ab-angle->ABCF C

Percentage Accurate: 80.2% → 80.0%

Time: 45.1s

Alternatives: 16

Speedup: 1.3×

• 37.2% 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: 80.2% 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: 80.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 1 + 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ {\left(a \cdot \mathsf{fma}\left(\sqrt{{\sin t\_0}^{2}}, \sin 1, \cos 1 \cdot \cos t\_0\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.005555555555555556 (* angle_m PI)))))
   (+
    (pow
     (* a (fma (sqrt (pow (sin t_0) 2.0)) (sin 1.0) (* (cos 1.0) (cos t_0))))
     2.0)
    (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = 1.0 + (0.005555555555555556 * (angle_m * ((double) M_PI)));
	return pow((a * fma(sqrt(pow(sin(t_0), 2.0)), sin(1.0), (cos(1.0) * cos(t_0)))), 2.0) + pow((b * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(1.0 + Float64(0.005555555555555556 * Float64(angle_m * pi)))
	return Float64((Float64(a * fma(sqrt((sin(t_0) ^ 2.0)), sin(1.0), Float64(cos(1.0) * cos(t_0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(1.0 + N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[(N[Sqrt[N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * N[Sin[1.0], $MachinePrecision] + N[(N[Cos[1.0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

   1. metadata-eval 75.1%

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

   2. div-inv 75.2%

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

   3. expm1-log1p-u 59.8%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. expm1-undefine 59.8%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. cos-diff 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{angle}{180} \cdot \pi}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. div-inv 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. metadata-eval 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\left(angle \cdot \color{blue}{0.005555555555555556}\right) \cdot \pi\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. associate-*l* 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 59.7%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. +-commutative 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \sin 1 + \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. fma-define 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{fma}\left(\sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. log1p-undefine 59.7%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(e^{\color{blue}{\log \left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. rem-exp-log 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \color{blue}{\left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}, \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. associate-*r* 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. associate-*r* 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. *-commutative 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \color{blue}{\cos 1 \cdot \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. log1p-undefine 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \cos 1 \cdot \cos \left(e^{\color{blue}{\log \left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. rem-exp-log 75.3%

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

   11. associate-*r* 75.3%

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

   12. *-commutative 75.3%

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

   13. associate-*r* 75.3%

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

8. Simplified 75.3%

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

   1. add-sqr-sqrt 57.6%

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

   2. sqrt-unprod 75.3%

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

   3. pow2 75.3%

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

   4. +-commutative 75.3%

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

   5. fma-define 75.3%

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

10. Applied egg-rr 75.3%

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

11. Taylor expanded in angle around inf 75.3%

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

12. Final simplification 75.3%

    \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sqrt{{\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}, \sin 1, \cos 1 \cdot \cos \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} \]
13. Add Preprocessing

Alternative 2: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle\_m, 1\right)\\ {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\sin t\_0, \sin 1, \cos 1 \cdot \cos t\_0\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (fma PI (* 0.005555555555555556 angle_m) 1.0)))
   (+
    (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0)
    (pow (* a (fma (sin t_0) (sin 1.0) (* (cos 1.0) (cos t_0)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = fma(((double) M_PI), (0.005555555555555556 * angle_m), 1.0);
	return pow((b * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0) + pow((a * fma(sin(t_0), sin(1.0), (cos(1.0) * cos(t_0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = fma(pi, Float64(0.005555555555555556 * angle_m), 1.0)
	return Float64((Float64(b * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0) + (Float64(a * fma(sin(t_0), sin(1.0), Float64(cos(1.0) * 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[(0.005555555555555556 * angle$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[1.0], $MachinePrecision] + N[(N[Cos[1.0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

   1. metadata-eval 75.1%

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

   2. div-inv 75.2%

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

   3. expm1-log1p-u 59.8%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. expm1-undefine 59.8%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. cos-diff 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{angle}{180} \cdot \pi}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. div-inv 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. metadata-eval 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\left(angle \cdot \color{blue}{0.005555555555555556}\right) \cdot \pi\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. associate-*l* 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 59.7%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. +-commutative 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \sin 1 + \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. fma-define 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{fma}\left(\sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. log1p-undefine 59.7%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(e^{\color{blue}{\log \left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. rem-exp-log 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \color{blue}{\left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}, \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. associate-*r* 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. associate-*r* 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. *-commutative 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \color{blue}{\cos 1 \cdot \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. log1p-undefine 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \cos 1 \cdot \cos \left(e^{\color{blue}{\log \left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. rem-exp-log 75.3%

