ab-angle->ABCF C

Percentage Accurate: 79.1% → 79.1%

Time: 6.6s

Alternatives: 14

Speedup: 1.0×

• 34.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 79.1

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

3. Applied rewrites 79.1%

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

Alternative 2: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-PI.f64 79.1

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

3. Applied rewrites 79.1%

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

Alternative 3: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 79.1

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

3. Applied rewrites 79.1%

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

4. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.1

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

6. Applied rewrites 79.1%

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

7. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.1

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

9. Applied rewrites 79.1%

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

    1. lift-sin.f64 N/A

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

    2. lift-+.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. sin-+PI/2 N/A

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

    6. lift-neg.f64 N/A

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

    7. cos-neg-rev N/A

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

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

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

    9. lower-sin.f64 N/A

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

    10. lift-PI.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. lift-/.f64 N/A

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

    13. lift-PI.f64 N/A

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

    14. lower-fma.f64 N/A

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

    15. lift-PI.f64 79.1

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

11. Applied rewrites 79.1%

    \[\leadsto {\left(a \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\pi, 0.005555555555555556 \cdot angle, \frac{\pi}{2}\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 4: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556, \pi \cdot angle, 0.5 \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (fma -0.005555555555555556 (* PI angle) (* 0.5 PI)))) 2.0)
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(fma(-0.005555555555555556, (((double) M_PI) * angle), (0.5 * ((double) M_PI))))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(fma(-0.005555555555555556, Float64(pi * angle), Float64(0.5 * pi)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.1%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 79.1

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

3. Applied rewrites 79.1%

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

4. Taylor expanded in angle around 0

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

   1. lower-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 79.0

      \[\leadsto {\left(a \cdot \sin \left(\mathsf{fma}\left(-0.005555555555555556, \pi \cdot angle, 0.5 \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied rewrites 79.0%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-0.005555555555555556, \pi \cdot angle, 0.5 \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 5: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. lift-pow.f64 N/A

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

3. Applied rewrites 79.1%

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

Alternative 6: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

Alternative 7: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ {\left(a \cdot \cos t\_0\right)}^{2} + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * 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 = Math.PI * (0.005555555555555556 * 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 = math.pi * (0.005555555555555556 * 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(pi * Float64(0.005555555555555556 * 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 = pi * (0.005555555555555556 * 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[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.1

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

4. Applied rewrites 79.1%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.1

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

7. Applied rewrites 79.1%

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

Alternative 8: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ {\left(\sin t\_0 \cdot b\right)}^{2} + {\left(\cos t\_0 \cdot a\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (+ (pow (* (sin t_0) b) 2.0) (pow (* (cos t_0) a) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	return pow((sin(t_0) * b), 2.0) + pow((cos(t_0) * a), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	return Math.pow((Math.sin(t_0) * b), 2.0) + Math.pow((Math.cos(t_0) * a), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (math.pi * angle) * 0.005555555555555556
	return math.pow((math.sin(t_0) * b), 2.0) + math.pow((math.cos(t_0) * a), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	return Float64((Float64(sin(t_0) * b) ^ 2.0) + (Float64(cos(t_0) * a) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = ((sin(t_0) * b) ^ 2.0) + ((cos(t_0) * a) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, N[(N[Power[N[(N[Sin[t$95$0], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Cos[t$95$0], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 79.1

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

4. Applied rewrites 79.1%

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

5. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 79.1

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

7. Applied rewrites 79.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 79.1

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 79.1

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 79.1

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

9. Applied rewrites 79.1%

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

Alternative 9: 79.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 79.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 79.1

