ab-angle->ABCF A

Percentage Accurate: 80.3% → 80.4%

Time: 34.1s

Alternatives: 11

Speedup: 1.3×

• 36.3% 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: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) + 1}\\ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\left(1 + t\_0\right) \cdot \left(t\_0 + -1\right)\right)} + -1\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (+ (cos (* angle (* PI 0.005555555555555556))) 1.0))))
   (+
    (pow (* a (sin (* angle (/ PI 180.0)))) 2.0)
    (pow (* b (+ (exp (log1p (* (+ 1.0 t_0) (+ t_0 -1.0)))) -1.0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = sqrt((cos((angle * (((double) M_PI) * 0.005555555555555556))) + 1.0));
	return pow((a * sin((angle * (((double) M_PI) / 180.0)))), 2.0) + pow((b * (exp(log1p(((1.0 + t_0) * (t_0 + -1.0)))) + -1.0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sqrt((Math.cos((angle * (Math.PI * 0.005555555555555556))) + 1.0));
	return Math.pow((a * Math.sin((angle * (Math.PI / 180.0)))), 2.0) + Math.pow((b * (Math.exp(Math.log1p(((1.0 + t_0) * (t_0 + -1.0)))) + -1.0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.sqrt((math.cos((angle * (math.pi * 0.005555555555555556))) + 1.0))
	return math.pow((a * math.sin((angle * (math.pi / 180.0)))), 2.0) + math.pow((b * (math.exp(math.log1p(((1.0 + t_0) * (t_0 + -1.0)))) + -1.0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = sqrt(Float64(cos(Float64(angle * Float64(pi * 0.005555555555555556))) + 1.0))
	return Float64((Float64(a * sin(Float64(angle * Float64(pi / 180.0)))) ^ 2.0) + (Float64(b * Float64(exp(log1p(Float64(Float64(1.0 + t_0) * Float64(t_0 + -1.0)))) + -1.0)) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sqrt[N[(N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[(N[Exp[N[Log[1 + N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

   1. associate-*r/ 74.2%

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

   2. associate-*l/ 74.2%

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

   3. expm1-log1p-u 74.2%

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

   4. expm1-undefine 74.2%

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

   5. associate-*l/ 74.2%

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

   6. associate-*r/ 74.3%

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

   7. div-inv 74.3%

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

   8. metadata-eval 74.3%

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

6. Applied egg-rr 74.3%

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

   1. expm1-log1p-u 74.3%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)} - 1\right)\right)}^{2} \]

   2. expm1-undefine 74.3%

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

   3. add-sqr-sqrt 73.5%

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

   4. fma-neg 74.3%

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

8. Applied egg-rr 74.3%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sqrt{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) + 1}, \sqrt{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) + 1}, -1\right)}\right)} - 1\right)\right)}^{2} \]
9. Step-by-step derivation

   1. fma-undefine 73.5%

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

   2. difference-of-sqr--1 74.3%

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

10. Applied egg-rr 74.3%

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

11. Final simplification 74.3%

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

Alternative 2: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \left(\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) + 1\right) + -1\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* PI (/ angle 180.0)))) 2.0)
  (pow
   (* b (+ (+ (cos (* angle (* PI 0.005555555555555556))) 1.0) -1.0))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * ((cos((angle * (((double) M_PI) * 0.005555555555555556))) + 1.0) + -1.0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((b * ((Math.cos((angle * (Math.PI * 0.005555555555555556))) + 1.0) + -1.0)), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((math.pi * (angle / 180.0)))), 2.0) + math.pow((b * ((math.cos((angle * (math.pi * 0.005555555555555556))) + 1.0) + -1.0)), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(b * Float64(Float64(cos(Float64(angle * Float64(pi * 0.005555555555555556))) + 1.0) + -1.0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((pi * (angle / 180.0)))) ^ 2.0) + ((b * ((cos((angle * (pi * 0.005555555555555556))) + 1.0) + -1.0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[(N[(N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. clear-num 74.1%

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

4. Applied egg-rr 74.1%

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

   1. clear-num 74.2%

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

   2. div-inv 74.2%

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

   3. metadata-eval 74.2%

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

   4. associate-*r* 74.3%

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

   5. expm1-log1p-u 74.3%

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

   6. log1p-define 74.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\color{blue}{\log \left(1 + \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}\right)\right)}^{2} \]

   7. +-commutative 74.3%

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\log \color{blue}{\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) + 1\right)}\right)\right)}^{2} \]

