ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.8%

Time: 26.4s

Alternatives: 9

Speedup: N/A×

• 36.4% 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: 79.9% 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: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{\sqrt{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)}{{\pi}^{-0.5}}\right)\right)}^{2} + {b}^{2} \end{array} \]

FPCore

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

C

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

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((Math.sqrt(Math.PI) * (0.005555555555555556 * angle)) / Math.pow(Math.PI, -0.5)))), 2.0) + Math.pow(b, 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((math.sqrt(math.pi) * (0.005555555555555556 * angle)) / math.pow(math.pi, -0.5)))), 2.0) + math.pow(b, 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((sqrt(pi) * (0.005555555555555556 * angle)) / (pi ^ -0.5)))) ^ 2.0) + (b ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

   1. associate-*r/ 81.5%

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

   2. *-commutative 81.5%

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

   3. associate-/l* 81.6%

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

   4. add-sqr-sqrt 81.4%

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

   5. associate-*l/ 81.4%

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

   6. associate-/r/ 81.4%

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

   7. div-inv 81.5%

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

   8. associate-/r* 81.6%

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

   9. clear-num 81.5%

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

   10. associate-/l/ 81.5%

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

   11. associate-/r/ 81.6%

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

   12. metadata-eval 81.6%

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

   13. *-commutative 81.6%

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

   14. pow1/2 81.6%

       \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}{\frac{1}{\color{blue}{{\pi}^{0.5}}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   15. pow-flip 81.6%

       \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}{\color{blue}{{\pi}^{\left(-0.5\right)}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   16. metadata-eval 81.6%

       \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}{{\pi}^{\color{blue}{-0.5}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Applied egg-rr 81.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}{{\pi}^{-0.5}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
8. Step-by-step derivation

   1. add-log-exp 41.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{\color{blue}{\log \left(e^{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}\right)}}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   2. associate-*r* 41.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{\log \left(e^{\color{blue}{\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}}}\right)}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   3. *-commutative 41.4%

      \[\leadsto {\left(a \cdot \sin \left(\frac{\log \left(e^{\color{blue}{\left(angle \cdot 0.005555555555555556\right)} \cdot \sqrt{\pi}}\right)}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   4. exp-prod 41.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{\log \color{blue}{\left({\left(e^{angle \cdot 0.005555555555555556}\right)}^{\left(\sqrt{\pi}\right)}\right)}}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   5. exp-prod 41.0%

      \[\leadsto {\left(a \cdot \sin \left(\frac{\log \left({\color{blue}{\left({\left(e^{angle}\right)}^{0.005555555555555556}\right)}}^{\left(\sqrt{\pi}\right)}\right)}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

9. Applied egg-rr 41.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{\color{blue}{\log \left({\left({\left(e^{angle}\right)}^{0.005555555555555556}\right)}^{\left(\sqrt{\pi}\right)}\right)}}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
10. Step-by-step derivation

    1. log-pow 41.0%

       \[\leadsto {\left(a \cdot \sin \left(\frac{\color{blue}{\sqrt{\pi} \cdot \log \left({\left(e^{angle}\right)}^{0.005555555555555556}\right)}}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

    2. log-pow 41.1%

       \[\leadsto {\left(a \cdot \sin \left(\frac{\sqrt{\pi} \cdot \color{blue}{\left(0.005555555555555556 \cdot \log \left(e^{angle}\right)\right)}}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

    3. rem-log-exp 81.6%

       \[\leadsto {\left(a \cdot \sin \left(\frac{\sqrt{\pi} \cdot \left(0.005555555555555556 \cdot \color{blue}{angle}\right)}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

11. Simplified 81.6%

    \[\leadsto {\left(a \cdot \sin \left(\frac{\color{blue}{\sqrt{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)}}{{\pi}^{-0.5}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

12. Final simplification 81.6%

    \[\leadsto {\left(a \cdot \sin \left(\frac{\sqrt{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)}{{\pi}^{-0.5}}\right)\right)}^{2} + {b}^{2} \]
13. Add Preprocessing

Alternative 2: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {b}^{2} + {\left(a \cdot \sin \left(\frac{0.005555555555555556 \cdot \left(\sqrt{\pi} \cdot angle\right)}{{\pi}^{-0.5}}\right)\right)}^{2} \end{array} \]

FPCore

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

C

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

Java

public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + Math.pow((a * Math.sin(((0.005555555555555556 * (Math.sqrt(Math.PI) * angle)) / Math.pow(Math.PI, -0.5)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow(b, 2.0) + math.pow((a * math.sin(((0.005555555555555556 * (math.sqrt(math.pi) * angle)) / math.pow(math.pi, -0.5)))), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + ((a * sin(((0.005555555555555556 * (sqrt(pi) * angle)) / (pi ^ -0.5)))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

   1. associate-*r/ 81.5%

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

   2. *-commutative 81.5%

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

   3. associate-/l* 81.6%

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

   4. add-sqr-sqrt 81.4%

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

   5. associate-*l/ 81.4%

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

   6. associate-/r/ 81.4%

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

   7. div-inv 81.5%

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

   8. associate-/r* 81.6%

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

   9. clear-num 81.5%

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

   10. associate-/l/ 81.5%

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

   11. associate-/r/ 81.6%

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

   12. metadata-eval 81.6%

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

   13. *-commutative 81.6%

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

   14. pow1/2 81.6%

       \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}{\frac{1}{\color{blue}{{\pi}^{0.5}}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   15. pow-flip 81.6%

