ab-angle->ABCF C

Percentage Accurate: 79.1% → 78.9%

Time: 1.1min

Alternatives: 7

Speedup: N/A×

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% 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: 78.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

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. Step-by-step derivation

   1. associate-*r/ 81.0%

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

   2. clear-num 81.1%

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

4. Applied egg-rr 81.1%

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

   1. clear-num 81.1%

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

   2. inv-pow 81.1%

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

   3. clear-num 81.1%

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

   4. add-sqr-sqrt 36.2%

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

   5. sqrt-unprod 64.9%

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

   6. associate-/r/ 64.9%

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

   7. associate-/r/ 64.9%

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

   8. swap-sqr 64.5%

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

   9. metadata-eval 64.5%

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

   10. metadata-eval 64.5%

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

   11. metadata-eval 64.5%

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

   12. metadata-eval 64.5%

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

   13. swap-sqr 64.9%

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

   14. sqrt-unprod 44.8%

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

   15. add-sqr-sqrt 81.1%

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

   16. *-commutative 81.1%

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

   17. associate-*r* 81.1%

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

6. Applied egg-rr 81.1%

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

   1. unpow-1 81.1%

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

   2. associate-*r* 81.1%

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

   3. associate-/r* 81.2%

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

8. Applied egg-rr 81.2%

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

9. Final simplification 81.2%

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

Alternative 2: 78.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

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. Step-by-step derivation

   1. associate-*r/ 81.0%

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

   2. clear-num 81.1%

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

4. Applied egg-rr 81.1%

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

5. Final simplification 81.1%

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

Alternative 3: 79.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. unpow2 79.7%

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

   2. swap-sqr 73.0%

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

   3. sqr-neg 73.0%

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

   4. swap-sqr 79.7%

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

   5. unpow2 79.7%

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

   6. distribute-lft-neg-out 79.7%

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

   7. distribute-rgt-neg-in 79.7%

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

   8. sin-neg 79.7%

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

   9. distribute-rgt-neg-out 79.7%

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

   10. distribute-frac-neg 79.7%

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

   11. unpow2 79.7%

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

   12. associate-*l* 79.1%

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

3. Simplified 79.8%

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

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. Final simplification 81.0%

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

Alternative 4: 79.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

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. Step-by-step derivation

   1. associate-*r/ 81.0%

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

4. Applied egg-rr 81.0%

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

5. Final simplification 81.0%

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

Alternative 5: 74.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

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 75.4%

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

   1. *-commutative 75.4%

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

5. Simplified 75.4%

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

6. Taylor expanded in angle around 0 69.5%

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

   1. *-commutative 69.5%

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

   2. *-commutative 69.5%

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

   3. unpow2 69.5%

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

   4. unpow2 69.5%

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

   5. swap-sqr 69.5%

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

   6. unpow2 69.5%

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

   7. swap-sqr 75.5%

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

   8. associate-*r* 75.4%

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

   9. associate-*r* 75.8%

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

   10. unpow2 75.8%

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

   11. associate-*r* 75.5%

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

   12. *-commutative 75.5%

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

8. Simplified 75.5%

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

9. Final simplification 75.5%

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

Alternative 6: 74.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

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 75.4%

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

   1. *-commutative 75.4%

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

5. Simplified 75.4%

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

   1. *-commutative 75.4%

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

   2. unpow-prod-down 75.5%

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

   3. *-commutative 75.5%

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

   4. associate-*l* 75.8%

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

   5. metadata-eval 75.8%

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

7. Applied egg-rr 75.8%

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

8. Final simplification 75.8%

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

Alternative 7: 74.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

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 75.4%

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

   1. *-commutative 75.4%

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

5. Simplified 75.4%

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

6. Taylor expanded in angle around 0 75.4%

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

   1. associate-*r* 75.8%

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

   2. *-commutative 75.8%

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

   3. *-commutative 75.8%

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

   4. *-commutative 75.8%

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

   5. associate-*l* 75.8%

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

   6. *-commutative 75.8%

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

   7. associate-*l* 75.8%

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

8. Simplified 75.8%

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

9. Final simplification 75.8%

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

Reproduce

herbie shell --seed 2023279 
(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)))