ab-angle->ABCF C

Percentage Accurate: 80.4% → 80.4%

Time: 3.7s

Alternatives: 13

Speedup: 1.0×

• 62.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* PI (/ angle 180))))
  (+ (pow (* a (cos t_0)) 2) (pow (* b (sin t_0)) 2))))

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), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Initial Program: 80.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* PI (/ angle 180))))
  (+ (pow (* a (cos t_0)) 2) (pow (* b (sin t_0)) 2))))

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), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 80.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (/ 1 (/ (/ 180 angle) PI))))
  (+ (pow (* a (cos t_0)) 2) (pow (* b (sin t_0)) 2))))

C

double code(double a, double b, double angle) {
	double t_0 = 1.0 / ((180.0 / angle) / ((double) M_PI));
	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 = 1.0 / ((180.0 / angle) / Math.PI);
	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 = 1.0 / ((180.0 / angle) / math.pi)
	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(1.0 / Float64(Float64(180.0 / angle) / pi))
	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 = 1.0 / ((180.0 / angle) / pi);
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(1 / N[(N[(180 / angle), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-flip N/A

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

   5. lower-unsound-/.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 80.3%

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

3. Applied rewrites 80.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-flip N/A

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

   5. lower-unsound-/.f64 N/A

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

   6. lower-unsound-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 80.3%

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

5. Applied rewrites 80.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 80.3%

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

7. Applied rewrites 80.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 80.3%

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

9. Applied rewrites 80.3%

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

Alternative 2: 80.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (* 1/180 PI) angle)))
  (+ (pow (* a (cos t_0)) 2) (pow (* b (sin t_0)) 2))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * ((double) M_PI)) * angle;
	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 = (0.005555555555555556 * Math.PI) * angle;
	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 = (0.005555555555555556 * math.pi) * angle
	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(Float64(0.005555555555555556 * pi) * angle)
	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 = (0.005555555555555556 * pi) * angle;
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(1/180 * Pi), $MachinePrecision] * angle), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 80.4%

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

3. Applied rewrites 80.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 80.4%

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

5. Applied rewrites 80.4%

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

Alternative 3: 80.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (* 1/180 angle) PI)))
  (+ (pow (* a (cos t_0)) 2) (pow (* b (sin t_0)) 2))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	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 = (0.005555555555555556 * angle) * Math.PI;
	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 = (0.005555555555555556 * angle) * math.pi
	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(Float64(0.005555555555555556 * angle) * pi)
	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 = (0.005555555555555556 * angle) * pi;
	tmp = ((a * cos(t_0)) ^ 2.0) + ((b * sin(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(1/180 * angle), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 80.4%

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.4%

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

3. Applied rewrites 80.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 80.4%

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

   4. lift-/.f64 N/A

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

   5. mult-flip N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 80.4%

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

5. Applied rewrites 80.4%

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

Alternative 4: 80.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(b \cdot \pi\right) \cdot \left|angle\right|\right) \cdot \frac{1}{180}\\ t_1 := \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot \left|angle\right|\right)\\ \mathbf{if}\;\left|angle\right| \leq \frac{3458764513820541}{4611686018427387904}:\\ \;\;\;\;\left(\left(\cos \left(\frac{-1}{90} \cdot \left(\left|angle\right| \cdot \pi\right)\right) + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{2} - t\_1\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + t\_1\right) \cdot a\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (* (* b PI) (fabs angle)) 1/180))
       (t_1 (* 1/2 (cos (* (* 1/90 PI) (fabs angle))))))
  (if (<= (fabs angle) 3458764513820541/4611686018427387904)
    (+
     (* (* (+ (cos (* -1/90 (* (fabs angle) PI))) 1) 1/2) (* a a))
     (* t_0 t_0))
    (+ (* (* (- 1/2 t_1) b) b) (* (* (+ 1/2 t_1) a) a)))))