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

   11. associate-*r* 75.3%

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

   12. *-commutative 75.3%

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

   13. associate-*r* 75.3%

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

8. Simplified 75.3%

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

   1. *-un-lft-identity 75.3%

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

   2. +-commutative 75.3%

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

   3. fma-define 75.3%

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

10. Applied egg-rr 75.3%

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

    1. *-lft-identity 75.3%

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

    2. fma-undefine 75.3%

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

    3. associate-*r* 75.3%

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

    4. *-commutative 75.3%

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

    5. fma-define 75.3%

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

    6. *-commutative 75.3%

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

12. Simplified 75.3%

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

    1. *-un-lft-identity 75.3%

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

    2. +-commutative 75.3%

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

    3. fma-define 75.3%

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

14. Applied egg-rr 75.3%

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

    1. *-lft-identity 75.3%

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

    2. fma-undefine 75.3%

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

    3. associate-*r* 75.4%

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

    4. *-commutative 75.4%

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

    5. fma-define 75.4%

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

    6. *-commutative 75.4%

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

16. Simplified 75.4%

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

17. Final simplification 75.4%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle, 1\right)\right), \sin 1, \cos 1 \cdot \cos \left(\mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle, 1\right)\right)\right)\right)}^{2} \]
18. Add Preprocessing

Alternative 3: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle\_m, 1\right)\right), \sin 1, \cos 1 \cdot \cos \left(1 + 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0)
  (pow
   (*
    a
    (fma
     (sin (fma PI (* 0.005555555555555556 angle_m) 1.0))
     (sin 1.0)
     (* (cos 1.0) (cos (+ 1.0 (* 0.005555555555555556 (* angle_m PI)))))))
   2.0)))

C

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0) + (Float64(a * fma(sin(fma(pi, Float64(0.005555555555555556 * angle_m), 1.0)), sin(1.0), Float64(cos(1.0) * cos(Float64(1.0 + Float64(0.005555555555555556 * Float64(angle_m * pi))))))) ^ 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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sin[1.0], $MachinePrecision] + N[(N[Cos[1.0], $MachinePrecision] * N[Cos[N[(1.0 + N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

   1. metadata-eval 75.1%

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

   2. div-inv 75.2%

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

   3. expm1-log1p-u 59.8%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. expm1-undefine 59.8%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. cos-diff 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{angle}{180} \cdot \pi}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. div-inv 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. metadata-eval 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\left(angle \cdot \color{blue}{0.005555555555555556}\right) \cdot \pi\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. associate-*l* 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 59.7%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. +-commutative 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \sin 1 + \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. fma-define 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{fma}\left(\sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. log1p-undefine 59.7%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(e^{\color{blue}{\log \left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. rem-exp-log 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \color{blue}{\left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}, \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. associate-*r* 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. associate-*r* 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. *-commutative 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \color{blue}{\cos 1 \cdot \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. log1p-undefine 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \cos 1 \cdot \cos \left(e^{\color{blue}{\log \left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. rem-exp-log 75.3%

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

   11. associate-*r* 75.3%

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

   12. *-commutative 75.3%

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

   13. associate-*r* 75.3%

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

8. Simplified 75.3%

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

   1. *-un-lft-identity 75.3%

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

   2. +-commutative 75.3%

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

   3. fma-define 75.3%

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

10. Applied egg-rr 75.3%

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

    1. *-lft-identity 75.3%

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

    2. fma-undefine 75.3%

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

    3. associate-*r* 75.3%

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

    4. *-commutative 75.3%

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

    5. fma-define 75.3%

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

    6. *-commutative 75.3%

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

12. Simplified 75.3%

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

13. Final simplification 75.3%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\sin \left(\mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle, 1\right)\right), \sin 1, \cos 1 \cdot \cos \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2} \]
14. Add Preprocessing

