ab-angle->ABCF C

Percentage Accurate: 79.2% → 79.1%

Time: 5.2s

Alternatives: 6

Speedup: N/A×

• 35.8% 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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-PI.f64 79.1

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

3. Applied rewrites 79.1%

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

Alternative 2: 79.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-PI.f64 79.1

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

3. Applied rewrites 79.1%

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

4. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. mult-flip N/A

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

   5. sin-PI/2 N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. metadata-eval N/A

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

   9. pow-neg N/A

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

   10. inv-pow N/A

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

   11. unpow-1 N/A

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

   12. unpow-prod-down N/A

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

   13. mult-flip N/A

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

   14. unpow-1 N/A

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

   15. inv-pow N/A

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

   16. pow-neg N/A

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

   17. metadata-eval N/A

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

   18. unpow1 79.1

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

6. Applied rewrites 79.1%

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

   1. lift-/.f64 N/A

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

   2. div-flip-rev N/A

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

   3. inv-pow N/A

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

   4. mult-flip N/A

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

   5. unpow-prod-down N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-/.f64 79.1

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

8. Applied rewrites 79.1%

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

Alternative 3: 79.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-PI.f64 79.1

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

3. Applied rewrites 79.1%

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

4. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. mult-flip N/A

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

   5. sin-PI/2 N/A

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

   6. metadata-eval N/A

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

   7. unpow1 N/A

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

   8. metadata-eval N/A

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

   9. pow-neg N/A

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

   10. inv-pow N/A

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

   11. unpow-1 N/A

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

   12. unpow-prod-down N/A

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

   13. mult-flip N/A

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

   14. unpow-1 N/A

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

   15. inv-pow N/A

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

   16. pow-neg N/A

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

   17. metadata-eval N/A

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

   18. unpow1 79.1

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

6. Applied rewrites 79.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 79.1

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

8. Applied rewrites 79.1%

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

Alternative 4: 66.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{-92}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.6e-92)
   (* a a)
   (+ (pow a 2.0) (pow (* b (* (* angle PI) 0.005555555555555556)) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.6e-92) {
		tmp = a * a;
	} else {
		tmp = pow(a, 2.0) + pow((b * ((angle * ((double) M_PI)) * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.6e-92) {
		tmp = a * a;
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((b * ((angle * Math.PI) * 0.005555555555555556)), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 5.6e-92:
		tmp = a * a
	else:
		tmp = math.pow(a, 2.0) + math.pow((b * ((angle * math.pi) * 0.005555555555555556)), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.6e-92)
		tmp = Float64(a * a);
	else
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5.6e-92)
		tmp = a * a;
	else
		tmp = (a ^ 2.0) + ((b * ((angle * pi) * 0.005555555555555556)) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 5.6e-92], N[(a * a), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.6e-92

   1. Initial program 79.2%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. lower-*.f64 56.7

         \[\leadsto a \cdot \color{blue}{a} \]

   4. Applied rewrites 56.7%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 5.6e-92 < b

   1. Initial program 79.2%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 79.1

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. sin-PI/2 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. inv-pow N/A

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

      11. unpow-1 N/A

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

      12. unpow-prod-down N/A

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

      13. mult-flip N/A

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

      14. unpow-1 N/A

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

      15. inv-pow N/A

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

      16. pow-neg N/A

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

      17. metadata-eval N/A

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

      18. unpow1 79.1

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

   6. Applied rewrites 79.1%

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

   7. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 74.0

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

   9. Applied rewrites 74.0%

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

Alternative 5: 61.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{-92}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(b \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.6e-92)
   (* a a)
   (+
    (pow a 2.0)
    (* (* 3.08641975308642e-5 (* angle angle)) (* (* PI PI) (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.6e-92) {
		tmp = a * a;
	} else {
		tmp = pow(a, 2.0) + ((3.08641975308642e-5 * (angle * angle)) * ((((double) M_PI) * ((double) M_PI)) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.6e-92) {
		tmp = a * a;
	} else {
		tmp = Math.pow(a, 2.0) + ((3.08641975308642e-5 * (angle * angle)) * ((Math.PI * Math.PI) * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 5.6e-92:
		tmp = a * a
	else:
		tmp = math.pow(a, 2.0) + ((3.08641975308642e-5 * (angle * angle)) * ((math.pi * math.pi) * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.6e-92)
		tmp = Float64(a * a);
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(3.08641975308642e-5 * Float64(angle * angle)) * Float64(Float64(pi * pi) * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 5.6e-92)
		tmp = a * a;
	else
		tmp = (a ^ 2.0) + ((3.08641975308642e-5 * (angle * angle)) * ((pi * pi) * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 5.6e-92], N[(a * a), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.6e-92

   1. Initial program 79.2%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{a} \]

      2. lower-*.f64 56.7

         \[\leadsto a \cdot \color{blue}{a} \]

   4. Applied rewrites 56.7%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 5.6e-92 < b

   1. Initial program 79.2%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-PI.f64 79.1

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

   3. Applied rewrites 79.1%

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

   4. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. sin-PI/2 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

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

      9. pow-neg N/A

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

      10. inv-pow N/A

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

      11. unpow-1 N/A

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

      12. unpow-prod-down N/A

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

      13. mult-flip N/A

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

      14. unpow-1 N/A

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

      15. inv-pow N/A

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

      16. pow-neg N/A

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

      17. metadata-eval N/A

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

      18. unpow1 79.1

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

   6. Applied rewrites 79.1%

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

   7. Taylor expanded in angle around 0

      \[\leadsto {a}^{2} + \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \]

      4. unpow2 N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({b}^{\color{blue}{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({b}^{\color{blue}{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \]

      6. *-commutative N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{{b}^{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{{b}^{2}}\right) \]

      8. unpow2 N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\color{blue}{b}}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\color{blue}{b}}^{2}\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot {b}^{2}\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {b}^{2}\right) \]

      12. pow2 N/A

          \[\leadsto {a}^{2} + \left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(b \cdot \color{blue}{b}\right)\right) \]

      13. lift-*.f64 63.6

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

   9. Applied rewrites 63.6%

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

Alternative 6: 56.7% accurate, 29.7× speedup

Math

\[\begin{array}{l} \\ a \cdot a \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* a a))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

public static double code(double a, double b, double angle) {
	return a * a;
}

Python

def code(a, b, angle):
	return a * a

Julia

function code(a, b, angle)
	return Float64(a * a)
end

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

2. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{a}^{2}} \]
3. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto a \cdot \color{blue}{a} \]

   2. lower-*.f64 56.7

      \[\leadsto a \cdot \color{blue}{a} \]

4. Applied rewrites 56.7%

   \[\leadsto \color{blue}{a \cdot a} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))