp = math.pow((a * math.cos(t_0)), 2.0)
	else:
		tmp = math.pow((b * math.sin(t_0)), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle_m))
	tmp = 0.0
	if (b <= 3.4e+128)
		tmp = Float64(a * cos(t_0)) ^ 2.0;
	else
		tmp = Float64(b * sin(t_0)) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = 0.005555555555555556 * (pi * angle_m);
	tmp = 0.0;
	if (b <= 3.4e+128)
		tmp = (a * cos(t_0)) ^ 2.0;
	else
		tmp = (b * sin(t_0)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3.4e+128], N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.3999999999999999e128

   1. Initial program 77.2%

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

      1. associate-*r/ 77.2%

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

      2. metadata-eval 77.2%

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

      3. metadata-eval 77.2%

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

      4. distribute-neg-frac2 77.2%

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

      5. distribute-frac-neg 77.2%

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

      6. distribute-rgt-neg-out 77.2%

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

      7. associate-/l* 77.2%

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

      8. neg-mul-1 77.2%

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

      9. *-commutative 77.2%

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

      10. associate-/l* 77.2%

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

      11. metadata-eval 77.2%

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

      12. metadata-eval 77.2%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in a around inf 63.5%

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

      1. *-commutative 63.5%

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

      2. unpow2 63.5%

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

      3. unpow2 63.5%

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

      4. swap-sqr 63.5%

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

      5. unpow2 63.5%

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

      6. *-commutative 63.5%

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

   7. Simplified 63.5%

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

   if 3.3999999999999999e128 < b

   1. Initial program 94.9%

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

      1. associate-*r/ 94.9%

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

      2. metadata-eval 94.9%

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

      3. metadata-eval 94.9%

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

      4. distribute-neg-frac2 94.9%

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

      5. distribute-frac-neg 94.9%

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

      6. distribute-rgt-neg-out 94.9%

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

      7. associate-/l* 94.9%

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

      8. neg-mul-1 94.9%

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

      9. *-commutative 94.9%

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

      10. associate-/l* 94.9%

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

      11. metadata-eval 94.9%

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

      12. metadata-eval 94.9%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in a around 0 77.2%

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

      1. unpow2 77.2%

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

      2. *-commutative 77.2%

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

      3. unpow2 77.2%

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

      4. swap-sqr 82.4%

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

      5. unpow2 82.4%

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

      6. *-commutative 82.4%

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

   7. Simplified 82.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+128}:\\ \;\;\;\;{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (pow (* a (cos (* 0.005555555555555556 (* PI angle_m)))) 2.0))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos((0.005555555555555556 * (((double) M_PI) * angle_m)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.cos((0.005555555555555556 * (Math.PI * angle_m)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.cos((0.005555555555555556 * (math.pi * angle_m)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(a * cos(Float64(0.005555555555555556 * Float64(pi * angle_m)))) ^ 2.0
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a * cos((0.005555555555555556 * (pi * angle_m)))) ^ 2.0;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. associate-*r/ 79.8%

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

   2. metadata-eval 79.8%

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

   3. metadata-eval 79.8%

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

   4. distribute-neg-frac2 79.8%

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

   5. distribute-frac-neg 79.8%

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

   6. distribute-rgt-neg-out 79.8%

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

   7. associate-/l* 79.8%

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

   8. neg-mul-1 79.8%

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

   9. *-commutative 79.8%

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

   10. associate-/l* 79.9%

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

   11. metadata-eval 79.9%

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

   12. metadata-eval 79.9%

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

3. Simplified 80.0%

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

5. Taylor expanded in a around inf 59.4%

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

   1. *-commutative 59.4%

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

   2. unpow2 59.4%

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

   3. unpow2 59.4%

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

   4. swap-sqr 59.4%

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

   5. unpow2 59.4%

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

   6. *-commutative 59.4%

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

7. Simplified 59.4%

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

8. Final simplification 59.4%

   \[\leadsto {\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 8: 56.5% accurate, 139.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* a a))

C

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

Fortran

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

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return a * a;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return a * a

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. associate-*r/ 79.8%

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

   2. metadata-eval 79.8%

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

   3. metadata-eval 79.8%

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

   4. distribute-neg-frac2 79.8%

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

   5. distribute-frac-neg 79.8%

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

   6. distribute-rgt-neg-out 79.8%

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

   7. associate-/l* 79.8%

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

   8. neg-mul-1 79.8%

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

   9. *-commutative 79.8%

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

   10. associate-/l* 79.9%

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

   11. metadata-eval 79.9%

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

   12. metadata-eval 79.9%

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

3. Simplified 80.0%

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

5. Taylor expanded in angle around 0 59.2%

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

   1. unpow2 59.2%

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

7. Applied egg-rr 59.2%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))