Alternative 4: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 1 + 0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\\ {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\sin t\_0, \sin 1, \cos 1 \cdot \cos t\_0\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.005555555555555556 (* angle_m PI)))))
   (+
    (pow (* b (sin (* PI (* 0.005555555555555556 angle_m)))) 2.0)
    (pow (* a (fma (sin t_0) (sin 1.0) (* (cos 1.0) (cos t_0)))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = 1.0 + (0.005555555555555556 * (angle_m * ((double) M_PI)));
	return pow((b * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0) + pow((a * fma(sin(t_0), sin(1.0), (cos(1.0) * cos(t_0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(1.0 + Float64(0.005555555555555556 * Float64(angle_m * pi)))
	return Float64((Float64(b * sin(Float64(pi * Float64(0.005555555555555556 * angle_m)))) ^ 2.0) + (Float64(a * fma(sin(t_0), sin(1.0), Float64(cos(1.0) * cos(t_0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(1.0 + N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[1.0], $MachinePrecision] + N[(N[Cos[1.0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

   1. metadata-eval 75.1%

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

   2. div-inv 75.2%

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

   3. expm1-log1p-u 59.8%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. expm1-undefine 59.8%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. cos-diff 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{angle}{180} \cdot \pi}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. div-inv 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \pi\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. metadata-eval 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\left(angle \cdot \color{blue}{0.005555555555555556}\right) \cdot \pi\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. associate-*l* 59.7%

      \[\leadsto {\left(a \cdot \left(\cos \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \frac{angle}{180}\right)}\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

6. Applied egg-rr 59.7%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1 + \sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \sin 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. +-commutative 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \sin 1 + \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   2. fma-define 59.7%

      \[\leadsto {\left(a \cdot \color{blue}{\mathsf{fma}\left(\sin \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   3. log1p-undefine 59.7%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(e^{\color{blue}{\log \left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   4. rem-exp-log 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \color{blue}{\left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}, \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   5. associate-*r* 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{\left(angle \cdot 0.005555555555555556\right) \cdot \pi}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   6. *-commutative 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{\left(0.005555555555555556 \cdot angle\right)} \cdot \pi\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   7. associate-*r* 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + \color{blue}{0.005555555555555556 \cdot \left(angle \cdot \pi\right)}\right), \sin 1, \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right) \cdot \cos 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   8. *-commutative 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \color{blue}{\cos 1 \cdot \cos \left(e^{\mathsf{log1p}\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. log1p-undefine 59.8%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \cos 1 \cdot \cos \left(e^{\color{blue}{\log \left(1 + angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. rem-exp-log 75.3%

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

   11. associate-*r* 75.3%

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

   12. *-commutative 75.3%

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

   13. associate-*r* 75.3%

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

8. Simplified 75.3%

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

9. Final simplification 75.3%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(a \cdot \mathsf{fma}\left(\sin \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin 1, \cos 1 \cdot \cos \left(1 + 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 5: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + ((a * cos((pi * (sqrt((0.005555555555555556 * angle_m)) ^ 2.0)))) ^ 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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[Power[N[Sqrt[N[(0.005555555555555556 * angle$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

   1. metadata-eval 75.1%

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

   2. div-inv 75.2%

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

   3. add-sqr-sqrt 36.0%

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

   4. pow2 36.0%

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

   5. div-inv 36.0%

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

   6. metadata-eval 36.0%

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

6. Applied egg-rr 36.0%

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

7. Final simplification 36.0%

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

Alternative 6: 80.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin((((double) M_PI) * (0.005555555555555556 * angle_m)))), 2.0) + pow((a * cos((((double) M_PI) * exp(log((0.005555555555555556 * angle_m)))))), 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 * (0.005555555555555556 * angle_m)))), 2.0) + Math.pow((a * Math.cos((Math.PI * Math.exp(Math.log((0.005555555555555556 * angle_m)))))), 2.0);
}

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + ((a * cos((pi * exp(log((0.005555555555555556 * angle_m)))))) ^ 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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[Exp[N[Log[N[(0.005555555555555556 * angle$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

