ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.6%

Time: 6.9s

Alternatives: 11

Speedup: N/A×

• 35.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% 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: 79.6% accurate, N/A× 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 80.8%

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

Alternative 2: 79.6% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 80.7

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

5. Applied rewrites 80.7%

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

Alternative 3: 79.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \pi, 0.5 \cdot \frac{\pi}{angle}\right)\right)\right)}^{2} + {\left(\frac{1}{{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot b\right)}^{-1}}\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (* a (sin (* angle (fma 0.005555555555555556 PI (* 0.5 (/ PI angle))))))
   2.0)
  (pow (/ 1.0 (pow (* (sin (* PI (/ angle 180.0))) b) -1.0)) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((angle * fma(0.005555555555555556, ((double) M_PI), (0.5 * (((double) M_PI) / angle)))))), 2.0) + pow((1.0 / pow((sin((((double) M_PI) * (angle / 180.0))) * b), -1.0)), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(angle * fma(0.005555555555555556, pi, Float64(0.5 * Float64(pi / angle)))))) ^ 2.0) + (Float64(1.0 / (Float64(sin(Float64(pi * Float64(angle / 180.0))) * b) ^ -1.0)) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(angle * N[(0.005555555555555556 * Pi + N[(0.5 * N[(Pi / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(1.0 / N[Power[N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.8%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 80.8

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

4. Applied rewrites 80.8%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. unpow1 N/A

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

   8. metadata-eval N/A

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

   9. pow-neg N/A

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

   10. lower-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. lift-sin.f64 80.8

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

6. Applied rewrites 80.8%

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

7. Taylor expanded in angle around inf

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

   1. associate-*r/ N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-/.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 80.6

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

9. Applied rewrites 80.6%

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

10. Taylor expanded in angle around inf

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

    1. lower-*.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lift-PI.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lift-PI.f64 80.6

       \[\leadsto {\left(a \cdot \sin \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \pi, 0.5 \cdot \frac{\pi}{angle}\right)\right)\right)}^{2} + {\left(\frac{1}{{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot b\right)}^{-1}}\right)}^{2} \]

12. Applied rewrites 80.6%

    \[\leadsto {\left(a \cdot \sin \left(angle \cdot \mathsf{fma}\left(0.005555555555555556, \pi, 0.5 \cdot \frac{\pi}{angle}\right)\right)\right)}^{2} + {\left(\frac{1}{{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot b\right)}^{-1}}\right)}^{2} \]
13. Add Preprocessing

Alternative 4: 79.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_1 := \sin t\_0 \cdot b\\ \mathsf{fma}\left({\sin \left(\mathsf{fma}\left(0.5, \pi, t\_0\right)\right)}^{2}, a \cdot a, t\_1 \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle))) (t_1 (* (sin t_0) b)))
   (fma (pow (sin (fma 0.5 PI t_0)) 2.0) (* a a) (* t_1 t_1))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_1 = sin(t_0) * b;
	return fma(pow(sin(fma(0.5, ((double) M_PI), t_0)), 2.0), (a * a), (t_1 * t_1));
}

Julia

function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_1 = Float64(sin(t_0) * b)
	return fma((sin(fma(0.5, pi, t_0)) ^ 2.0), Float64(a * a), Float64(t_1 * t_1))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[t$95$0], $MachinePrecision] * b), $MachinePrecision]}, N[(N[Power[N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 80.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. pow-to-exp N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 80.6%

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

8. Final simplification 80.6%

   \[\leadsto \mathsf{fma}\left({\sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}, a \cdot a, \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b\right)\right) \]
9. Add Preprocessing

Alternative 5: 79.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_1 := \sin t\_0 \cdot b\\ t_2 := \left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, t\_0\right)\right)\right) \cdot -1\\ \mathsf{fma}\left(t\_1, t\_1, t\_2 \cdot t\_2\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle)))
        (t_1 (* (sin t_0) b))
        (t_2 (* (* (pow (/ -1.0 a) -1.0) (sin (fma 0.5 PI t_0))) -1.0)))
   (fma t_1 t_1 (* t_2 t_2))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_1 = sin(t_0) * b;
	double t_2 = (pow((-1.0 / a), -1.0) * sin(fma(0.5, ((double) M_PI), t_0))) * -1.0;
	return fma(t_1, t_1, (t_2 * t_2));
}