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

6. Applied rewrites 79.1%

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

Alternative 10: 57.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot b\right) \cdot b, \pi \cdot \pi, \left(\pi \cdot \pi\right) \cdot \left(\left(-3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - \cos \left(\left(\frac{angle}{180} \cdot \pi\right) \cdot 2\right) \cdot 0.5\right) \cdot b, b, \left(1 \cdot a\right) \cdot \left(1 \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.5e-33)
   (fma
    a
    a
    (*
     (* angle angle)
     (fma
      (* (* 3.08641975308642e-5 b) b)
      (* PI PI)
      (* (* PI PI) (* (* -3.08641975308642e-5 a) a)))))
   (fma
    (* (- 0.5 (* (cos (* (* (/ angle 180.0) PI) 2.0)) 0.5)) b)
    b
    (* (* 1.0 a) (* 1.0 a)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.5e-33) {
		tmp = fma(a, a, ((angle * angle) * fma(((3.08641975308642e-5 * b) * b), (((double) M_PI) * ((double) M_PI)), ((((double) M_PI) * ((double) M_PI)) * ((-3.08641975308642e-5 * a) * a)))));
	} else {
		tmp = fma(((0.5 - (cos((((angle / 180.0) * ((double) M_PI)) * 2.0)) * 0.5)) * b), b, ((1.0 * a) * (1.0 * a)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.5e-33)
		tmp = fma(a, a, Float64(Float64(angle * angle) * fma(Float64(Float64(3.08641975308642e-5 * b) * b), Float64(pi * pi), Float64(Float64(pi * pi) * Float64(Float64(-3.08641975308642e-5 * a) * a)))));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(cos(Float64(Float64(Float64(angle / 180.0) * pi) * 2.0)) * 0.5)) * b), b, Float64(Float64(1.0 * a) * Float64(1.0 * a)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.5e-33], N[(a * a + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(3.08641975308642e-5 * b), $MachinePrecision] * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(N[Cos[N[(N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(N[(1.0 * a), $MachinePrecision] * N[(1.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.50000000000000014e-33

   1. Initial program 79.1%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

   7. Applied rewrites 40.9%

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

   if 2.50000000000000014e-33 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 79.1%

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

   6. Applied rewrites 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(\frac{angle}{180} \cdot \pi\right) \cdot 2\right) \cdot 0.5\right) \cdot b, b, \left(1 \cdot a\right) \cdot \left(1 \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 54.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.42 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot b\right) \cdot b, \pi \cdot \pi, \left(\pi \cdot \pi\right) \cdot \left(\left(-3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(3.175328964080679 \cdot 10^{-10} \cdot \left(angle \cdot angle\right), \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi, \left(\pi \cdot \pi\right) \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, 1\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.42e-14)
   (fma
    a
    a
    (*
     (* angle angle)
     (fma
      (* (* 3.08641975308642e-5 b) b)
      (* PI PI)
      (* (* PI PI) (* (* -3.08641975308642e-5 a) a)))))
   (*
    (fma
     (fma
      (* 3.175328964080679e-10 (* angle angle))
      (* (* (* PI PI) PI) PI)
      (* (* PI PI) -3.08641975308642e-5))
     (* angle angle)
     1.0)
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.42e-14) {
		tmp = fma(a, a, ((angle * angle) * fma(((3.08641975308642e-5 * b) * b), (((double) M_PI) * ((double) M_PI)), ((((double) M_PI) * ((double) M_PI)) * ((-3.08641975308642e-5 * a) * a)))));
	} else {
		tmp = fma(fma((3.175328964080679e-10 * (angle * angle)), (((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * ((double) M_PI)), ((((double) M_PI) * ((double) M_PI)) * -3.08641975308642e-5)), (angle * angle), 1.0) * (a * a);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.42e-14)
		tmp = fma(a, a, Float64(Float64(angle * angle) * fma(Float64(Float64(3.08641975308642e-5 * b) * b), Float64(pi * pi), Float64(Float64(pi * pi) * Float64(Float64(-3.08641975308642e-5 * a) * a)))));
	else
		tmp = Float64(fma(fma(Float64(3.175328964080679e-10 * Float64(angle * angle)), Float64(Float64(Float64(pi * pi) * pi) * pi), Float64(Float64(pi * pi) * -3.08641975308642e-5)), Float64(angle * angle), 1.0) * Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.42e-14], N[(a * a + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(3.08641975308642e-5 * b), $MachinePrecision] * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(3.175328964080679e-10 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle * angle), $MachinePrecision] + 1.0), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.42000000000000004e-14

