ab-angle->ABCF A

Percentage Accurate: 79.3% → 79.3%

Time: 57.5s

Alternatives: 7

Speedup: N/A×

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. unpow2 76.9%

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

   2. swap-sqr 76.9%

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

   3. *-commutative 76.9%

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

   4. associate-*r/ 76.8%

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

   5. associate-*l/ 76.9%

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

   6. *-commutative 76.9%

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

   7. swap-sqr 76.9%

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

   8. unpow2 76.9%

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

   9. *-commutative 76.9%

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

   10. associate-*r/ 76.9%

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

   11. associate-*l/ 77.0%

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

   12. *-commutative 77.0%

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

3. Simplified 77.0%

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

   1. associate-*r/ 76.9%

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

   2. associate-*l/ 76.9%

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

   3. add-log-exp 66.7%

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

   4. associate-/r/ 66.8%

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

   5. div-inv 66.8%

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

   6. clear-num 66.8%

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

   7. div-inv 66.8%

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

   8. metadata-eval 66.8%

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

6. Applied egg-rr 66.8%

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

7. Taylor expanded in angle around 0 66.9%

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

   1. rem-log-exp 77.1%

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

9. Applied egg-rr 77.1%

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

    1. log1p-expm1-u 77.1%

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

11. Applied egg-rr 77.1%

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

12. Final simplification 77.1%

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

Alternative 2: 79.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. unpow2 76.9%

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

   2. swap-sqr 76.9%

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

   3. *-commutative 76.9%

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

   4. associate-*r/ 76.8%

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

   5. associate-*l/ 76.9%

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

   6. *-commutative 76.9%

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

   7. swap-sqr 76.9%

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

   8. unpow2 76.9%

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

   9. *-commutative 76.9%

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

   10. associate-*r/ 76.9%

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

   11. associate-*l/ 77.0%

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

   12. *-commutative 77.0%

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

3. Simplified 77.0%

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

   1. associate-*r/ 76.9%

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

   2. associate-*l/ 76.9%

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

   3. add-log-exp 66.7%

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

   4. associate-/r/ 66.8%

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

   5. div-inv 66.8%

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

   6. clear-num 66.8%

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

   7. div-inv 66.8%

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

   8. metadata-eval 66.8%

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

6. Applied egg-rr 66.8%

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

7. Taylor expanded in angle around 0 66.9%

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

   1. rem-log-exp 77.1%

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

9. Applied egg-rr 77.1%

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

10. Final simplification 77.1%

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

Alternative 3: 65.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{-88}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + angle_m \cdot \left(\left(a \cdot \pi\right) \cdot \left(angle_m \cdot \left(0.005555555555555556 \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 2.4e-88)
   (pow b 2.0)
   (+
    (pow b 2.0)
    (*
     angle_m
     (*
      (* a PI)
      (*
       angle_m
       (* 0.005555555555555556 (* a (* PI 0.005555555555555556)))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.4e-88) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + (angle_m * ((a * ((double) M_PI)) * (angle_m * (0.005555555555555556 * (a * (((double) M_PI) * 0.005555555555555556))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.4e-88) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (angle_m * ((a * Math.PI) * (angle_m * (0.005555555555555556 * (a * (Math.PI * 0.005555555555555556))))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 2.4e-88:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + (angle_m * ((a * math.pi) * (angle_m * (0.005555555555555556 * (a * (math.pi * 0.005555555555555556))))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 2.4e-88)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(angle_m * Float64(Float64(a * pi) * Float64(angle_m * Float64(0.005555555555555556 * Float64(a * Float64(pi * 0.005555555555555556)))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 2.4e-88)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + (angle_m * ((a * pi) * (angle_m * (0.005555555555555556 * (a * (pi * 0.005555555555555556))))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 2.4e-88], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(a * Pi), $MachinePrecision] * N[(angle$95$m * N[(0.005555555555555556 * N[(a * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.4e-88

   1. Initial program 75.1%

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

      1. unpow2 75.1%

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

      2. swap-sqr 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r/ 75.1%

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

      5. associate-*l/ 75.1%

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

      6. *-commutative 75.1%

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

      7. swap-sqr 75.1%

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

      8. unpow2 75.1%

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

      9. *-commutative 75.1%

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

      10. associate-*r/ 75.1%

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

      11. associate-*l/ 75.2%

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

      12. *-commutative 75.2%

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

   3. Simplified 75.2%

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

      1. expm1-log1p-u 74.7%

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

      2. expm1-udef 69.5%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in angle around 0 69.6%