   1. add-exp-log 36.0%

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

6. Applied egg-rr 36.0%

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

7. Final simplification 36.0%

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

Alternative 7: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (0.005555555555555556 * angle_m)))) ^ 2.0) + ((a * cos((0.005555555555555556 * (angle_m * pi)))) ^ 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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

5. Taylor expanded in angle around inf 75.1%

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

6. Final simplification 75.1%

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

Alternative 8: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = pi * (0.005555555555555556 * angle_m);
	tmp = ((b * sin(t_0)) ^ 2.0) + ((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[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

5. Final simplification 75.1%

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

Alternative 9: 80.1% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle_m 180.0)))) 2.0)
  (pow (* b (sin (* 0.005555555555555556 (* angle_m PI)))) 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 / 180.0)))), 2.0) + pow((b * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))), 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 / 180.0)))), 2.0) + Math.pow((b * Math.sin((0.005555555555555556 * (angle_m * Math.PI)))), 2.0);
}

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * cos((pi * (angle_m / 180.0)))) ^ 2.0) + ((b * sin((0.005555555555555556 * (angle_m * pi)))) ^ 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 / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Taylor expanded in b around 0 67.1%

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

   1. *-commutative 67.1%

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

   2. associate-*r* 67.1%

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

   3. *-commutative 67.1%

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

   4. associate-*r* 67.1%

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

   5. unpow2 67.1%

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

   6. unpow2 67.1%

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

   7. swap-sqr 75.2%

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

   8. unpow2 75.2%

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

   9. *-commutative 75.2%

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

   10. associate-*r* 75.2%

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

   11. *-commutative 75.2%

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

   12. associate-*r* 75.2%

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

5. Simplified 75.2%

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

6. Final simplification 75.2%

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

Alternative 10: 80.2% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (/ angle_m 180.0))))
   (+ (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 = ((double) M_PI) * (angle_m / 180.0);
	return pow((a * cos(t_0)), 2.0) + pow((b * sin(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 / 180.0);
	return Math.pow((a * Math.cos(t_0)), 2.0) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 75.2%

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

3. Final simplification 75.2%

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

Alternative 11: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin((0.005555555555555556 * (angle_m * ((double) M_PI))))), 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((0.005555555555555556 * (angle_m * Math.PI)))), 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((0.005555555555555556 * (angle_m * math.pi)))), 2.0) + math.pow(a, 2.0)

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((0.005555555555555556 * (angle_m * pi)))) ^ 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[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

5. Taylor expanded in angle around 0 75.1%

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

6. Taylor expanded in angle around inf 75.1%

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

7. Final simplification 75.1%

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

Alternative 12: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin((pi * (angle_m / 180.0)))) ^ 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 / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

5. Taylor expanded in angle around 0 75.1%

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

   1. metadata-eval 75.1%

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

   2. div-inv 75.1%

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

7. Applied egg-rr 75.1%

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

8. Final simplification 75.1%

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

Alternative 13: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\left(angle\_m \cdot \pi\right) \cdot b\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.1e-62)
   (pow a 2.0)
   (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* (* angle_m PI) b) 2.0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.1e-62) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (3.08641975308642e-5 * pow(((angle_m * ((double) M_PI)) * b), 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 <= 1.1e-62) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow(((angle_m * Math.PI) * b), 2.0));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 1.1e-62:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow(((angle_m * math.pi) * b), 2.0))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.1e-62)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(Float64(angle_m * pi) * b) ^ 2.0)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.1e-62)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (3.08641975308642e-5 * (((angle_m * pi) * b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.1e-62], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(N[(angle$95$m * Pi), $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.10000000000000009e-62

   1. Initial program 74.5%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in angle around 0 74.5%

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

   6. Taylor expanded in angle around 0 60.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right)} \]

      2. unpow2 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right) \]

      3. unpow2 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {\pi}^{2}\right) \]

      4. swap-sqr 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot {\pi}^{2}\right) \]

      5. unpow2 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \]

      6. swap-sqr 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right)} \]

      7. *-commutative 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)} \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]

      8. *-commutative 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)}\right) \]