Julia

function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_1 = Float64(sin(t_0) * b)
	t_2 = Float64(Float64((Float64(-1.0 / a) ^ -1.0) * sin(fma(0.5, pi, t_0))) * -1.0)
	return fma(t_1, t_1, Float64(t_2 * t_2))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[t$95$0], $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[N[(-1.0 / a), $MachinePrecision], -1.0], $MachinePrecision] * N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]}, N[(t$95$1 * t$95$1 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 80.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. pow-to-exp N/A

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

   8. exp-prod N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 33.8%

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

5. Taylor expanded in a around -inf

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

7. Applied rewrites 80.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right)\right)} \]

8. Final simplification 80.5%

   \[\leadsto \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right)\right) \]
9. Add Preprocessing

Alternative 6: 74.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_1 := \sin t\_0 \cdot b\\ \mathbf{if}\;a \leq 1.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_1, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, t\_0\right)\right)\right) \cdot -1\right) \cdot \left(\left(\left(-1 \cdot a\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle))) (t_1 (* (sin t_0) b)))
   (if (<= a 1.4e+153)
     (fma
      t_1
      t_1
      (*
       (* (* (pow (/ -1.0 a) -1.0) (sin (fma 0.5 PI t_0))) -1.0)
       (* (* (* -1.0 a) (sin (* 0.5 PI))) -1.0)))
     (exp (* (log a) 2.0)))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_1 = sin(t_0) * b;
	double tmp;
	if (a <= 1.4e+153) {
		tmp = fma(t_1, t_1, (((pow((-1.0 / a), -1.0) * sin(fma(0.5, ((double) M_PI), t_0))) * -1.0) * (((-1.0 * a) * sin((0.5 * ((double) M_PI)))) * -1.0)));
	} else {
		tmp = exp((log(a) * 2.0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_1 = Float64(sin(t_0) * b)
	tmp = 0.0
	if (a <= 1.4e+153)
		tmp = fma(t_1, t_1, Float64(Float64(Float64((Float64(-1.0 / a) ^ -1.0) * sin(fma(0.5, pi, t_0))) * -1.0) * Float64(Float64(Float64(-1.0 * a) * sin(Float64(0.5 * pi))) * -1.0)));
	else
		tmp = exp(Float64(log(a) * 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[t$95$0], $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[a, 1.4e+153], N[(t$95$1 * t$95$1 + N[(N[(N[(N[Power[N[(-1.0 / a), $MachinePrecision], -1.0], $MachinePrecision] * N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] * N[(N[(N[(-1.0 * a), $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Log[a], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.39999999999999993e153

   1. Initial program 77.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. pow-to-exp N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

   4. Applied rewrites 24.5%

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

   5. Taylor expanded in a around -inf

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

   7. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right)\right)} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 71.0

         \[\leadsto \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left(\left(-1 \cdot a\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot -1\right)\right) \]

   10. Applied rewrites 71.0%

       \[\leadsto \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left(\left(-1 \cdot a\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot -1\right)\right) \]

