ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.4%

Time: 1.3min

Alternatives: 6

Speedup: N/A×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. swap-sqr 71.9%

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

   3. sqr-neg 71.9%

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

   4. swap-sqr 80.6%

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

   5. unpow2 80.6%

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

   6. distribute-lft-neg-out 80.6%

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

   7. distribute-rgt-neg-in 80.6%

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

   8. sin-neg 80.6%

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

   9. distribute-rgt-neg-out 80.6%

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

   10. distribute-frac-neg 80.6%

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

   11. unpow2 80.6%

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

   12. associate-*l* 79.9%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 81.0%

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

5. Taylor expanded in angle around inf 81.0%

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

   1. associate-*r* 81.0%

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

   2. *-commutative 81.0%

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

   3. *-commutative 81.0%

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

   4. unpow1 81.0%

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

   5. sqr-pow 42.1%

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

   6. fabs-sqr 42.1%

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

   7. sqr-pow 81.0%

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

   8. unpow1 81.0%

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

   9. associate-*r* 81.0%

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

   10. *-commutative 81.0%

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

   11. associate-*r* 81.1%

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

   12. fabs-mul 81.1%

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

   13. fabs-mul 81.1%

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

   14. metadata-eval 81.1%

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

   15. unpow1 81.1%

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

   16. sqr-pow 80.7%

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

   17. fabs-sqr 80.7%

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

   18. sqr-pow 81.1%

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

   19. unpow1 81.1%

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

   20. unpow1 81.1%

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

   21. sqr-pow 38.9%

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

   22. fabs-sqr 38.9%

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

   23. sqr-pow 81.1%

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

   24. unpow1 81.1%

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

7. Simplified 81.1%

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

8. Final simplification 81.1%

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

Alternative 2: 79.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

2. Taylor expanded in angle around 0 81.0%

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

3. Taylor expanded in b around 0 81.0%

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

4. Final simplification 81.0%

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

Alternative 3: 74.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

2. Taylor expanded in angle around 0 81.0%

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

3. Taylor expanded in angle around 0 77.3%

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

   1. associate-*r* 77.3%

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

5. Simplified 77.3%

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

   1. unpow2 77.3%

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

   2. *-commutative 77.3%

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

   3. associate-*r* 77.3%

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

   4. associate-*r* 77.3%

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

   5. *-commutative 77.3%

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

   6. *-commutative 77.3%

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

   7. associate-*r* 77.3%

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

   8. associate-*l* 77.3%

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

   9. *-commutative 77.3%

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

7. Applied egg-rr 77.3%

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

8. Final simplification 77.3%

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

Alternative 4: 74.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* b 0.005555555555555556))))
   (+ (pow a 2.0) (* PI (* t_0 (* PI t_0))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

2. Taylor expanded in angle around 0 81.0%

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

3. Taylor expanded in angle around 0 77.3%

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

   1. associate-*r* 77.3%

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

5. Simplified 77.3%

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

   1. unpow2 77.3%

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

   2. associate-*r* 77.3%

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

   3. associate-*r* 77.3%

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

   4. associate-*r* 77.3%

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

   5. *-commutative 77.3%

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

   6. *-commutative 77.3%

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

   7. associate-*r* 77.3%

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

   8. *-commutative 77.3%

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

   9. associate-*r* 77.3%

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

7. Applied egg-rr 77.3%

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

8. Final simplification 77.3%

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

Alternative 5: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(b \cdot angle\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* PI (* b angle)) 2.0))))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + (3.08641975308642e-5 * pow((((double) M_PI) * (b * angle)), 2.0));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow((Math.PI * (b * angle)), 2.0));
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow((math.pi * (b * angle)), 2.0))

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(pi * Float64(b * angle)) ^ 2.0)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((pi * (b * angle)) ^ 2.0));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(Pi * N[(b * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.6%

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

   1. unpow2 80.6%

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

   2. swap-sqr 71.9%

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

   3. sqr-neg 71.9%

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

   4. swap-sqr 80.6%

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

   5. unpow2 80.6%

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

   6. distribute-lft-neg-out 80.6%

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

   7. distribute-rgt-neg-in 80.6%

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

   8. sin-neg 80.6%

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

   9. distribute-rgt-neg-out 80.6%

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

   10. distribute-frac-neg 80.6%

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

   11. unpow2 80.6%

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

   12. associate-*l* 79.9%

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

3. Simplified 80.6%

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

4. Taylor expanded in angle around 0 81.0%

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

5. Taylor expanded in angle around inf 81.0%

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

   1. associate-*r* 81.0%

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

   2. *-commutative 81.0%

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

   3. *-commutative 81.0%

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

   4. unpow1 81.0%

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

   5. sqr-pow 42.1%

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

   6. fabs-sqr 42.1%

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

   7. sqr-pow 81.0%

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

   8. unpow1 81.0%

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

   9. associate-*r* 81.0%

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

   10. *-commutative 81.0%

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

   11. associate-*r* 81.1%

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

   12. fabs-mul 81.1%

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

   13. fabs-mul 81.1%

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

   14. metadata-eval 81.1%

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

   15. unpow1 81.1%

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

   16. sqr-pow 80.7%

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

   17. fabs-sqr 80.7%

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

   18. sqr-pow 81.1%

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

   19. unpow1 81.1%

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

   20. unpow1 81.1%

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

   21. sqr-pow 38.9%

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

   22. fabs-sqr 38.9%

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

   23. sqr-pow 81.1%

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

   24. unpow1 81.1%

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

7. Simplified 81.1%

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

8. Taylor expanded in angle around 0 67.4%

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

   1. unpow2 67.4%

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

   2. unpow2 67.4%

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

   3. *-commutative 67.4%

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

   4. unpow2 67.4%

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

   5. unswap-sqr 67.4%

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

   6. swap-sqr 77.3%

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

   7. unpow2 77.3%

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

   8. *-commutative 77.3%

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

   9. associate-*r* 77.3%

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

   10. *-commutative 77.3%

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

   11. *-commutative 77.3%

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

10. Simplified 77.3%

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

11. Final simplification 77.3%

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

Alternative 6: 74.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.6%

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

2. Taylor expanded in angle around 0 81.0%

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

3. Taylor expanded in angle around 0 77.3%

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

   1. associate-*r* 77.3%

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

5. Simplified 77.3%

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

   1. expm1-log1p-u 59.2%

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

   2. expm1-udef 54.4%

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

   3. associate-*r* 54.4%

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

   4. *-commutative 54.4%

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

   5. *-commutative 54.4%

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

   6. associate-*r* 54.4%

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

7. Applied egg-rr 54.4%

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

   1. expm1-def 59.2%

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

   2. expm1-log1p 77.3%

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

   3. associate-*l* 77.3%

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

9. Simplified 77.3%

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

10. Final simplification 77.3%

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

Reproduce

herbie shell --seed 2023275 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))