ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.5%

Time: 5.4s

Alternatives: 9

Speedup: N/A×

• 37.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + \left(0.5 + 0.5 \cdot \cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(b \cdot b\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (/ (* PI angle) 180.0))) 2.0)
  (* (+ 0.5 (* 0.5 (cos (* (* 0.011111111111111112 angle) PI)))) (* b b))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(Pi * angle), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(0.5 + N[(0.5 * N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 79.5

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

3. Applied rewrites 79.5%

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

4. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. sin-+PI/2 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 79.5%

   \[\leadsto {\left(a \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 79.5

       \[\leadsto {\left(a \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + \left(0.5 + 0.5 \cdot \cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(b \cdot b\right) \]

8. Applied rewrites 79.5%

   \[\leadsto {\left(a \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} + \left(0.5 + 0.5 \cdot \cos \left(\left(0.011111111111111112 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(b \cdot b\right) \]
9. Add Preprocessing

Alternative 2: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 79.5

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

3. Applied rewrites 79.5%

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

4. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. sin-+PI/2 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 79.4

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

6. Applied rewrites 79.4%

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

Alternative 3: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 79.5

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

4. Applied rewrites 79.5%

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

Alternative 4: 66.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-94}:\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos \left(\mathsf{fma}\left(0.011111111111111112 \cdot angle, \pi, \left(0.5 \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.6e-94)
   (*
    (-
     0.5
     (* 0.5 (cos (fma (* 0.011111111111111112 angle) PI (* (* 0.5 PI) 2.0)))))
    (* b b))
   (+ (pow (* (* (* PI angle) a) 0.005555555555555556) 2.0) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.6e-94) {
		tmp = (0.5 - (0.5 * cos(fma((0.011111111111111112 * angle), ((double) M_PI), ((0.5 * ((double) M_PI)) * 2.0))))) * (b * b);
	} else {
		tmp = pow((((((double) M_PI) * angle) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.6e-94)
		tmp = Float64(Float64(0.5 - Float64(0.5 * cos(fma(Float64(0.011111111111111112 * angle), pi, Float64(Float64(0.5 * pi) * 2.0))))) * Float64(b * b));
	else
		tmp = Float64((Float64(Float64(Float64(pi * angle) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.6e-94], N[(N[(0.5 - N[(0.5 * N[Cos[N[(N[(0.011111111111111112 * angle), $MachinePrecision] * Pi + N[(N[(0.5 * Pi), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.59999999999999998e-94

   1. Initial program 79.3%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 79.2

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. sin-+PI/2 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, \frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \frac{1}{180} \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right) \]

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

   8. Applied rewrites 61.4%

      \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(\mathsf{fma}\left(0.011111111111111112 \cdot angle, \pi, \left(0.5 \cdot \pi\right) \cdot 2\right)\right)\right) \cdot \left(b \cdot b\right) \]

   if 1.59999999999999998e-94 < a

   1. Initial program 80.2%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. sin-+PI/2 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

Alternative 5: 66.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-94}:\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.6e-94)
   (*
    (-
     0.5
     (* 0.5 (cos (* 2.0 (fma 0.5 PI (* 0.005555555555555556 (* PI angle)))))))
    (* b b))
   (+ (pow (* (* (* PI angle) a) 0.005555555555555556) 2.0) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.6e-94) {
		tmp = (0.5 - (0.5 * cos((2.0 * fma(0.5, ((double) M_PI), (0.005555555555555556 * (((double) M_PI) * angle))))))) * (b * b);
	} else {
		tmp = pow((((((double) M_PI) * angle) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.6e-94)
		tmp = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * fma(0.5, pi, Float64(0.005555555555555556 * Float64(pi * angle))))))) * Float64(b * b));
	else
		tmp = Float64((Float64(Float64(Float64(pi * angle) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.6e-94], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * Pi + N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.59999999999999998e-94

   1. Initial program 79.3%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-PI.f64 79.2

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

   3. Applied rewrites 79.2%

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

   4. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. sin-+PI/2 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 61.4%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)} \]

   if 1.59999999999999998e-94 < a

   1. Initial program 80.2%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. sin-+PI/2 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

Alternative 6: 66.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-94}:\\ \;\;\;\;\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.6e-94)
   (* (* (fma (cos (* 0.011111111111111112 (* PI angle))) 0.5 0.5) b) b)
   (+ (pow (* (* (* PI angle) a) 0.005555555555555556) 2.0) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.6e-94) {
		tmp = (fma(cos((0.011111111111111112 * (((double) M_PI) * angle))), 0.5, 0.5) * b) * b;
	} else {
		tmp = pow((((((double) M_PI) * angle) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.6e-94)
		tmp = Float64(Float64(fma(cos(Float64(0.011111111111111112 * Float64(pi * angle))), 0.5, 0.5) * b) * b);
	else
		tmp = Float64((Float64(Float64(Float64(pi * angle) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.6e-94], N[(N[(N[(N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision], N[(N[Power[N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.59999999999999998e-94

   1. Initial program 79.3%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.6

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

   4. Applied rewrites 61.6%

      \[\leadsto \color{blue}{b \cdot b} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

         \[\leadsto e^{\log b \cdot 2} \]