       \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}{\color{blue}{{\pi}^{\left(-0.5\right)}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   16. metadata-eval 81.6%

       \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}{{\pi}^{\color{blue}{-0.5}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Applied egg-rr 81.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{0.005555555555555556 \cdot \left(angle \cdot \sqrt{\pi}\right)}{{\pi}^{-0.5}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

8. Final simplification 81.6%

   \[\leadsto {b}^{2} + {\left(a \cdot \sin \left(\frac{0.005555555555555556 \cdot \left(\sqrt{\pi} \cdot angle\right)}{{\pi}^{-0.5}}\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 3: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

6. Taylor expanded in angle around 0 81.4%

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

7. Final simplification 81.4%

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

Alternative 4: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

6. Final simplification 81.5%

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

Alternative 5: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

   1. clear-num 81.5%

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

   2. div-inv 81.5%

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

7. Applied egg-rr 81.5%

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

8. Final simplification 81.5%

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

Alternative 6: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

   1. associate-*r/ 81.5%

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

   2. *-commutative 81.5%

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

   3. associate-/l* 81.6%

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

7. Applied egg-rr 81.6%

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

8. Final simplification 81.6%

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

Alternative 7: 77.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ {b}^{2} + 0.005555555555555556 \cdot \left(\left(angle \cdot \left(-1 + \left(1 + a \cdot \pi\right)\right)\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

6. Taylor expanded in angle around 0 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r* 77.2%

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

8. Simplified 77.2%

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

   1. unpow2 77.2%

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

   2. associate-*l* 77.2%

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

   3. *-commutative 77.2%

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

   4. *-commutative 77.2%

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

   5. *-commutative 77.2%

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

   6. associate-*l* 77.2%

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

   7. *-commutative 77.2%

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

   8. associate-*l* 77.2%

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

   9. associate-*r* 77.2%

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

   10. *-commutative 77.2%

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

10. Applied egg-rr 77.2%

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

    1. *-commutative 77.2%

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

    2. *-commutative 77.2%

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

    3. associate-*r* 77.2%

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

12. Simplified 77.2%

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

    1. expm1-log1p-u 55.9%

       \[\leadsto 0.005555555555555556 \cdot \left(\left(angle \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot a\right)\right)}\right) \cdot \left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]

    2. expm1-def 59.1%

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

    3. sub-neg 59.1%

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

    4. distribute-lft-in 59.1%

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

    5. log1p-udef 59.1%

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

    6. rem-exp-log 80.4%

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

    7. metadata-eval 80.4%

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

14. Applied egg-rr 80.4%

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

    1. distribute-lft-out 80.5%

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

    2. +-commutative 80.5%

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

16. Simplified 80.5%

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

17. Final simplification 80.5%

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

Alternative 8: 74.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

6. Taylor expanded in angle around 0 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r* 77.2%

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

8. Simplified 77.2%

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

   1. unpow2 77.2%

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

   2. associate-*l* 77.2%

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

   3. *-commutative 77.2%

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

   4. *-commutative 77.2%

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

   5. *-commutative 77.2%

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

   6. associate-*l* 77.2%

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

   7. *-commutative 77.2%

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

   8. associate-*l* 77.2%

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

   9. associate-*r* 77.2%

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

   10. *-commutative 77.2%

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

10. Applied egg-rr 77.2%

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

    1. *-commutative 77.2%

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

    2. *-commutative 77.2%

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

    3. associate-*r* 77.2%

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

12. Simplified 77.2%

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

13. Final simplification 77.2%

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

Alternative 9: 74.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.0%

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

   1. unpow2 81.0%

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

   2. swap-sqr 81.0%

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

   3. *-commutative 81.0%

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

   4. associate-*r/ 81.0%

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

   5. associate-*l/ 81.1%

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

   6. *-commutative 81.1%

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

   7. swap-sqr 81.1%

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

   8. unpow2 81.1%

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

   9. *-commutative 81.1%

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

   10. associate-*r/ 81.1%

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

   11. associate-*l/ 81.2%

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

   12. *-commutative 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.5%

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

6. Taylor expanded in angle around 0 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r* 77.2%

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

8. Simplified 77.2%

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

   1. unpow2 77.2%

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

   2. associate-*l* 77.2%

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

   3. *-commutative 77.2%

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

   4. *-commutative 77.2%

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

   5. *-commutative 77.2%

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

   6. associate-*l* 77.2%

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

   7. *-commutative 77.2%

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

   8. associate-*l* 77.2%

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

   9. associate-*r* 77.2%

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

   10. *-commutative 77.2%

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

10. Applied egg-rr 77.2%

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

    1. *-commutative 77.2%

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

    2. *-commutative 77.2%

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

    3. associate-*r* 77.2%

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

12. Simplified 77.2%

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

    1. *-commutative 77.2%

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

    2. *-commutative 77.2%

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

    3. metadata-eval 77.2%

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

    4. div-inv 77.2%

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

    5. associate-/r/ 77.2%

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

14. Applied egg-rr 77.2%

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

15. Final simplification 77.2%

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

Reproduce

herbie shell --seed 2024024 
(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)))