   if 1.39999999999999993e153 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{a \cdot a} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto e^{\log a \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left(\left(-1 \cdot a\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log a \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ t_1 := \sin \left(0.5 \cdot \pi\right)\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\\ t_4 := \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b\\ t_5 := \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\\ \mathbf{if}\;a \leq 1.35 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(t\_4, t\_4, \mathsf{fma}\left(angle, \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot t\_5\right) \cdot t\_1\right) + angle \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot angle, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot t\_5\right) \cdot t\_1\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(a \cdot a\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), t\_2, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(t\_5 \cdot t\_5\right)\right)\right), \left(a \cdot a\right) \cdot t\_2\right)\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0 \cdot t\_0}{a} \cdot \frac{b \cdot b}{a}, a \cdot a, t\_3 \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* (* PI angle) 0.005555555555555556)))
        (t_1 (sin (* 0.5 PI)))
        (t_2 (* t_1 t_1))
        (t_3 (* (sin (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))) a))
        (t_4 (* (sin (* 0.005555555555555556 (* PI angle))) b))
        (t_5 (sin (fma 0.5 PI (/ PI 2.0)))))
   (if (<= a 1.35e-121)
     (fma
      t_4
      t_4
      (fma
       angle
       (+
        (* (* 0.011111111111111112 (* a a)) (* (* PI t_5) t_1))
        (*
         angle
         (fma
          (* (* a a) angle)
          (* (* (* (* (* PI PI) PI) t_5) t_1) -2.2862368541380886e-7)
          (*
           (* a a)
           (fma
            (* -3.08641975308642e-5 (* PI PI))
            t_2
            (* (* 3.08641975308642e-5 (* PI PI)) (* t_5 t_5)))))))
       (* (* a a) t_2)))
     (if (<= a 6.2e+142)
       (fma (* (/ (* t_0 t_0) a) (/ (* b b) a)) (* a a) (* t_3 t_3))
       (* a a)))))

C

double code(double a, double b, double angle) {
	double t_0 = sin(((((double) M_PI) * angle) * 0.005555555555555556));
	double t_1 = sin((0.5 * ((double) M_PI)));
	double t_2 = t_1 * t_1;
	double t_3 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0))) * a;
	double t_4 = sin((0.005555555555555556 * (((double) M_PI) * angle))) * b;
	double t_5 = sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0)));
	double tmp;
	if (a <= 1.35e-121) {
		tmp = fma(t_4, t_4, fma(angle, (((0.011111111111111112 * (a * a)) * ((((double) M_PI) * t_5) * t_1)) + (angle * fma(((a * a) * angle), (((((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)) * t_5) * t_1) * -2.2862368541380886e-7), ((a * a) * fma((-3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI))), t_2, ((3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI))) * (t_5 * t_5))))))), ((a * a) * t_2)));
	} else if (a <= 6.2e+142) {
		tmp = fma((((t_0 * t_0) / a) * ((b * b) / a)), (a * a), (t_3 * t_3));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = sin(Float64(Float64(pi * angle) * 0.005555555555555556))
	t_1 = sin(Float64(0.5 * pi))
	t_2 = Float64(t_1 * t_1)
	t_3 = Float64(sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0))) * a)
	t_4 = Float64(sin(Float64(0.005555555555555556 * Float64(pi * angle))) * b)
	t_5 = sin(fma(0.5, pi, Float64(pi / 2.0)))
	tmp = 0.0
	if (a <= 1.35e-121)
		tmp = fma(t_4, t_4, fma(angle, Float64(Float64(Float64(0.011111111111111112 * Float64(a * a)) * Float64(Float64(pi * t_5) * t_1)) + Float64(angle * fma(Float64(Float64(a * a) * angle), Float64(Float64(Float64(Float64(Float64(pi * pi) * pi) * t_5) * t_1) * -2.2862368541380886e-7), Float64(Float64(a * a) * fma(Float64(-3.08641975308642e-5 * Float64(pi * pi)), t_2, Float64(Float64(3.08641975308642e-5 * Float64(pi * pi)) * Float64(t_5 * t_5))))))), Float64(Float64(a * a) * t_2)));
	elseif (a <= 6.2e+142)
		tmp = fma(Float64(Float64(Float64(t_0 * t_0) / a) * Float64(Float64(b * b) / a)), Float64(a * a), Float64(t_3 * t_3));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, 1.35e-121], N[(t$95$4 * t$95$4 + N[(angle * N[(N[(N[(0.011111111111111112 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(Pi * t$95$5), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(angle * N[(N[(N[(a * a), $MachinePrecision] * angle), $MachinePrecision] * N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision] * t$95$5), $MachinePrecision] * t$95$1), $MachinePrecision] * -2.2862368541380886e-7), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 6.2e+142], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.3500000000000001e-121