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

   8. Taylor expanded in a around 0 59.7%

      \[\leadsto \left(\color{blue}{1} - 1\right) + {b}^{2} \]

   if 2.4e-88 < a

   1. Initial program 82.1%

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

      1. unpow2 82.1%

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

      2. swap-sqr 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r/ 82.0%

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

      5. associate-*l/ 82.3%

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

      6. *-commutative 82.3%

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

      7. swap-sqr 82.3%

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

      8. unpow2 82.3%

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

      9. *-commutative 82.3%

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

      10. associate-*r/ 82.1%

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

      11. associate-*l/ 82.2%

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

      12. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in angle around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in angle around 0 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r* 78.2%

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

      3. associate-*l* 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*r* 75.4%

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

      6. *-commutative 75.4%

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

      7. *-commutative 75.4%

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

      8. associate-*l* 75.4%

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

      9. *-commutative 75.4%

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

   10. Applied egg-rr 75.4%

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

       1. associate-*r* 78.3%

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

       2. *-commutative 78.3%

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

       3. associate-*l* 75.5%

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

       4. *-commutative 75.5%

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

       5. associate-*l* 75.5%

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

       6. *-commutative 75.5%

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

       7. associate-*l* 75.5%

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

   12. Simplified 75.5%

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

       1. expm1-log1p-u 49.8%

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

       2. expm1-udef 44.4%

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

       3. *-commutative 44.4%

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

       4. associate-*l* 44.4%

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

       5. associate-*r* 44.4%

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

       6. *-commutative 44.4%

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

   14. Applied egg-rr 44.4%

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

       1. expm1-def 49.8%

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

       2. expm1-log1p 75.5%

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

       3. *-commutative 75.5%

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

   16. Simplified 75.5%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.4 \cdot 10^{-88}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + angle \cdot \left(\left(a \cdot \pi\right) \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-88}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(\left(angle_m \cdot \left(a \cdot \pi\right)\right) \cdot \left(angle_m \cdot 0.005555555555555556\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 2.8e-88)
   (pow b 2.0)
   (+
    (pow b 2.0)
    (*
     (* PI (* a 0.005555555555555556))
     (* (* angle_m (* a PI)) (* angle_m 0.005555555555555556))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.8e-88) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + ((((double) M_PI) * (a * 0.005555555555555556)) * ((angle_m * (a * ((double) M_PI))) * (angle_m * 0.005555555555555556)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.8e-88) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + ((Math.PI * (a * 0.005555555555555556)) * ((angle_m * (a * Math.PI)) * (angle_m * 0.005555555555555556)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 2.8e-88:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + ((math.pi * (a * 0.005555555555555556)) * ((angle_m * (a * math.pi)) * (angle_m * 0.005555555555555556)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 2.8e-88)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(pi * Float64(a * 0.005555555555555556)) * Float64(Float64(angle_m * Float64(a * pi)) * Float64(angle_m * 0.005555555555555556))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 2.8e-88)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + ((pi * (a * 0.005555555555555556)) * ((angle_m * (a * pi)) * (angle_m * 0.005555555555555556)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 2.8e-88], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(Pi * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * N[(N[(angle$95$m * N[(a * Pi), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.79999999999999976e-88

   1. Initial program 75.1%

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

      1. unpow2 75.1%

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

      2. swap-sqr 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r/ 75.1%

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

      5. associate-*l/ 75.1%

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

      6. *-commutative 75.1%

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

      7. swap-sqr 75.1%

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

      8. unpow2 75.1%

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

      9. *-commutative 75.1%

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

      10. associate-*r/ 75.1%

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

      11. associate-*l/ 75.2%

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

      12. *-commutative 75.2%

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

   3. Simplified 75.2%

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

      1. expm1-log1p-u 74.7%

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

      2. expm1-udef 69.5%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in angle around 0 69.6%