   1. Initial program 79.1%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

   7. Applied rewrites 40.9%

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

   if 1.42000000000000004e-14 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + {a}^{2} \cdot \left(\frac{1}{12597120000} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{4199040000} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right) + {a}^{2}} \]

   4. Applied rewrites 30.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), \pi \cdot \pi, \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right), \pi \cdot \pi, \mathsf{fma}\left(-3.175328964080679 \cdot 10^{-10} \cdot \left(b \cdot b\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.175328964080679 \cdot 10^{-10}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(angle \cdot angle\right)\right)\right), angle \cdot angle, a \cdot a\right)} \]

   5. Taylor expanded in a around inf

      \[\leadsto {a}^{2} \cdot \color{blue}{\left(1 + {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{3149280000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{3149280000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot {a}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{1}{3149280000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right) \cdot {a}^{\color{blue}{2}} \]

   7. Applied rewrites 57.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(3.175328964080679 \cdot 10^{-10} \cdot \left(angle \cdot angle\right), \left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi, \left(\pi \cdot \pi\right) \cdot -3.08641975308642 \cdot 10^{-5}\right), angle \cdot angle, 1\right) \cdot \color{blue}{\left(a \cdot a\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 53.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3600000:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot b\right) \cdot b, \pi \cdot \pi, \left(\pi \cdot \pi\right) \cdot \left(\left(-3.08641975308642 \cdot 10^{-5} \cdot a\right) \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3600000.0)
   (fma
    a
    a
    (*
     (* angle angle)
     (fma
      (* (* 3.08641975308642e-5 b) b)
      (* PI PI)
      (* (* PI PI) (* (* -3.08641975308642e-5 a) a)))))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3600000.0) {
		tmp = fma(a, a, ((angle * angle) * fma(((3.08641975308642e-5 * b) * b), (((double) M_PI) * ((double) M_PI)), ((((double) M_PI) * ((double) M_PI)) * ((-3.08641975308642e-5 * a) * a)))));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3600000.0)
		tmp = fma(a, a, Float64(Float64(angle * angle) * fma(Float64(Float64(3.08641975308642e-5 * b) * b), Float64(pi * pi), Float64(Float64(pi * pi) * Float64(Float64(-3.08641975308642e-5 * a) * a)))));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3600000.0], N[(a * a + N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(3.08641975308642e-5 * b), $MachinePrecision] * b), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.6e6

   1. Initial program 79.1%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

   7. Applied rewrites 40.9%

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

   if 3.6e6 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

Alternative 13: 53.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3600000:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), \pi \cdot \pi, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3600000.0)
   (fma
    a
    a
    (*
     (* angle angle)
     (fma
      (* -3.08641975308642e-5 (* a a))
      (* PI PI)
      (* (* 3.08641975308642e-5 (* b b)) (* PI PI)))))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3600000.0) {
		tmp = fma(a, a, ((angle * angle) * fma((-3.08641975308642e-5 * (a * a)), (((double) M_PI) * ((double) M_PI)), ((3.08641975308642e-5 * (b * b)) * (((double) M_PI) * ((double) M_PI))))));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3600000.0)
		tmp = fma(a, a, Float64(Float64(angle * angle) * fma(Float64(-3.08641975308642e-5 * Float64(a * a)), Float64(pi * pi), Float64(Float64(3.08641975308642e-5 * Float64(b * b)) * Float64(pi * pi)))));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3600000.0], N[(a * a + N[(N[(angle * angle), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.6e6

   1. Initial program 79.1%

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

   2. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 40.7%

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

   if 3.6e6 < a

   1. Initial program 79.1%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.4

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

   4. Applied rewrites 57.4%

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

Alternative 14: 52.9% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.4

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

4. Applied rewrites 57.4%

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

Reproduce

herbie shell --seed 2025131 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))