C

double code(double a, double b, double angle) {
	double t_0 = ((b * ((double) M_PI)) * fabs(angle)) * 0.005555555555555556;
	double t_1 = 0.5 * cos(((0.011111111111111112 * ((double) M_PI)) * fabs(angle)));
	double tmp;
	if (fabs(angle) <= 0.00075) {
		tmp = (((cos((-0.011111111111111112 * (fabs(angle) * ((double) M_PI)))) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	} else {
		tmp = (((0.5 - t_1) * b) * b) + (((0.5 + t_1) * a) * a);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = ((b * Math.PI) * Math.abs(angle)) * 0.005555555555555556;
	double t_1 = 0.5 * Math.cos(((0.011111111111111112 * Math.PI) * Math.abs(angle)));
	double tmp;
	if (Math.abs(angle) <= 0.00075) {
		tmp = (((Math.cos((-0.011111111111111112 * (Math.abs(angle) * Math.PI))) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	} else {
		tmp = (((0.5 - t_1) * b) * b) + (((0.5 + t_1) * a) * a);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = ((b * math.pi) * math.fabs(angle)) * 0.005555555555555556
	t_1 = 0.5 * math.cos(((0.011111111111111112 * math.pi) * math.fabs(angle)))
	tmp = 0
	if math.fabs(angle) <= 0.00075:
		tmp = (((math.cos((-0.011111111111111112 * (math.fabs(angle) * math.pi))) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0)
	else:
		tmp = (((0.5 - t_1) * b) * b) + (((0.5 + t_1) * a) * a)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(Float64(b * pi) * abs(angle)) * 0.005555555555555556)
	t_1 = Float64(0.5 * cos(Float64(Float64(0.011111111111111112 * pi) * abs(angle))))
	tmp = 0.0
	if (abs(angle) <= 0.00075)
		tmp = Float64(Float64(Float64(Float64(cos(Float64(-0.011111111111111112 * Float64(abs(angle) * pi))) + 1.0) * 0.5) * Float64(a * a)) + Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 - t_1) * b) * b) + Float64(Float64(Float64(0.5 + t_1) * a) * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = ((b * pi) * abs(angle)) * 0.005555555555555556;
	t_1 = 0.5 * cos(((0.011111111111111112 * pi) * abs(angle)));
	tmp = 0.0;
	if (abs(angle) <= 0.00075)
		tmp = (((cos((-0.011111111111111112 * (abs(angle) * pi))) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	else
		tmp = (((0.5 - t_1) * b) * b) + (((0.5 + t_1) * a) * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(b * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * 1/180), $MachinePrecision]}, Block[{t$95$1 = N[(1/2 * N[Cos[N[(N[(1/90 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[angle], $MachinePrecision], 3458764513820541/4611686018427387904], N[(N[(N[(N[(N[Cos[N[(-1/90 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1), $MachinePrecision] * 1/2), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1/2 - t$95$1), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] + N[(N[(N[(1/2 + t$95$1), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.5000000000000002e-4

   1. Initial program 80.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 75.8%

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\left(\left(\cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right)} \]

   if 7.5000000000000002e-4 < angle

   1. Initial program 80.4%

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

   3. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot \frac{1}{90}\right)\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\pi \cdot \frac{1}{90}\right) \cdot angle\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\pi \cdot \frac{1}{90}\right) \cdot angle\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\color{blue}{\left(\frac{1}{90} \cdot \pi\right)} \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      7. lower-*.f64 68.2%

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\color{blue}{\left(\frac{1}{90} \cdot \pi\right)} \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

   5. Applied rewrites 68.2%

      \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)}\right) \cdot a\right) \cdot a \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      3. associate-*l* N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(angle \cdot \left(\pi \cdot \frac{1}{90}\right)\right)}\right) \cdot a\right) \cdot a \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\pi \cdot \frac{1}{90}\right) \cdot angle\right)}\right) \cdot a\right) \cdot a \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\pi \cdot \frac{1}{90}\right) \cdot angle\right)}\right) \cdot a\right) \cdot a \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\color{blue}{\left(\frac{1}{90} \cdot \pi\right)} \cdot angle\right)\right) \cdot a\right) \cdot a \]

      7. lower-*.f64 68.3%

         \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\color{blue}{\left(\frac{1}{90} \cdot \pi\right)} \cdot angle\right)\right) \cdot a\right) \cdot a \]

   7. Applied rewrites 68.3%

      \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\left(\frac{1}{90} \cdot \pi\right) \cdot angle\right)}\right) \cdot a\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.3% accurate, 1.3× speedup

Math

\[\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (* (+ 1/2 (* 1/2 (cos (* (* angle PI) 1/90)))) (* a a))
 (pow (* b (sin (* PI (/ angle 180)))) 2)))