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

   8. Taylor expanded in a around 0 59.7%

      \[\leadsto \left(\color{blue}{1} - 1\right) + {b}^{2} \]

   if 2.79999999999999976e-88 < a

   1. Initial program 82.1%

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

      1. unpow2 82.1%

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

      2. swap-sqr 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r/ 82.0%

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

      5. associate-*l/ 82.3%

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

      6. *-commutative 82.3%

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

      7. swap-sqr 82.3%

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

      8. unpow2 82.3%

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

      9. *-commutative 82.3%

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

      10. associate-*r/ 82.1%

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

      11. associate-*l/ 82.2%

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

      12. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in angle around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in angle around 0 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r* 78.2%

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

      3. associate-*l* 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*r* 75.4%

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

      6. *-commutative 75.4%

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

      7. *-commutative 75.4%

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

      8. associate-*l* 75.4%

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

      9. *-commutative 75.4%

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

   10. Applied egg-rr 75.4%

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

       1. associate-*l* 75.4%

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

       2. associate-*r* 75.4%

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

       3. *-commutative 75.4%

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

   12. Simplified 75.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-88}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(a \cdot \pi\right) \cdot \left(angle_m \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 2.8 \cdot 10^{-88}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + t_0 \cdot t_0\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* a PI) (* angle_m 0.005555555555555556))))
   (if (<= a 2.8e-88) (pow b 2.0) (+ (pow b 2.0) (* t_0 t_0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (a * ((double) M_PI)) * (angle_m * 0.005555555555555556);
	double tmp;
	if (a <= 2.8e-88) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + (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 = (a * Math.PI) * (angle_m * 0.005555555555555556);
	double tmp;
	if (a <= 2.8e-88) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (t_0 * t_0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (a * math.pi) * (angle_m * 0.005555555555555556)
	tmp = 0
	if a <= 2.8e-88:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + (t_0 * t_0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(a * pi) * Float64(angle_m * 0.005555555555555556))
	tmp = 0.0
	if (a <= 2.8e-88)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = (a * pi) * (angle_m * 0.005555555555555556);
	tmp = 0.0;
	if (a <= 2.8e-88)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + (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[(N[(a * Pi), $MachinePrecision] * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.8e-88], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 2.79999999999999976e-88

   1. Initial program 75.1%

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

      1. unpow2 75.1%

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

      2. swap-sqr 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r/ 75.1%

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

      5. associate-*l/ 75.1%

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

      6. *-commutative 75.1%

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

      7. swap-sqr 75.1%

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

      8. unpow2 75.1%

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

      9. *-commutative 75.1%

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

      10. associate-*r/ 75.1%

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

      11. associate-*l/ 75.2%

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

      12. *-commutative 75.2%

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

   3. Simplified 75.2%

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

      1. expm1-log1p-u 74.7%

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

      2. expm1-udef 69.5%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in angle around 0 69.6%

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

   8. Taylor expanded in a around 0 59.7%

      \[\leadsto \left(\color{blue}{1} - 1\right) + {b}^{2} \]

   if 2.79999999999999976e-88 < a

   1. Initial program 82.1%

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

      1. unpow2 82.1%

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

      2. swap-sqr 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r/ 82.0%

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

      5. associate-*l/ 82.3%

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

      6. *-commutative 82.3%

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

      7. swap-sqr 82.3%

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

      8. unpow2 82.3%

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

      9. *-commutative 82.3%

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

      10. associate-*r/ 82.1%

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

      11. associate-*l/ 82.2%

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

      12. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in angle around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in angle around 0 78.2%

