ab-angle->ABCF A

Percentage Accurate: 79.0% → 79.0%

Time: 5.8s

Alternatives: 9

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.0

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

5. Applied rewrites 79.0%

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

Alternative 2: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(t\_0 \cdot 2\right), 0.5, 0.5\right), b \cdot b, {\left(\sin t\_0 \cdot a\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (fma (fma (cos (* t_0 2.0)) 0.5 0.5) (* b b) (pow (* (sin t_0) a) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	return fma(fma(cos((t_0 * 2.0)), 0.5, 0.5), (b * b), pow((sin(t_0) * a), 2.0));
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	return fma(fma(cos(Float64(t_0 * 2.0)), 0.5, 0.5), Float64(b * b), (Float64(sin(t_0) * a) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[Power[N[(N[Sin[t$95$0], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.0%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

4. Applied rewrites 79.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. sqr-cos-a N/A

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

   14. lower-+.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lower-*.f64 79.1

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

6. Applied rewrites 79.1%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

8. Applied rewrites 79.1%

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

Alternative 3: 79.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in angle around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 79.0

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

8. Applied rewrites 79.0%

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

Alternative 4: 65.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot angle\right)}^{2} \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e-42)
   (pow (* (cos (* (* PI angle) 0.005555555555555556)) b) 2.0)
   (+ (* (* (pow (* a angle) 2.0) (* PI PI)) 3.08641975308642e-5) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.2e-42) {
		tmp = pow((cos(((((double) M_PI) * angle) * 0.005555555555555556)) * b), 2.0);
	} else {
		tmp = ((pow((a * angle), 2.0) * (((double) M_PI) * ((double) M_PI))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.2e-42) {
		tmp = Math.pow((Math.cos(((Math.PI * angle) * 0.005555555555555556)) * b), 2.0);
	} else {
		tmp = ((Math.pow((a * angle), 2.0) * (Math.PI * Math.PI)) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.2e-42:
		tmp = math.pow((math.cos(((math.pi * angle) * 0.005555555555555556)) * b), 2.0)
	else:
		tmp = ((math.pow((a * angle), 2.0) * (math.pi * math.pi)) * 3.08641975308642e-5) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.2e-42)
		tmp = Float64(cos(Float64(Float64(pi * angle) * 0.005555555555555556)) * b) ^ 2.0;
	else
		tmp = Float64(Float64(Float64((Float64(a * angle) ^ 2.0) * Float64(pi * pi)) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.2e-42)
		tmp = (cos(((pi * angle) * 0.005555555555555556)) * b) ^ 2.0;
	else
		tmp = ((((a * angle) ^ 2.0) * (pi * pi)) * 3.08641975308642e-5) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.2e-42], N[Power[N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(N[(N[Power[N[(a * angle), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.20000000000000001e-42

   1. Initial program 77.8%

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

   3. Taylor expanded in a around 0

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 60.8

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

   5. Applied rewrites 60.8%

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

   if 1.20000000000000001e-42 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in angle around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow-prod-down N/A

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

      9. pow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 78.9

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

   10. Applied rewrites 78.9%

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

Alternative 5: 65.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(a \cdot angle\right)}^{2} \cdot \left(\pi \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.2e-42)
   (* (fma (cos (* 0.011111111111111112 (* PI angle))) 0.5 0.5) (* b b))
   (+ (* (* (pow (* a angle) 2.0) (* PI PI)) 3.08641975308642e-5) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.2e-42) {
		tmp = fma(cos((0.011111111111111112 * (((double) M_PI) * angle))), 0.5, 0.5) * (b * b);
	} else {
		tmp = ((pow((a * angle), 2.0) * (((double) M_PI) * ((double) M_PI))) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.2e-42)
		tmp = Float64(fma(cos(Float64(0.011111111111111112 * Float64(pi * angle))), 0.5, 0.5) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64((Float64(a * angle) ^ 2.0) * Float64(pi * pi)) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.2e-42], N[(N[(N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[N[(a * angle), $MachinePrecision], 2.0], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.20000000000000001e-42

   1. Initial program 77.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   4. Applied rewrites 77.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. sqr-cos-a N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 77.8

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

   6. Applied rewrites 77.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 60.8%

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

   if 1.20000000000000001e-42 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in angle around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow-prod-down N/A