C

double code(double a, double b, double angle) {
	return ((0.5 + (0.5 * cos(((angle * ((double) M_PI)) * 0.011111111111111112)))) * (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 ((0.5 + (0.5 * Math.cos(((angle * Math.PI) * 0.011111111111111112)))) * (a * a)) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return ((0.5 + (0.5 * math.cos(((angle * math.pi) * 0.011111111111111112)))) * (a * a)) + math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(Float64(angle * pi) * 0.011111111111111112)))) * Float64(a * a)) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((0.5 + (0.5 * cos(((angle * pi) * 0.011111111111111112)))) * (a * a)) + ((b * sin((pi * (angle / 180.0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(1/2 + N[(1/2 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. swap-sqr N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.4%

   \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
4. Add Preprocessing

Alternative 6: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(b \cdot \pi\right) \cdot \left|angle\right|\right) \cdot \frac{1}{180}\\ t_1 := \frac{-1}{90} \cdot \left(\left|angle\right| \cdot \pi\right)\\ \mathbf{if}\;\left|angle\right| \leq 350:\\ \;\;\;\;\left(\left(\cos t\_1 + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(t\_1 + \frac{1}{2} \cdot \pi\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (* (* b PI) (fabs angle)) 1/180))
       (t_1 (* -1/90 (* (fabs angle) PI))))
  (if (<= (fabs angle) 350)
    (+ (* (* (+ (cos t_1) 1) 1/2) (* a a)) (* t_0 t_0))
    (+
     (pow a 2)
     (* (- 1/2 (* 1/2 (sin (+ t_1 (* 1/2 PI))))) (* b b))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((b * ((double) M_PI)) * fabs(angle)) * 0.005555555555555556;
	double t_1 = -0.011111111111111112 * (fabs(angle) * ((double) M_PI));
	double tmp;
	if (fabs(angle) <= 350.0) {
		tmp = (((cos(t_1) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	} else {
		tmp = pow(a, 2.0) + ((0.5 - (0.5 * sin((t_1 + (0.5 * ((double) M_PI)))))) * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = ((b * Math.PI) * Math.abs(angle)) * 0.005555555555555556;
	double t_1 = -0.011111111111111112 * (Math.abs(angle) * Math.PI);
	double tmp;
	if (Math.abs(angle) <= 350.0) {
		tmp = (((Math.cos(t_1) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	} else {
		tmp = Math.pow(a, 2.0) + ((0.5 - (0.5 * Math.sin((t_1 + (0.5 * Math.PI))))) * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = ((b * math.pi) * math.fabs(angle)) * 0.005555555555555556
	t_1 = -0.011111111111111112 * (math.fabs(angle) * math.pi)
	tmp = 0
	if math.fabs(angle) <= 350.0:
		tmp = (((math.cos(t_1) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0)
	else:
		tmp = math.pow(a, 2.0) + ((0.5 - (0.5 * math.sin((t_1 + (0.5 * math.pi))))) * (b * b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(Float64(b * pi) * abs(angle)) * 0.005555555555555556)
	t_1 = Float64(-0.011111111111111112 * Float64(abs(angle) * pi))
	tmp = 0.0
	if (abs(angle) <= 350.0)
		tmp = Float64(Float64(Float64(Float64(cos(t_1) + 1.0) * 0.5) * Float64(a * a)) + Float64(t_0 * t_0));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * sin(Float64(t_1 + Float64(0.5 * pi))))) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = ((b * pi) * abs(angle)) * 0.005555555555555556;
	t_1 = -0.011111111111111112 * (abs(angle) * pi);
	tmp = 0.0;
	if (abs(angle) <= 350.0)
		tmp = (((cos(t_1) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	else
		tmp = (a ^ 2.0) + ((0.5 - (0.5 * sin((t_1 + (0.5 * pi))))) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(b * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * 1/180), $MachinePrecision]}, Block[{t$95$1 = N[(-1/90 * N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[angle], $MachinePrecision], 350], N[(N[(N[(N[(N[Cos[t$95$1], $MachinePrecision] + 1), $MachinePrecision] * 1/2), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2], $MachinePrecision] + N[(N[(1/2 - N[(1/2 * N[Sin[N[(t$95$1 + N[(1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 350

   1. Initial program 80.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 75.8%

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\left(\left(\cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right)} \]

   if 350 < angle

   1. Initial program 80.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. swap-sqr N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)}\right) \cdot \left(b \cdot b\right) \]

      2. cos-neg-rev N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)}\right) \cdot \left(b \cdot b\right) \]