   1. Initial program 78.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. pow-to-exp N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

   4. Applied rewrites 13.6%

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

   5. Taylor expanded in a around -inf

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

   7. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right)\right)} \]

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(\frac{1}{180} \cdot \left(\pi \cdot angle\right)\right) \cdot b, angle \cdot \left(\frac{1}{90} \cdot \left({a}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) + angle \cdot \left({a}^{2} \cdot \left(angle \cdot \left(\frac{-1}{5832000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{17496000} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + {a}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) + \frac{1}{32400} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right)\right)\right)\right) + {a}^{2} \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}\right) \]

   10. Applied rewrites 55.7%

       \[\leadsto \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \mathsf{fma}\left(angle, \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) + angle \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot angle, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(a \cdot a\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), \sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right), \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right), \left(a \cdot a\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right) \]

   if 1.3500000000000001e-121 < a < 6.1999999999999998e142

   1. Initial program 71.9%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 59.3%

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

   if 6.1999999999999998e142 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \mathsf{fma}\left(angle, \left(0.011111111111111112 \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\pi \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) + angle \cdot \mathsf{fma}\left(\left(a \cdot a\right) \cdot angle, \left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right) \cdot \sin \left(0.5 \cdot \pi\right)\right) \cdot -2.2862368541380886 \cdot 10^{-7}, \left(a \cdot a\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), \sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right), \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right)\right)\right)\right), \left(a \cdot a\right) \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}{a} \cdot \frac{b \cdot b}{a}, a \cdot a, \left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ t_1 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\\ t_2 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_3 := \left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, t\_2\right)\right)\right) \cdot -1\\ \mathbf{if}\;angle \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\sin t\_2 \cdot b, 0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right), t\_3 \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0 \cdot t\_0}{a} \cdot \frac{b \cdot b}{a}, a \cdot a, t\_1 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* (* PI angle) 0.005555555555555556)))
        (t_1 (* (sin (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))) a))
        (t_2 (* 0.005555555555555556 (* PI angle)))
        (t_3 (* (* (pow (/ -1.0 a) -1.0) (sin (fma 0.5 PI t_2))) -1.0)))
   (if (<= angle 1.85e+55)
     (fma
      (* (sin t_2) b)
      (* 0.005555555555555556 (* angle (* b PI)))
      (* t_3 t_3))
     (fma (* (/ (* t_0 t_0) a) (/ (* b b) a)) (* a a) (* t_1 t_1)))))

C

double code(double a, double b, double angle) {
	double t_0 = sin(((((double) M_PI) * angle) * 0.005555555555555556));
	double t_1 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0))) * a;
	double t_2 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_3 = (pow((-1.0 / a), -1.0) * sin(fma(0.5, ((double) M_PI), t_2))) * -1.0;
	double tmp;
	if (angle <= 1.85e+55) {
		tmp = fma((sin(t_2) * b), (0.005555555555555556 * (angle * (b * ((double) M_PI)))), (t_3 * t_3));
	} else {
		tmp = fma((((t_0 * t_0) / a) * ((b * b) / a)), (a * a), (t_1 * t_1));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = sin(Float64(Float64(pi * angle) * 0.005555555555555556))
	t_1 = Float64(sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0))) * a)
	t_2 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_3 = Float64(Float64((Float64(-1.0 / a) ^ -1.0) * sin(fma(0.5, pi, t_2))) * -1.0)
	tmp = 0.0
	if (angle <= 1.85e+55)
		tmp = fma(Float64(sin(t_2) * b), Float64(0.005555555555555556 * Float64(angle * Float64(b * pi))), Float64(t_3 * t_3));
	else
		tmp = fma(Float64(Float64(Float64(t_0 * t_0) / a) * Float64(Float64(b * b) / a)), Float64(a * a), Float64(t_1 * t_1));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[N[(-1.0 / a), $MachinePrecision], -1.0], $MachinePrecision] * N[Sin[N[(0.5 * Pi + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision]}, If[LessEqual[angle, 1.85e+55], N[(N[(N[Sin[t$95$2], $MachinePrecision] * b), $MachinePrecision] * N[(0.005555555555555556 * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if angle < 1.8500000000000001e55