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

      1. unpow2 78.2%

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

      2. *-commutative 78.2%

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

      3. *-commutative 78.2%

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

      4. associate-*l* 78.2%

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

      5. *-commutative 78.2%

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

      6. associate-*l* 78.3%

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

      7. *-commutative 78.3%

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

   10. Applied egg-rr 78.3%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.8 \cdot 10^{-88}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(a \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(a \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{-88}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + angle_m \cdot \left(\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \left(angle_m \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.15e-88)
   (pow b 2.0)
   (+
    (pow b 2.0)
    (* angle_m (* (* a PI) (* (* a (* angle_m PI)) 3.08641975308642e-5))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.15e-88) {
		tmp = pow(b, 2.0);
	} else {
		tmp = pow(b, 2.0) + (angle_m * ((a * ((double) M_PI)) * ((a * (angle_m * ((double) M_PI))) * 3.08641975308642e-5)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.15e-88) {
		tmp = Math.pow(b, 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + (angle_m * ((a * Math.PI) * ((a * (angle_m * Math.PI)) * 3.08641975308642e-5)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.15e-88:
		tmp = math.pow(b, 2.0)
	else:
		tmp = math.pow(b, 2.0) + (angle_m * ((a * math.pi) * ((a * (angle_m * math.pi)) * 3.08641975308642e-5)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.15e-88)
		tmp = b ^ 2.0;
	else
		tmp = Float64((b ^ 2.0) + Float64(angle_m * Float64(Float64(a * pi) * Float64(Float64(a * Float64(angle_m * pi)) * 3.08641975308642e-5))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.15e-88)
		tmp = b ^ 2.0;
	else
		tmp = (b ^ 2.0) + (angle_m * ((a * pi) * ((a * (angle_m * pi)) * 3.08641975308642e-5)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.15e-88], N[Power[b, 2.0], $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(angle$95$m * N[(N[(a * Pi), $MachinePrecision] * N[(N[(a * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.14999999999999993e-88

   1. Initial program 75.1%

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

      1. unpow2 75.1%

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

      2. swap-sqr 75.1%

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

      3. *-commutative 75.1%

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

      4. associate-*r/ 75.1%

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

      5. associate-*l/ 75.1%

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

      6. *-commutative 75.1%

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

      7. swap-sqr 75.1%

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

      8. unpow2 75.1%

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

      9. *-commutative 75.1%

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

      10. associate-*r/ 75.1%

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

      11. associate-*l/ 75.2%

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

      12. *-commutative 75.2%

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

   3. Simplified 75.2%

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

      1. expm1-log1p-u 74.7%

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

      2. expm1-udef 69.5%

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

   6. Applied egg-rr 69.5%

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

   7. Taylor expanded in angle around 0 69.6%

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

   8. Taylor expanded in a around 0 59.7%

      \[\leadsto \left(\color{blue}{1} - 1\right) + {b}^{2} \]

   if 1.14999999999999993e-88 < a

   1. Initial program 82.1%

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

      1. unpow2 82.1%

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

      2. swap-sqr 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-*r/ 82.0%

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

      5. associate-*l/ 82.3%

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

      6. *-commutative 82.3%

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

      7. swap-sqr 82.3%

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

      8. unpow2 82.3%

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

      9. *-commutative 82.3%

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

      10. associate-*r/ 82.1%

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

      11. associate-*l/ 82.2%

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

      12. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in angle around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r* 78.1%

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

   7. Simplified 78.1%

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

   8. Taylor expanded in angle around 0 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*r* 78.2%

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

      3. associate-*l* 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*r* 75.4%

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

      6. *-commutative 75.4%

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

      7. *-commutative 75.4%

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

      8. associate-*l* 75.4%

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

      9. *-commutative 75.4%

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

   10. Applied egg-rr 75.4%

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

       1. associate-*r* 78.3%

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

       2. *-commutative 78.3%

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

       3. associate-*l* 75.5%

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

       4. *-commutative 75.5%

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

       5. associate-*l* 75.5%

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

       6. *-commutative 75.5%

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

       7. associate-*l* 75.5%

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

   12. Simplified 75.5%

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

   13. Taylor expanded in angle around 0 75.4%

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

       1. *-commutative 75.4%

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

   15. Simplified 75.4%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{-88}:\\ \;\;\;\;{b}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + angle \cdot \left(\left(a \cdot \pi\right) \cdot \left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(b, 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 = b ** 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

   1. unpow2 76.9%

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

   2. swap-sqr 76.9%

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

   3. *-commutative 76.9%

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

   4. associate-*r/ 76.8%

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

   5. associate-*l/ 76.9%

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

   6. *-commutative 76.9%

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

   7. swap-sqr 76.9%

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

   8. unpow2 76.9%

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

   9. *-commutative 76.9%

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

   10. associate-*r/ 76.9%

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

   11. associate-*l/ 77.0%

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

   12. *-commutative 77.0%

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

3. Simplified 77.0%

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

   1. expm1-log1p-u 76.5%

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

   2. expm1-udef 70.7%

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

6. Applied egg-rr 70.7%

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

7. Taylor expanded in angle around 0 70.8%

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

8. Taylor expanded in a around 0 57.9%

   \[\leadsto \left(\color{blue}{1} - 1\right) + {b}^{2} \]

9. Final simplification 57.9%

   \[\leadsto {b}^{2} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024014 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))