   8. expm1-undefine 74.3%

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

   9. add-exp-log 74.3%

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

6. Applied egg-rr 74.3%

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

7. Final simplification 74.3%

   \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \left(\left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) + 1\right) + -1\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in angle around inf 74.2%

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

6. Final simplification 74.2%

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

Alternative 4: 80.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (/ PI 180.0))))
   (+ (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 * (((double) M_PI) / 180.0);
	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 * (Math.PI / 180.0);
	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 * (math.pi / 180.0)
	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(angle * Float64(pi / 180.0))
	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 * (pi / 180.0);
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi / 180.0), $MachinePrecision]), $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]]

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

5. Final simplification 74.3%

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

Alternative 5: 67.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.3 \cdot 10^{-57}:\\ \;\;\;\;{\left(a \cdot 0\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(0.005555555555555556 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(a \cdot angle\right)\right)\right)\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.3e-57)
   (+ (pow (* a 0.0) 2.0) (pow (* b (cos (* PI (/ angle 180.0)))) 2.0))
   (+
    (pow b 2.0)
    (*
     (* 0.005555555555555556 (* PI (* 0.005555555555555556 (* a angle))))
     (* angle (* a PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.3e-57) {
		tmp = pow((a * 0.0), 2.0) + pow((b * cos((((double) M_PI) * (angle / 180.0)))), 2.0);
	} else {
		tmp = pow(b, 2.0) + ((0.005555555555555556 * (((double) M_PI) * (0.005555555555555556 * (a * angle)))) * (angle * (a * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.3e-57) {
		tmp = Math.pow((a * 0.0), 2.0) + Math.pow((b * Math.cos((Math.PI * (angle / 180.0)))), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + ((0.005555555555555556 * (Math.PI * (0.005555555555555556 * (a * angle)))) * (angle * (a * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 4.3e-57:
		tmp = math.pow((a * 0.0), 2.0) + math.pow((b * math.cos((math.pi * (angle / 180.0)))), 2.0)
	else:
		tmp = math.pow(b, 2.0) + ((0.005555555555555556 * (math.pi * (0.005555555555555556 * (a * angle)))) * (angle * (a * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.3e-57)
		tmp = Float64((Float64(a * 0.0) ^ 2.0) + (Float64(b * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(0.005555555555555556 * Float64(pi * Float64(0.005555555555555556 * Float64(a * angle)))) * Float64(angle * Float64(a * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 4.3e-57)
		tmp = ((a * 0.0) ^ 2.0) + ((b * cos((pi * (angle / 180.0)))) ^ 2.0);
	else
		tmp = (b ^ 2.0) + ((0.005555555555555556 * (pi * (0.005555555555555556 * (a * angle)))) * (angle * (a * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 4.3e-57], N[(N[Power[N[(a * 0.0), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(0.005555555555555556 * N[(Pi * N[(0.005555555555555556 * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.30000000000000022e-57

   1. Initial program 72.0%

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

      1. add-cube-cbrt 71.7%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\frac{angle}{180} \cdot \pi} \cdot \sqrt[3]{\frac{angle}{180} \cdot \pi}\right) \cdot \sqrt[3]{\frac{angle}{180} \cdot \pi}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. pow3 71.7%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\frac{angle}{180} \cdot \pi}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. associate-*l/ 71.6%

         \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{\color{blue}{\frac{angle \cdot \pi}{180}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. associate-*r/ 71.7%

         \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{\color{blue}{angle \cdot \frac{\pi}{180}}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. div-inv 71.7%

         \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \color{blue}{\left(\pi \cdot \frac{1}{180}\right)}}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. metadata-eval 71.7%

         \[\leadsto {\left(a \cdot \sin \left({\left(\sqrt[3]{angle \cdot \left(\pi \cdot \color{blue}{0.005555555555555556}\right)}\right)}^{3}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied egg-rr 71.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left({\left(\sqrt[3]{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)}^{3}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0 56.9%

      \[\leadsto {\left(a \cdot \color{blue}{0}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   if 4.30000000000000022e-57 < a

   1. Initial program 79.7%

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

      1. associate-*l/ 79.8%

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

      2. associate-/l* 79.9%

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

      3. associate-*l/ 80.0%

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

      4. associate-/l* 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in angle around 0 80.1%