   1. Initial program 86.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. pow-to-exp N/A

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

      8. exp-prod N/A

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

      9. lower-pow.f64 N/A

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

   4. Applied rewrites 39.0%

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

   5. Taylor expanded in a around -inf

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

   7. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right)\right)} \]

   8. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 75.3

         \[\leadsto \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, 0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right), \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right)\right) \]

   10. Applied rewrites 75.3%

       \[\leadsto \mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, 0.005555555555555556 \cdot \color{blue}{\left(angle \cdot \left(b \cdot \pi\right)\right)}, \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right)\right) \]

   if 1.8500000000000001e55 < angle

   1. Initial program 63.3%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 38.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}{a} \cdot \frac{b \cdot b}{a}, a \cdot a, \left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot b, 0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right), \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right) \cdot \left(\left({\left(\frac{-1}{a}\right)}^{-1} \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right) \cdot -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}{a} \cdot \frac{b \cdot b}{a}, a \cdot a, \left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, \frac{\pi}{2}\right)\right) \cdot a\\ t_1 := \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 6.2 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1 \cdot t\_1}{a} \cdot \frac{b \cdot b}{a}, a \cdot a, t\_0 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sin (fma (* 0.005555555555555556 angle) PI (/ PI 2.0))) a))
        (t_1 (sin (* (* PI angle) 0.005555555555555556))))
   (if (<= a 6.2e+142)
     (fma (* (/ (* t_1 t_1) a) (/ (* b b) a)) (* a a) (* t_0 t_0))
     (* a a))))

C

double code(double a, double b, double angle) {
	double t_0 = sin(fma((0.005555555555555556 * angle), ((double) M_PI), (((double) M_PI) / 2.0))) * a;
	double t_1 = sin(((((double) M_PI) * angle) * 0.005555555555555556));
	double tmp;
	if (a <= 6.2e+142) {
		tmp = fma((((t_1 * t_1) / a) * ((b * b) / a)), (a * a), (t_0 * t_0));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(pi / 2.0))) * a)
	t_1 = sin(Float64(Float64(pi * angle) * 0.005555555555555556))
	tmp = 0.0
	if (a <= 6.2e+142)
		tmp = fma(Float64(Float64(Float64(t_1 * t_1) / a) * Float64(Float64(b * b) / a)), Float64(a * a), Float64(t_0 * t_0));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, 6.2e+142], N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] / a), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.1999999999999998e142

   1. Initial program 77.0%

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

   3. Taylor expanded in a around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 47.6%

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

   if 6.1999999999999998e142 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 10: 53.6% accurate, N/A× 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 80.8%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

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

Alternative 11: 27.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ e^{\log a \cdot 2} \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (exp (* (log a) 2.0)))

C

double code(double a, double b, double angle) {
	return exp((log(a) * 2.0));
}

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 = exp((log(a) * 2.0d0))
end function

Java

public static double code(double a, double b, double angle) {
	return Math.exp((Math.log(a) * 2.0));
}

Python

def code(a, b, angle):
	return math.exp((math.log(a) * 2.0))

Julia

function code(a, b, angle)
	return exp(Float64(log(a) * 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = exp((log(a) * 2.0));
end

Wolfram

code[a_, b_, angle_] := N[Exp[N[(N[Log[a], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 80.8%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.0

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

5. Applied rewrites 55.0%

   \[\leadsto \color{blue}{a \cdot a} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-log.f64 29.8

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

7. Applied rewrites 29.8%

   \[\leadsto e^{\log a \cdot 2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))