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

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \left(b \cdot b\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \left(b \cdot b\right) \]

      5. lower-+.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \left(b \cdot b\right) \]

      6. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(angle \cdot \pi\right) \cdot \frac{1}{90}}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

      7. *-commutative N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{1}{90} \cdot \left(angle \cdot \pi\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

      8. distribute-lft-neg-in N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{90}\right)\right) \cdot \left(angle \cdot \pi\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\frac{1}{90}\right)\right) \cdot \left(angle \cdot \pi\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

      10. metadata-eval N/A

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(b \cdot b\right) \]

      11. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \frac{\color{blue}{\pi}}{2}\right)\right) \cdot \left(b \cdot b\right) \]

      12. mult-flip N/A

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)\right) \cdot \left(b \cdot b\right) \]

      13. metadata-eval N/A

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \left(b \cdot b\right) \]

      14. *-commutative N/A

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)\right) \cdot \left(b \cdot b\right) \]

      15. lower-*.f64 62.7%

          \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)\right) \cdot \left(b \cdot b\right) \]

   5. Applied rewrites 62.7%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \frac{1}{2} \cdot \pi\right)}\right) \cdot \left(b \cdot b\right) \]

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \frac{1}{2} \cdot \pi\right)\right) \cdot \left(b \cdot b\right) \]
   7. Step-by-step derivation

      1. lower-pow.f64 62.6%

         \[\leadsto {a}^{\color{blue}{2}} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \frac{1}{2} \cdot \pi\right)\right) \cdot \left(b \cdot b\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto \color{blue}{{a}^{2}} + \left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \frac{1}{2} \cdot \pi\right)\right) \cdot \left(b \cdot b\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(b \cdot \pi\right) \cdot \left|angle\right|\right) \cdot \frac{1}{180}\\ t_1 := \left|angle\right| \cdot \pi\\ \mathbf{if}\;\left|angle\right| \leq 350:\\ \;\;\;\;\left(\left(\cos \left(\frac{-1}{90} \cdot t\_1\right) + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(t\_1 \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (* (* b PI) (fabs angle)) 1/180))
       (t_1 (* (fabs angle) PI)))
  (if (<= (fabs angle) 350)
    (+ (* (* (+ (cos (* -1/90 t_1)) 1) 1/2) (* a a)) (* t_0 t_0))
    (+ (pow a 2) (* (- 1/2 (* 1/2 (cos (* t_1 1/90)))) (* b b))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((b * ((double) M_PI)) * fabs(angle)) * 0.005555555555555556;
	double t_1 = fabs(angle) * ((double) M_PI);
	double tmp;
	if (fabs(angle) <= 350.0) {
		tmp = (((cos((-0.011111111111111112 * t_1)) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	} else {
		tmp = pow(a, 2.0) + ((0.5 - (0.5 * cos((t_1 * 0.011111111111111112)))) * (b * b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = ((b * Math.PI) * Math.abs(angle)) * 0.005555555555555556;
	double t_1 = Math.abs(angle) * Math.PI;
	double tmp;
	if (Math.abs(angle) <= 350.0) {
		tmp = (((Math.cos((-0.011111111111111112 * t_1)) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	} else {
		tmp = Math.pow(a, 2.0) + ((0.5 - (0.5 * Math.cos((t_1 * 0.011111111111111112)))) * (b * b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = ((b * math.pi) * math.fabs(angle)) * 0.005555555555555556
	t_1 = math.fabs(angle) * math.pi
	tmp = 0
	if math.fabs(angle) <= 350.0:
		tmp = (((math.cos((-0.011111111111111112 * t_1)) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0)
	else:
		tmp = math.pow(a, 2.0) + ((0.5 - (0.5 * math.cos((t_1 * 0.011111111111111112)))) * (b * b))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(Float64(b * pi) * abs(angle)) * 0.005555555555555556)
	t_1 = Float64(abs(angle) * pi)
	tmp = 0.0
	if (abs(angle) <= 350.0)
		tmp = Float64(Float64(Float64(Float64(cos(Float64(-0.011111111111111112 * t_1)) + 1.0) * 0.5) * Float64(a * a)) + Float64(t_0 * t_0));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(t_1 * 0.011111111111111112)))) * Float64(b * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = ((b * pi) * abs(angle)) * 0.005555555555555556;
	t_1 = abs(angle) * pi;
	tmp = 0.0;
	if (abs(angle) <= 350.0)
		tmp = (((cos((-0.011111111111111112 * t_1)) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	else
		tmp = (a ^ 2.0) + ((0.5 - (0.5 * cos((t_1 * 0.011111111111111112)))) * (b * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(b * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * 1/180), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[angle], $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[Abs[angle], $MachinePrecision], 350], N[(N[(N[(N[(N[Cos[N[(-1/90 * t$95$1), $MachinePrecision]], $MachinePrecision] + 1), $MachinePrecision] * 1/2), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2], $MachinePrecision] + N[(N[(1/2 - N[(1/2 * N[Cos[N[(t$95$1 * 1/90), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 350