      4. lower-exp.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      6. lower-log.f64 29.4

         \[\leadsto e^{\log b \cdot 2} \]

   6. Applied rewrites 29.4%

      \[\leadsto e^{\log b \cdot 2} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      2. lift-log.f64 N/A

         \[\leadsto e^{\log b \cdot 2} \]

      3. *-commutative N/A

         \[\leadsto e^{2 \cdot \log b} \]

      4. log-pow-rev N/A

         \[\leadsto e^{\log \left({b}^{2}\right)} \]

      5. lower-log.f64 N/A

         \[\leadsto e^{\log \left({b}^{2}\right)} \]

      6. unpow2 N/A

         \[\leadsto e^{\log \left(b \cdot b\right)} \]

      7. lower-*.f64 59.8

         \[\leadsto e^{\log \left(b \cdot b\right)} \]

   8. Applied rewrites 59.8%

      \[\leadsto e^{\log \left(b \cdot b\right)} \]

   9. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

   11. Applied rewrites 61.4%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot b\right) \cdot b} \]

   if 1.59999999999999998e-94 < a

   1. Initial program 80.2%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. sin-+PI/2 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 76.5

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

   7. Applied rewrites 76.5%

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

Alternative 7: 66.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{-29}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\left(\pi \cdot angle\right) \cdot a\right) \cdot 0.005555555555555556\right)}^{2} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.6e-29)
   (* b b)
   (+ (pow (* (* (* PI angle) a) 0.005555555555555556) 2.0) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.6e-29) {
		tmp = b * b;
	} else {
		tmp = pow((((((double) M_PI) * angle) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.6e-29) {
		tmp = b * b;
	} else {
		tmp = Math.pow((((Math.PI * angle) * a) * 0.005555555555555556), 2.0) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.6e-29:
		tmp = b * b
	else:
		tmp = math.pow((((math.pi * angle) * a) * 0.005555555555555556), 2.0) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.6e-29)
		tmp = Float64(b * b);
	else
		tmp = Float64((Float64(Float64(Float64(pi * angle) * a) * 0.005555555555555556) ^ 2.0) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.6e-29)
		tmp = b * b;
	else
		tmp = ((((pi * angle) * a) * 0.005555555555555556) ^ 2.0) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.6e-29], N[(b * b), $MachinePrecision], N[(N[Power[N[(N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision] * 0.005555555555555556), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.6000000000000002e-29

   1. Initial program 78.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   4. Applied rewrites 61.7%

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

   if 2.6000000000000002e-29 < a

   1. Initial program 81.7%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 78.5

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in angle around 0

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. sin-+PI/2 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 78.4

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

   7. Applied rewrites 78.4%

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

Alternative 8: 65.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.6 \cdot 10^{+83}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(angle \cdot angle\right)\right) + b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 6.6e+83)
   (+
    (* (* 3.08641975308642e-5 (* a a)) (* (* PI PI) (* angle angle)))
    (* b b))
   (* b b)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+83) {
		tmp = ((3.08641975308642e-5 * (a * a)) * ((((double) M_PI) * ((double) M_PI)) * (angle * angle))) + (b * b);
	} else {
		tmp = b * b;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 6.6e+83) {
		tmp = ((3.08641975308642e-5 * (a * a)) * ((Math.PI * Math.PI) * (angle * angle))) + (b * b);
	} else {
		tmp = b * b;
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 6.6e+83:
		tmp = ((3.08641975308642e-5 * (a * a)) * ((math.pi * math.pi) * (angle * angle))) + (b * b)
	else:
		tmp = b * b
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 6.6e+83)
		tmp = Float64(Float64(Float64(3.08641975308642e-5 * Float64(a * a)) * Float64(Float64(pi * pi) * Float64(angle * angle))) + Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 6.6e+83)
		tmp = ((3.08641975308642e-5 * (a * a)) * ((pi * pi) * (angle * angle))) + (b * b);
	else
		tmp = b * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 6.6e+83], N[(N[(N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.59999999999999969e83

   1. Initial program 77.1%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{32400} \cdot {a}^{2}\right) \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot {a}^{2}\right) \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot {a}^{2}\right) \cdot \left(\color{blue}{{angle}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      4. unpow2 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left({angle}^{\color{blue}{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left({angle}^{\color{blue}{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{{angle}^{2}}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{{angle}^{2}}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\color{blue}{angle}}^{2}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot {\color{blue}{angle}}^{2}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      10. lift-PI.f64 N/A

          \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot {angle}^{2}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      11. lift-PI.f64 N/A

          \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {angle}^{2}\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      12. unpow2 N/A

          \[\leadsto \left(\frac{1}{32400} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(angle \cdot \color{blue}{angle}\right)\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      13. lower-*.f64 61.2

          \[\leadsto \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(angle \cdot \color{blue}{angle}\right)\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(angle \cdot angle\right)\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. sin-+PI/2 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

   if 6.59999999999999969e83 < b

   1. Initial program 90.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.2

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

   4. Applied rewrites 85.2%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.8% accurate, 29.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle) {
	return 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 = b * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.8

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

4. Applied rewrites 56.8%

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

Reproduce

herbie shell --seed 2025116 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))