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

      9. pow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. lift-PI.f64 78.9

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

   10. Applied rewrites 78.9%

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

Alternative 6: 65.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot a\\ \mathbf{if}\;a \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) a)))
   (if (<= a 1.2e-42)
     (* (fma (cos (* 0.011111111111111112 (* PI angle))) 0.5 0.5) (* b b))
     (+ (* (* t_0 t_0) 3.08641975308642e-5) (* b b)))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * a;
	double tmp;
	if (a <= 1.2e-42) {
		tmp = fma(cos((0.011111111111111112 * (((double) M_PI) * angle))), 0.5, 0.5) * (b * b);
	} else {
		tmp = ((t_0 * t_0) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * a)
	tmp = 0.0
	if (a <= 1.2e-42)
		tmp = Float64(fma(cos(Float64(0.011111111111111112 * Float64(pi * angle))), 0.5, 0.5) * Float64(b * b));
	else
		tmp = Float64(Float64(Float64(t_0 * t_0) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, 1.2e-42], N[(N[(N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.20000000000000001e-42

   1. Initial program 77.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   4. Applied rewrites 77.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. sqr-cos-a N/A

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

      14. lower-+.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. lower-*.f64 77.8

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

   6. Applied rewrites 77.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 60.8%

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

   if 1.20000000000000001e-42 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in angle around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 78.8

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

   10. Applied rewrites 78.8%

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

Alternative 7: 65.9% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot a\\ \mathbf{if}\;a \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot t\_0\right) \cdot 3.08641975308642 \cdot 10^{-5} + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) a)))
   (if (<= a 1.2e-42)
     (* b b)
     (+ (* (* t_0 t_0) 3.08641975308642e-5) (* b b)))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * a;
	double tmp;
	if (a <= 1.2e-42) {
		tmp = b * b;
	} else {
		tmp = ((t_0 * t_0) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.PI * angle) * a;
	double tmp;
	if (a <= 1.2e-42) {
		tmp = b * b;
	} else {
		tmp = ((t_0 * t_0) * 3.08641975308642e-5) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (math.pi * angle) * a
	tmp = 0
	if a <= 1.2e-42:
		tmp = b * b
	else:
		tmp = ((t_0 * t_0) * 3.08641975308642e-5) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * a)
	tmp = 0.0
	if (a <= 1.2e-42)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(t_0 * t_0) * 3.08641975308642e-5) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (pi * angle) * a;
	tmp = 0.0;
	if (a <= 1.2e-42)
		tmp = b * b;
	else
		tmp = ((t_0 * t_0) * 3.08641975308642e-5) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, 1.2e-42], N[(b * b), $MachinePrecision], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.20000000000000001e-42

   1. Initial program 77.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if 1.20000000000000001e-42 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in angle around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 78.8

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

   10. Applied rewrites 78.8%

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

Alternative 8: 66.0% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot a\\ \mathbf{if}\;a \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right) + b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) a)))
   (if (<= a 1.2e-42)
     (* b b)
     (+ (* t_0 (* t_0 3.08641975308642e-5)) (* b b)))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * a;
	double tmp;
	if (a <= 1.2e-42) {
		tmp = b * b;
	} else {
		tmp = (t_0 * (t_0 * 3.08641975308642e-5)) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.PI * angle) * a;
	double tmp;
	if (a <= 1.2e-42) {
		tmp = b * b;
	} else {
		tmp = (t_0 * (t_0 * 3.08641975308642e-5)) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (math.pi * angle) * a
	tmp = 0
	if a <= 1.2e-42:
		tmp = b * b
	else:
		tmp = (t_0 * (t_0 * 3.08641975308642e-5)) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * a)
	tmp = 0.0
	if (a <= 1.2e-42)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(t_0 * Float64(t_0 * 3.08641975308642e-5)) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (pi * angle) * a;
	tmp = 0.0;
	if (a <= 1.2e-42)
		tmp = b * b;
	else
		tmp = (t_0 * (t_0 * 3.08641975308642e-5)) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, 1.2e-42], N[(b * b), $MachinePrecision], N[(N[(t$95$0 * N[(t$95$0 * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.20000000000000001e-42

   1. Initial program 77.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if 1.20000000000000001e-42 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in angle around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 78.8

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

   8. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 79.1

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

   10. Applied rewrites 79.1%

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

Alternative 9: 56.6% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.0%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 56.6

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

5. Applied rewrites 56.6%

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

Reproduce

herbie shell --seed 2025089 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))