   1. Initial program 80.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 75.8%

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\left(\left(\cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right)} \]

   if 350 < angle

   1. Initial program 80.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. swap-sqr N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 62.8%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right)} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{2}} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]
   5. Step-by-step derivation

      1. lower-pow.f64 62.7%

         \[\leadsto {a}^{\color{blue}{2}} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]

   6. Applied rewrites 62.7%

      \[\leadsto \color{blue}{{a}^{2}} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot \left(b \cdot b\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.3% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (+ (pow a 2) (pow (* b (sin (* PI (/ angle 180)))) 2)))

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(pi * Float64(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], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 80.3%

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

4. Applied rewrites 80.3%

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

Alternative 9: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(\left|b\right| \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\\ \mathbf{if}\;\left|b\right| \leq \frac{7436652464262241}{8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296}:\\ \;\;\;\;{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + t\_0 \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (* (* (fabs b) PI) angle) 1/180)))
  (if (<=
       (fabs b)
       7436652464262241/8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296)
    (* (pow a 2) (+ 1/2 (* 1/2 (cos (* 1/90 (* angle PI))))))
    (+
     (* (* (+ (cos (* -1/90 (* angle PI))) 1) 1/2) (* a a))
     (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = ((fabs(b) * ((double) M_PI)) * angle) * 0.005555555555555556;
	double tmp;
	if (fabs(b) <= 8.5e-85) {
		tmp = pow(a, 2.0) * (0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))));
	} else {
		tmp = (((cos((-0.011111111111111112 * (angle * ((double) M_PI)))) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = ((Math.abs(b) * Math.PI) * angle) * 0.005555555555555556;
	double tmp;
	if (Math.abs(b) <= 8.5e-85) {
		tmp = Math.pow(a, 2.0) * (0.5 + (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)))));
	} else {
		tmp = (((Math.cos((-0.011111111111111112 * (angle * Math.PI))) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = ((math.fabs(b) * math.pi) * angle) * 0.005555555555555556
	tmp = 0
	if math.fabs(b) <= 8.5e-85:
		tmp = math.pow(a, 2.0) * (0.5 + (0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))))
	else:
		tmp = (((math.cos((-0.011111111111111112 * (angle * math.pi))) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(Float64(abs(b) * pi) * angle) * 0.005555555555555556)
	tmp = 0.0
	if (abs(b) <= 8.5e-85)
		tmp = Float64((a ^ 2.0) * Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))));
	else
		tmp = Float64(Float64(Float64(Float64(cos(Float64(-0.011111111111111112 * Float64(angle * pi))) + 1.0) * 0.5) * Float64(a * a)) + Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = ((abs(b) * pi) * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (abs(b) <= 8.5e-85)
		tmp = (a ^ 2.0) * (0.5 + (0.5 * cos((0.011111111111111112 * (angle * pi)))));
	else
		tmp = (((cos((-0.011111111111111112 * (angle * pi))) + 1.0) * 0.5) * (a * a)) + (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision] * angle), $MachinePrecision] * 1/180), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 7436652464262241/8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296], N[(N[Power[a, 2], $MachinePrecision] * N[(1/2 + N[(1/2 * N[Cos[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[N[(-1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1), $MachinePrecision] * 1/2), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5000000000000005e-85

   1. Initial program 80.4%

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

   3. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-PI.f64 25.6%

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \]

   6. Applied rewrites 25.6%

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   8. Applied rewrites 25.6%

      \[\leadsto \color{blue}{\left(\frac{1}{2} - \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

       2. lower-pow.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       3. lower-+.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       8. lower-PI.f64 57.2%

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \]

   11. Applied rewrites 57.2%

       \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   if 8.5000000000000005e-85 < b