      9. unpow2 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   8. Simplified 68.6%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   9. Taylor expanded in a around inf 60.2%

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

   if 1.10000000000000009e-62 < b

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in angle around 0 76.4%

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

   6. Taylor expanded in angle around 0 62.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right)} \]

      2. unpow2 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right) \]

      3. unpow2 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {\pi}^{2}\right) \]

      4. swap-sqr 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot {\pi}^{2}\right) \]

      5. unpow2 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \]

      6. swap-sqr 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right)} \]

      7. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)} \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]

      8. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)}\right) \]

      9. unpow2 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   8. Simplified 73.3%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   9. Taylor expanded in angle around 0 73.3%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}}^{2} \]
   10. Step-by-step derivation

       1. *-commutative 73.3%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \color{blue}{\left(\pi \cdot b\right)}\right)}^{2} \]

       2. associate-*r* 73.3%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot b\right)}}^{2} \]

   11. Simplified 73.3%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot b\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\left(angle \cdot \pi\right) \cdot b\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 67.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := angle\_m \cdot \left(\pi \cdot b\right)\\ \mathbf{if}\;b \leq 1.22 \cdot 10^{-64}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (* PI b))))
   (if (<= b 1.22e-64)
     (pow a 2.0)
     (+ (pow a 2.0) (* 3.08641975308642e-5 (* t_0 t_0))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (((double) M_PI) * b);
	double tmp;
	if (b <= 1.22e-64) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (3.08641975308642e-5 * (t_0 * t_0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = angle_m * (Math.PI * b);
	double tmp;
	if (b <= 1.22e-64) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (3.08641975308642e-5 * (t_0 * t_0));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = angle_m * (math.pi * b)
	tmp = 0
	if b <= 1.22e-64:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (3.08641975308642e-5 * (t_0 * t_0))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(angle_m * Float64(pi * b))
	tmp = 0.0
	if (b <= 1.22e-64)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * Float64(t_0 * t_0)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = angle_m * (pi * b);
	tmp = 0.0;
	if (b <= 1.22e-64)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (3.08641975308642e-5 * (t_0 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.22e-64], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.22000000000000003e-64

   1. Initial program 74.5%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in angle around 0 74.5%

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

   6. Taylor expanded in angle around 0 60.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right)} \]

      2. unpow2 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right) \]

      3. unpow2 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {\pi}^{2}\right) \]

      4. swap-sqr 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot {\pi}^{2}\right) \]

      5. unpow2 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \]

      6. swap-sqr 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right)} \]

      7. *-commutative 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)} \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]

      8. *-commutative 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)}\right) \]

      9. unpow2 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   8. Simplified 68.6%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   9. Taylor expanded in a around inf 60.2%

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

   if 1.22000000000000003e-64 < b

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in angle around 0 76.4%

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

   6. Taylor expanded in angle around 0 62.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right)} \]

      2. unpow2 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right) \]

      3. unpow2 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {\pi}^{2}\right) \]

      4. swap-sqr 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot {\pi}^{2}\right) \]

      5. unpow2 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \]

      6. swap-sqr 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right)} \]

      7. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)} \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]

      8. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)}\right) \]

      9. unpow2 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   8. Simplified 73.3%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)} \]

      2. associate-*r* 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\left(\pi \cdot angle\right) \cdot b\right)}\right) \]

      3. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot b\right)\right) \]

      4. associate-*r* 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot b\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot b\right)\right) \]

      5. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot \pi\right)} \cdot b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot b\right)\right) \]

      6. associate-*l* 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot b\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot b\right)\right) \]

      7. associate-*l* 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot b\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot b\right)\right)}\right) \]

