ab-angle->ABCF C

Percentage Accurate: 79.3% → 79.3%

Time: 19.6s

Alternatives: 13

Speedup: 0.7×

• 35.6% 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: 79.3% 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: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_1 := b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathsf{fma}\left(t\_1, t\_1, a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, t\_0, 0 - \mathsf{fma}\left(\cos \left(\left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right)\right), 0.5\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (* PI (* angle 0.005555555555555556))))
        (t_1 (* b (sin (* (* PI angle) 0.005555555555555556)))))
   (fma
    t_1
    t_1
    (*
     a
     (*
      a
      (fma
       0.5
       (fma
        t_0
        t_0
        (-
         0.0
         (fma
          (cos
           (*
            (* angle (* (sqrt (* PI (sqrt PI))) (sqrt (sqrt PI))))
            0.011111111111111112))
          -0.5
          0.5)))
       0.5))))))

C

double code(double a, double b, double angle) {
	double t_0 = cos((((double) M_PI) * (angle * 0.005555555555555556)));
	double t_1 = b * sin(((((double) M_PI) * angle) * 0.005555555555555556));
	return fma(t_1, t_1, (a * (a * fma(0.5, fma(t_0, t_0, (0.0 - fma(cos(((angle * (sqrt((((double) M_PI) * sqrt(((double) M_PI)))) * sqrt(sqrt(((double) M_PI))))) * 0.011111111111111112)), -0.5, 0.5))), 0.5))));
}

Julia

function code(a, b, angle)
	t_0 = cos(Float64(pi * Float64(angle * 0.005555555555555556)))
	t_1 = Float64(b * sin(Float64(Float64(pi * angle) * 0.005555555555555556)))
	return fma(t_1, t_1, Float64(a * Float64(a * fma(0.5, fma(t_0, t_0, Float64(0.0 - fma(cos(Float64(Float64(angle * Float64(sqrt(Float64(pi * sqrt(pi))) * sqrt(sqrt(pi)))) * 0.011111111111111112)), -0.5, 0.5))), 0.5))))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$1 * t$95$1 + N[(a * N[(a * N[(0.5 * N[(t$95$0 * t$95$0 + N[(0.0 - N[(N[Cos[N[(N[(angle * N[(N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 74.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. exp-lowering-exp.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. PI-lowering-PI.f64 41.8

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

4. Applied egg-rr 41.8%

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

6. Applied egg-rr 74.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\right)} \]

8. Applied egg-rr 74.5%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), -\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right)\right)}, 0.5\right)\right)\right) \]
9. Step-by-step derivation

   1. add-sqr-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. pow3 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), a \cdot \left(a \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \mathsf{neg}\left(\mathsf{fma}\left(\cos \left(\left(\sqrt{\color{blue}{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{3}} \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot angle\right) \cdot \frac{1}{90}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)\right), \frac{1}{2}\right)\right)\right) \]

   7. sqrt-prod N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), a \cdot \left(a \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \mathsf{neg}\left(\mathsf{fma}\left(\cos \left(\left(\color{blue}{\left(\sqrt{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{3}} \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right)} \cdot angle\right) \cdot \frac{1}{90}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)\right), \frac{1}{2}\right)\right)\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), a \cdot \left(a \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \mathsf{neg}\left(\mathsf{fma}\left(\cos \left(\left(\left(\sqrt{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{3}} \cdot \color{blue}{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}}\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)\right), \frac{1}{2}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), a \cdot \left(a \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \mathsf{neg}\left(\mathsf{fma}\left(\cos \left(\left(\color{blue}{\left(\sqrt{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{3}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)} \cdot angle\right) \cdot \frac{1}{90}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)\right), \frac{1}{2}\right)\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), b \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{180}\right), a \cdot \left(a \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{fma}\left(\cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \mathsf{neg}\left(\mathsf{fma}\left(\cos \left(\left(\left(\color{blue}{\sqrt{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{3}}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{-1}{2}, \frac{1}{2}\right)\right)\right), \frac{1}{2}\right)\right)\right) \]

   11. pow3 N/A

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

   12. add-sqr-sqrt N/A

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

   13. *-lowering-*.f64 N/A

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

   14. PI-lowering-PI.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. pow1/2 N/A

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

   18. sqrt-lowering-sqrt.f64 N/A

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

   19. sqrt-lowering-sqrt.f64 N/A

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

10. Applied egg-rr 74.5%

    \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), -\mathsf{fma}\left(\cos \left(\left(\color{blue}{\left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)} \cdot angle\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right)\right), 0.5\right)\right)\right) \]

11. Final simplification 74.5%

    \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0 - \mathsf{fma}\left(\cos \left(\left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right)\right), 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 2: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_1 := b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathsf{fma}\left(t\_1, t\_1, a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(t\_0, t\_0, 0 - \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right)\right), 0.5\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (* PI (* angle 0.005555555555555556))))
        (t_1 (* b (sin (* (* PI angle) 0.005555555555555556)))))
   (fma
    t_1
    t_1
    (*
     a
     (*
      a
      (fma
       0.5
       (fma
        t_0
        t_0
        (- 0.0 (fma (cos (* (* PI angle) 0.011111111111111112)) -0.5 0.5)))
       0.5))))))