   1. Initial program 80.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 75.8%

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\left(\left(\cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq \frac{7436652464262241}{8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296}:\\ \;\;\;\;{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot 1\right)}^{2} + {\left(\frac{1}{180} \cdot \left(angle \cdot \left(\left|b\right| \cdot \pi\right)\right)\right)}^{2}\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<=
     (fabs b)
     7436652464262241/8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296)
  (* (pow a 2) (+ 1/2 (* 1/2 (cos (* 1/90 (* angle PI))))))
  (+ (pow (* a 1) 2) (pow (* 1/180 (* angle (* (fabs b) PI))) 2))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 8.5e-85) {
		tmp = pow(a, 2.0) * (0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))))));
	} else {
		tmp = pow((a * 1.0), 2.0) + pow((0.005555555555555556 * (angle * (fabs(b) * ((double) M_PI)))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 8.5e-85) {
		tmp = Math.pow(a, 2.0) * (0.5 + (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)))));
	} else {
		tmp = Math.pow((a * 1.0), 2.0) + Math.pow((0.005555555555555556 * (angle * (Math.abs(b) * Math.PI))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 8.5e-85:
		tmp = math.pow(a, 2.0) * (0.5 + (0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))))
	else:
		tmp = math.pow((a * 1.0), 2.0) + math.pow((0.005555555555555556 * (angle * (math.fabs(b) * math.pi))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 8.5e-85)
		tmp = Float64((a ^ 2.0) * Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))));
	else
		tmp = Float64((Float64(a * 1.0) ^ 2.0) + (Float64(0.005555555555555556 * Float64(angle * Float64(abs(b) * pi))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 8.5e-85)
		tmp = (a ^ 2.0) * (0.5 + (0.5 * cos((0.011111111111111112 * (angle * pi)))));
	else
		tmp = ((a * 1.0) ^ 2.0) + ((0.005555555555555556 * (angle * (abs(b) * pi))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 7436652464262241/8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296], N[(N[Power[a, 2], $MachinePrecision] * N[(1/2 + N[(1/2 * N[Cos[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(a * 1), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(1/180 * N[(angle * N[(N[Abs[b], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.5000000000000005e-85

   1. Initial program 80.4%

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

   3. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-PI.f64 25.6%

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \]

   6. Applied rewrites 25.6%

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   8. Applied rewrites 25.6%

      \[\leadsto \color{blue}{\left(\frac{1}{2} - \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right)} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

       2. lower-pow.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       3. lower-+.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

       8. lower-PI.f64 57.2%

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \]

   11. Applied rewrites 57.2%

       \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   if 8.5000000000000005e-85 < b

   1. Initial program 80.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 75.8%

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in angle around 0

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

   7. Applied rewrites 75.5%

      \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\\ t_1 := \pi \cdot \frac{angle}{180}\\ t_2 := \left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\\ \mathbf{if}\;{\left(a \cdot \cos t\_1\right)}^{2} + {\left(b \cdot \sin t\_1\right)}^{2} \leq \frac{5890680864316837}{2945340432158418383223693624588738123559693482299075088767878449688292160397327779966295692450325070170031945807812908771881611572255401942922812303597144053805349165872996110766935565946816006053119311086960734516644260779498911850068592403100913453684334767056261910363295677456051671938422478104563288264146944}:\\ \;\;\;\;\left(\frac{1}{2} - \left(-\sin \left(\frac{-1}{2} \cdot \pi - t\_0\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos t\_0 + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + t\_2 \cdot t\_2\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* -1/90 (* angle PI)))
       (t_1 (* PI (/ angle 180)))
       (t_2 (* (* (* b PI) angle) 1/180)))
  (if (<=
       (+ (pow (* a (cos t_1)) 2) (pow (* b (sin t_1)) 2))
       5890680864316837/2945340432158418383223693624588738123559693482299075088767878449688292160397327779966295692450325070170031945807812908771881611572255401942922812303597144053805349165872996110766935565946816006053119311086960734516644260779498911850068592403100913453684334767056261910363295677456051671938422478104563288264146944)
    (* (- 1/2 (* (- (sin (- (* -1/2 PI) t_0))) 1/2)) (* b b))
    (+ (* (* (+ (cos t_0) 1) 1/2) (* a a)) (* t_2 t_2)))))