   10. Applied egg-rr 73.3%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(angle \cdot \left(\pi \cdot b\right)\right) \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.22 \cdot 10^{-64}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot b\right)\right) \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-63}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right)\right) \cdot \left(angle\_m \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 8.8e-63)
   (pow a 2.0)
   (+
    (pow a 2.0)
    (* 3.08641975308642e-5 (* (* PI (* angle_m (* PI b))) (* angle_m b))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 8.8e-63) {
		tmp = pow(a, 2.0);
	} else {
		tmp = pow(a, 2.0) + (3.08641975308642e-5 * ((((double) M_PI) * (angle_m * (((double) M_PI) * b))) * (angle_m * b)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 8.8e-63) {
		tmp = Math.pow(a, 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (3.08641975308642e-5 * ((Math.PI * (angle_m * (Math.PI * b))) * (angle_m * b)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 8.8e-63:
		tmp = math.pow(a, 2.0)
	else:
		tmp = math.pow(a, 2.0) + (3.08641975308642e-5 * ((math.pi * (angle_m * (math.pi * b))) * (angle_m * b)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 8.8e-63)
		tmp = a ^ 2.0;
	else
		tmp = Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * Float64(Float64(pi * Float64(angle_m * Float64(pi * b))) * Float64(angle_m * b))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 8.8e-63)
		tmp = a ^ 2.0;
	else
		tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((pi * (angle_m * (pi * b))) * (angle_m * b)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 8.8e-63], N[Power[a, 2.0], $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[(Pi * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.7999999999999998e-63

   1. Initial program 74.5%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in angle around 0 74.5%

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

   6. Taylor expanded in angle around 0 60.9%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right)} \]

      2. unpow2 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right) \]

      3. unpow2 60.9%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {\pi}^{2}\right) \]

      4. swap-sqr 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot {\pi}^{2}\right) \]

      5. unpow2 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \]

      6. swap-sqr 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right)} \]

      7. *-commutative 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)} \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]

      8. *-commutative 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)}\right) \]

      9. unpow2 68.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   8. Simplified 68.6%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   9. Taylor expanded in a around inf 60.2%

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

   if 8.7999999999999998e-63 < b

   1. Initial program 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in angle around 0 76.4%

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

   6. Taylor expanded in angle around 0 62.5%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right)} \]

      2. unpow2 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right) \]

      3. unpow2 63.6%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {\pi}^{2}\right) \]

      4. swap-sqr 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot {\pi}^{2}\right) \]

      5. unpow2 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \]

      6. swap-sqr 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right)} \]

      7. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)} \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]

      8. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)}\right) \]

      9. unpow2 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

   8. Simplified 73.3%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)} \]

      2. associate-*r* 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)} \]

      3. associate-*r* 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot b\right)} \cdot \pi\right) \cdot \left(angle \cdot b\right)\right) \]

      4. *-commutative 73.3%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(\color{blue}{\left(angle \cdot \pi\right)} \cdot b\right) \cdot \pi\right) \cdot \left(angle \cdot b\right)\right) \]

      5. associate-*l* 73.4%

         \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot \left(\pi \cdot b\right)\right)} \cdot \pi\right) \cdot \left(angle \cdot b\right)\right) \]

   10. Applied egg-rr 73.4%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \left(\pi \cdot b\right)\right) \cdot \pi\right) \cdot \left(angle \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-63}:\\ \;\;\;\;{a}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {a}^{2} \end{array} \]

FPCore

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

C

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

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 ** 2.0d0
end function

Java

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

Python

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

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return a ^ 2.0
end

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[Power[a, 2.0], $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Simplified 75.1%

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

5. Taylor expanded in angle around 0 75.1%

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

6. Taylor expanded in angle around 0 61.4%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 61.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right)} \]

   2. unpow2 61.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {b}^{2}\right) \cdot {\pi}^{2}\right) \]

   3. unpow2 61.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(b \cdot b\right)}\right) \cdot {\pi}^{2}\right) \]

   4. swap-sqr 70.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right)} \cdot {\pi}^{2}\right) \]

   5. unpow2 70.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot b\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \]

   6. swap-sqr 70.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right)} \]

   7. *-commutative 70.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)} \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \]

   8. *-commutative 70.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(\pi \cdot \left(angle \cdot b\right)\right)}\right) \]

   9. unpow2 70.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

8. Simplified 70.1%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot b\right)\right)}^{2}} \]

9. Taylor expanded in a around inf 54.1%

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

10. Final simplification 54.1%

    \[\leadsto {a}^{2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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)))