C

double code(double a, double b, double angle) {
	double t_0 = cos((((double) M_PI) * (angle * 0.005555555555555556)));
	double t_1 = b * sin(((((double) M_PI) * angle) * 0.005555555555555556));
	return fma(t_1, t_1, (a * (a * fma(0.5, fma(t_0, t_0, (0.0 - fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), -0.5, 0.5))), 0.5))));
}

Julia

function code(a, b, angle)
	t_0 = cos(Float64(pi * Float64(angle * 0.005555555555555556)))
	t_1 = Float64(b * sin(Float64(Float64(pi * angle) * 0.005555555555555556)))
	return fma(t_1, t_1, Float64(a * Float64(a * fma(0.5, fma(t_0, t_0, Float64(0.0 - fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), -0.5, 0.5))), 0.5))))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$1 * t$95$1 + N[(a * N[(a * N[(0.5 * N[(t$95$0 * t$95$0 + N[(0.0 - N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 74.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. exp-lowering-exp.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. PI-lowering-PI.f64 41.8

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

4. Applied egg-rr 41.8%

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

6. Applied egg-rr 74.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\right)} \]

8. Applied egg-rr 74.5%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \color{blue}{\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), -\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right)\right)}, 0.5\right)\right)\right) \]

9. Final simplification 74.5%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0 - \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right)\right), 0.5\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 79.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\\ \mathsf{fma}\left(t\_0, t\_0, a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* b (sin (* (* PI angle) 0.005555555555555556)))))
   (fma
    t_0
    t_0
    (* a (* a (fma 0.5 (cos (* (* PI angle) 0.011111111111111112)) 0.5))))))

C

double code(double a, double b, double angle) {
	double t_0 = b * sin(((((double) M_PI) * angle) * 0.005555555555555556));
	return fma(t_0, t_0, (a * (a * fma(0.5, cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5))));
}

Julia

function code(a, b, angle)
	t_0 = Float64(b * sin(Float64(Float64(pi * angle) * 0.005555555555555556)))
	return fma(t_0, t_0, Float64(a * Float64(a * fma(0.5, cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5))))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(a * N[(a * N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 74.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. exp-lowering-exp.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. PI-lowering-PI.f64 41.8

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

4. Applied egg-rr 41.8%

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

6. Applied egg-rr 74.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\right)} \]
7. Add Preprocessing

Alternative 4: 79.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. inv-pow N/A

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

   4. pow-to-exp N/A

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

   5. exp-lowering-exp.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. PI-lowering-PI.f64 41.8

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

4. Applied egg-rr 41.8%

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

6. Applied egg-rr 74.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\right)} \]

7. Taylor expanded in angle around 0

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

9. Simplified 73.6%

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

10. Final simplification 73.6%

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

Alternative 5: 79.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 73.6

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

5. Simplified 73.6%

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

Alternative 6: 64.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 3.5 \cdot 10^{-162}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 6600000:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 3.5e-162)
   (* a a)
   (if (<= angle 6600000.0)
     (fma
      a
      (* a (fma 0.5 (cos (* (* PI angle) 0.011111111111111112)) 0.5))
      (* (* b b) (* (* angle angle) (* PI (* PI 3.08641975308642e-5)))))
     (fma
      a
      a
      (*
       (* b b)
       (- 0.5 (* 0.5 (cos (* (* PI (* angle 0.005555555555555556)) 2.0)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 3.5e-162) {
		tmp = a * a;
	} else if (angle <= 6600000.0) {
		tmp = fma(a, (a * fma(0.5, cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5)), ((b * b) * ((angle * angle) * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)))));
	} else {
		tmp = fma(a, a, ((b * b) * (0.5 - (0.5 * cos(((((double) M_PI) * (angle * 0.005555555555555556)) * 2.0))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 3.5e-162)
		tmp = Float64(a * a);
	elseif (angle <= 6600000.0)
		tmp = fma(a, Float64(a * fma(0.5, cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5)), Float64(Float64(b * b) * Float64(Float64(angle * angle) * Float64(pi * Float64(pi * 3.08641975308642e-5)))));
	else
		tmp = fma(a, a, Float64(Float64(b * b) * Float64(0.5 - Float64(0.5 * cos(Float64(Float64(pi * Float64(angle * 0.005555555555555556)) * 2.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 3.5e-162], N[(a * a), $MachinePrecision], If[LessEqual[angle, 6600000.0], N[(a * N[(a * N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * a + N[(N[(b * b), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 3.4999999999999999e-162

   1. Initial program 79.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.2

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

   5. Simplified 57.2%

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

   if 3.4999999999999999e-162 < angle < 6.6e6

   1. Initial program 99.3%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. exp-lowering-exp.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 99.0