C

double code(double a, double b, double angle) {
	double t_0 = -0.011111111111111112 * (angle * ((double) M_PI));
	double t_1 = ((double) M_PI) * (angle / 180.0);
	double t_2 = ((b * ((double) M_PI)) * angle) * 0.005555555555555556;
	double tmp;
	if ((pow((a * cos(t_1)), 2.0) + pow((b * sin(t_1)), 2.0)) <= 2e-297) {
		tmp = (0.5 - (-sin(((-0.5 * ((double) M_PI)) - t_0)) * 0.5)) * (b * b);
	} else {
		tmp = (((cos(t_0) + 1.0) * 0.5) * (a * a)) + (t_2 * t_2);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = -0.011111111111111112 * (angle * Math.PI);
	double t_1 = Math.PI * (angle / 180.0);
	double t_2 = ((b * Math.PI) * angle) * 0.005555555555555556;
	double tmp;
	if ((Math.pow((a * Math.cos(t_1)), 2.0) + Math.pow((b * Math.sin(t_1)), 2.0)) <= 2e-297) {
		tmp = (0.5 - (-Math.sin(((-0.5 * Math.PI) - t_0)) * 0.5)) * (b * b);
	} else {
		tmp = (((Math.cos(t_0) + 1.0) * 0.5) * (a * a)) + (t_2 * t_2);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = -0.011111111111111112 * (angle * math.pi)
	t_1 = math.pi * (angle / 180.0)
	t_2 = ((b * math.pi) * angle) * 0.005555555555555556
	tmp = 0
	if (math.pow((a * math.cos(t_1)), 2.0) + math.pow((b * math.sin(t_1)), 2.0)) <= 2e-297:
		tmp = (0.5 - (-math.sin(((-0.5 * math.pi) - t_0)) * 0.5)) * (b * b)
	else:
		tmp = (((math.cos(t_0) + 1.0) * 0.5) * (a * a)) + (t_2 * t_2)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(-0.011111111111111112 * Float64(angle * pi))
	t_1 = Float64(pi * Float64(angle / 180.0))
	t_2 = Float64(Float64(Float64(b * pi) * angle) * 0.005555555555555556)
	tmp = 0.0
	if (Float64((Float64(a * cos(t_1)) ^ 2.0) + (Float64(b * sin(t_1)) ^ 2.0)) <= 2e-297)
		tmp = Float64(Float64(0.5 - Float64(Float64(-sin(Float64(Float64(-0.5 * pi) - t_0))) * 0.5)) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(Float64(cos(t_0) + 1.0) * 0.5) * Float64(a * a)) + Float64(t_2 * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = -0.011111111111111112 * (angle * pi);
	t_1 = pi * (angle / 180.0);
	t_2 = ((b * pi) * angle) * 0.005555555555555556;
	tmp = 0.0;
	if ((((a * cos(t_1)) ^ 2.0) + ((b * sin(t_1)) ^ 2.0)) <= 2e-297)
		tmp = (0.5 - (-sin(((-0.5 * pi) - t_0)) * 0.5)) * (b * b);
	else
		tmp = (((cos(t_0) + 1.0) * 0.5) * (a * a)) + (t_2 * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(-1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(angle / 180), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b * Pi), $MachinePrecision] * angle), $MachinePrecision] * 1/180), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(a * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision], 5890680864316837/2945340432158418383223693624588738123559693482299075088767878449688292160397327779966295692450325070170031945807812908771881611572255401942922812303597144053805349165872996110766935565946816006053119311086960734516644260779498911850068592403100913453684334767056261910363295677456051671938422478104563288264146944], N[(N[(1/2 - N[((-N[Sin[N[(N[(-1/2 * Pi), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]) * 1/2), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[Cos[t$95$0], $MachinePrecision] + 1), $MachinePrecision] * 1/2), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 2.0000000000000001e-297

   1. Initial program 80.4%

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

   3. Applied rewrites 68.2%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

   4. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-PI.f64 25.6%

         \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \]

   6. Applied rewrites 25.6%

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

   8. Applied rewrites 25.6%

      \[\leadsto \color{blue}{\left(\frac{1}{2} - \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right)} \]
   9. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \left(\frac{1}{2} - \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