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

   6. Taylor expanded in angle around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*l* 76.6%

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

   8. Simplified 76.6%

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

      1. unpow2 76.6%

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

      2. associate-*r* 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-*l* 76.6%

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

      5. associate-*r* 76.6%

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

      6. *-commutative 76.6%

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

      7. associate-*l* 76.6%

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

   10. Applied egg-rr 76.6%

       \[\leadsto \color{blue}{\left(\left(\pi \cdot \left(\left(angle \cdot a\right) \cdot 0.005555555555555556\right)\right) \cdot 0.005555555555555556\right) \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.3 \cdot 10^{-57}:\\ \;\;\;\;{\left(a \cdot 0\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(0.005555555555555556 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(a \cdot angle\right)\right)\right)\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} + {b}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow (* a (sin (/ 1.0 (/ 180.0 (* angle PI))))) 2.0) (pow b 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((1.0 / (180.0 / (angle * ((double) M_PI)))))), 2.0) + pow(b, 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((1.0 / (180.0 / (angle * Math.PI))))), 2.0) + Math.pow(b, 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((1.0 / (180.0 / (angle * math.pi))))), 2.0) + math.pow(b, 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(1.0 / Float64(180.0 / Float64(angle * pi))))) ^ 2.0) + (b ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin((1.0 / (180.0 / (angle * pi))))) ^ 2.0) + (b ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(1.0 / N[(180.0 / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in angle around 0 73.4%

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

   1. associate-*r/ 73.4%

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

   2. clear-num 73.4%

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

7. Applied egg-rr 73.4%

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

8. Final simplification 73.4%

   \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} + {b}^{2} \]
9. Add Preprocessing

Alternative 7: 80.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in angle around 0 73.4%

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

6. Final simplification 73.4%

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

Alternative 8: 75.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in angle around 0 73.4%

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

6. Taylor expanded in angle around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. *-commutative 69.0%

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

   3. associate-*l* 69.0%

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

8. Simplified 69.0%

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

   1. unpow2 69.0%

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

   2. associate-*r* 69.0%

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

   3. *-commutative 69.0%

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

   4. associate-*l* 69.0%

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

   5. *-commutative 69.0%

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

   6. associate-*l* 69.0%

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

10. Applied egg-rr 69.0%

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

    1. associate-*l* 69.0%

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

    2. associate-*r* 69.0%

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

    3. *-commutative 69.0%

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

    4. associate-*r* 69.0%

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

    5. *-commutative 69.0%

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

    6. associate-*r* 69.0%

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

    7. *-commutative 69.0%

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

    8. associate-*l* 69.0%

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

    9. *-commutative 69.0%

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

12. Simplified 69.0%

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

13. Final simplification 69.0%

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

Alternative 9: 75.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in angle around 0 73.4%

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

6. Taylor expanded in angle around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. *-commutative 69.0%

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

   3. associate-*l* 69.0%

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

8. Simplified 69.0%

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

   1. unpow2 69.0%

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

   2. associate-*r* 69.0%

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

   3. *-commutative 69.0%

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

   4. associate-*l* 69.0%

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

   5. *-commutative 69.0%

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

   6. associate-*l* 69.0%

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

10. Applied egg-rr 69.0%

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

11. Final simplification 69.0%

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

Alternative 10: 75.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in angle around 0 73.4%

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

6. Taylor expanded in angle around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. *-commutative 69.0%

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

   3. associate-*l* 69.0%

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

8. Simplified 69.0%

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

   1. unpow2 69.0%

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

   2. associate-*r* 69.0%

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

   3. *-commutative 69.0%

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

   4. associate-*l* 69.0%

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

   5. associate-*r* 69.0%

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

   6. *-commutative 69.0%

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

   7. associate-*l* 69.0%

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

10. Applied egg-rr 69.0%

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

11. Final simplification 69.0%

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

Alternative 11: 57.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ {b}^{2} \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (pow b 2.0))

C

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

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = b ** 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[Power[b, 2.0], $MachinePrecision]

Derivation

1. Initial program 74.2%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

   3. associate-*l/ 74.2%

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

   4. associate-/l* 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in angle around 0 73.4%

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

6. Taylor expanded in angle around 0 69.0%

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

   1. *-commutative 69.0%

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

   2. *-commutative 69.0%

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

   3. associate-*l* 69.0%

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

8. Simplified 69.0%

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

9. Taylor expanded in angle around 0 51.8%

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

10. Final simplification 51.8%

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

Reproduce

herbie shell --seed 2024044 
(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)))