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

         \[\leadsto \left(\frac{1}{2} - \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      3. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{2} - \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \frac{\pi}{2}\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      4. mult-flip N/A

         \[\leadsto \left(\frac{1}{2} - \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \pi \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(\frac{1}{2} - \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \pi \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} - \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) + \frac{1}{2} \cdot \pi\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      7. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(\frac{1}{2} - \sin \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right) - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \pi\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      8. sub-negate-rev N/A

         \[\leadsto \left(\frac{1}{2} - \sin \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \pi - \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      9. sin-neg N/A

         \[\leadsto \left(\frac{1}{2} - \left(\mathsf{neg}\left(\sin \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \pi - \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(\frac{1}{2} - \left(-\sin \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \pi - \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      11. lower-sin.f64 N/A

          \[\leadsto \left(\frac{1}{2} - \left(-\sin \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \pi - \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      12. lower--.f64 N/A

          \[\leadsto \left(\frac{1}{2} - \left(-\sin \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \pi - \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} - \left(-\sin \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \pi - \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

      14. metadata-eval 25.6%

          \[\leadsto \left(\frac{1}{2} - \left(-\sin \left(\frac{-1}{2} \cdot \pi - \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

   10. Applied rewrites 25.6%

       \[\leadsto \left(\frac{1}{2} - \left(-\sin \left(\frac{-1}{2} \cdot \pi - \frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

   if 2.0000000000000001e-297 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 80.4%

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

   2. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 75.8%

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

   4. Applied rewrites 75.8%

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

   6. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\left(\left(\cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) + 1\right) \cdot \frac{1}{2}\right) \cdot \left(a \cdot a\right) + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{180}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 25.9% accurate, 3.5× speedup

Math

\[\left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot b\right) \cdot b \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (* (- 1/2 (* (cos (* 1/90 (* angle PI))) 1/2)) b) b))

C

double code(double a, double b, double angle) {
	return ((0.5 - (cos((0.011111111111111112 * (angle * ((double) M_PI)))) * 0.5)) * b) * b;
}

Java

public static double code(double a, double b, double angle) {
	return ((0.5 - (Math.cos((0.011111111111111112 * (angle * Math.PI))) * 0.5)) * b) * b;
}

Python

def code(a, b, angle):
	return ((0.5 - (math.cos((0.011111111111111112 * (angle * math.pi))) * 0.5)) * b) * b

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(angle * pi))) * 0.5)) * b) * b)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((0.5 - (cos((0.011111111111111112 * (angle * pi))) * 0.5)) * b) * b;
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(1/2 - N[(N[Cos[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

3. Applied rewrites 68.2%

   \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

4. Taylor expanded in a around 0

   \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   2. lower-pow.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower--.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   5. lower-cos.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lower-PI.f64 25.6%

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \]

6. Applied rewrites 25.6%

   \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

8. Applied rewrites 25.6%

   \[\leadsto \color{blue}{\left(\frac{1}{2} - \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right)} \]

10. Applied rewrites 25.9%

    \[\leadsto \left(\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot b\right) \cdot \color{blue}{b} \]
11. Add Preprocessing

Alternative 13: 10.6% accurate, 32.0× speedup

Math

\[\left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (* (- 1/2 1/2) (* b b)))

C

double code(double a, double b, double angle) {
	return (0.5 - 0.5) * (b * b);
}

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 = (0.5d0 - 0.5d0) * (b * b)
end function

Java

public static double code(double a, double b, double angle) {
	return (0.5 - 0.5) * (b * b);
}

Python

def code(a, b, angle):
	return (0.5 - 0.5) * (b * b)

Julia

function code(a, b, angle)
	return Float64(Float64(0.5 - 0.5) * Float64(b * b))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (0.5 - 0.5) * (b * b);
end

Wolfram

code[a_, b_, angle_] := N[(N[(1/2 - 1/2), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.4%

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

3. Applied rewrites 68.2%

   \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

4. Taylor expanded in a around 0

   \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

   2. lower-pow.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   3. lower--.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

   5. lower-cos.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lower-PI.f64 25.6%

      \[\leadsto {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \]

6. Applied rewrites 25.6%

   \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \]

8. Applied rewrites 25.6%

   \[\leadsto \color{blue}{\left(\frac{1}{2} - \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right)} \]

9. Taylor expanded in angle around 0

   \[\leadsto \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]
10. Step-by-step derivation

11. Applied rewrites 10.6%

    \[\leadsto \left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left(b \cdot b\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180)))) 2) (pow (* b (sin (* PI (/ angle 180)))) 2)))