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

   4. Applied egg-rr 99.0%

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

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\right)} \]

   7. Taylor expanded in b around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. cos-lowering-cos.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

   9. Simplified 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right), \left(b \cdot b\right) \cdot {\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}\right)} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left({angle}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right)\right) \]

       11. PI-lowering-PI.f64 N/A

           \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right)\right) \]

       13. PI-lowering-PI.f64 96.7

           \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\color{blue}{\pi} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \]

   12. Simplified 96.7%

       \[\leadsto \mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right), \left(b \cdot b\right) \cdot \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)}\right) \]

   if 6.6e6 < angle

   1. Initial program 51.2%

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

      1. rem-exp-log N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. log-prod N/A

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

      6. exp-sum N/A

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

      7. rem-exp-log N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 20.2%

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

   5. Taylor expanded in angle around 0

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

   7. Simplified 51.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{a}, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.5 \cdot 10^{-162}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 6600000:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), \left(b \cdot b\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 1.8 \cdot 10^{-162}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 470:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 1.8e-162)
   (* a a)
   (if (<= angle 470.0)
     (fma
      (* angle angle)
      (* PI (* PI (* (* b b) 3.08641975308642e-5)))
      (* a a))
     (fma
      a
      a
      (*
       (* b b)
       (- 0.5 (* 0.5 (cos (* (* PI (* angle 0.005555555555555556)) 2.0)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 1.8e-162) {
		tmp = a * a;
	} else if (angle <= 470.0) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * ((b * b) * 3.08641975308642e-5))), (a * a));
	} else {
		tmp = fma(a, a, ((b * b) * (0.5 - (0.5 * cos(((((double) M_PI) * (angle * 0.005555555555555556)) * 2.0))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 1.8e-162)
		tmp = Float64(a * a);
	elseif (angle <= 470.0)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * Float64(Float64(b * b) * 3.08641975308642e-5))), Float64(a * a));
	else
		tmp = fma(a, a, Float64(Float64(b * b) * Float64(0.5 - Float64(0.5 * cos(Float64(Float64(pi * Float64(angle * 0.005555555555555556)) * 2.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 1.8e-162], N[(a * a), $MachinePrecision], If[LessEqual[angle, 470.0], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a + N[(N[(b * b), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if angle < 1.7999999999999999e-162

   1. Initial program 79.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.2

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

   5. Simplified 57.2%

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

   if 1.7999999999999999e-162 < angle < 470

   1. Initial program 99.8%

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

   3. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 67.6%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

   if 470 < angle

   1. Initial program 52.3%

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

      1. rem-exp-log N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. log-prod N/A

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

      6. exp-sum N/A

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

      7. rem-exp-log N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 20.8%

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

   5. Taylor expanded in angle around 0

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

   7. Simplified 50.6%

      \[\leadsto \mathsf{fma}\left(\color{blue}{a}, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.8 \cdot 10^{-162}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 470:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot \left(a \cdot -3.08641975308642 \cdot 10^{-5}\right)\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.7e+54)
   (fma
    (*
     (* PI angle)
     (* PI (fma b (* b 3.08641975308642e-5) (* a (* a -3.08641975308642e-5)))))
    angle
    (* a a))
   (* (* a a) (fma 0.5 (cos (* (* PI angle) 0.011111111111111112)) 0.5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.7e+54) {
		tmp = fma(((((double) M_PI) * angle) * (((double) M_PI) * fma(b, (b * 3.08641975308642e-5), (a * (a * -3.08641975308642e-5))))), angle, (a * a));
	} else {
		tmp = (a * a) * fma(0.5, cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.7e+54)
		tmp = fma(Float64(Float64(pi * angle) * Float64(pi * fma(b, Float64(b * 3.08641975308642e-5), Float64(a * Float64(a * -3.08641975308642e-5))))), angle, Float64(a * a));
	else
		tmp = Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.7e+54], N[(N[(N[(Pi * angle), $MachinePrecision] * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision] + N[(a * N[(a * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7e54

   1. Initial program 70.0%

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

   3. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 40.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 44.8%

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

   if 1.7e54 < a

   1. Initial program 87.5%

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

      1. associate-*r/ N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. exp-lowering-exp.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 48.1

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

   4. Applied egg-rr 48.1%

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

   6. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), b \cdot \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), a \cdot \left(a \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\right)} \]

   7. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 84.4

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

   9. Simplified 84.4%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.7 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot \left(a \cdot -3.08641975308642 \cdot 10^{-5}\right)\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot \left(a \cdot -3.08641975308642 \cdot 10^{-5}\right)\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 4.8e+54)
   (fma
    (*
     (* PI angle)
     (* PI (fma b (* b 3.08641975308642e-5) (* a (* a -3.08641975308642e-5)))))
    angle
    (* a a))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 4.8e+54) {
		tmp = fma(((((double) M_PI) * angle) * (((double) M_PI) * fma(b, (b * 3.08641975308642e-5), (a * (a * -3.08641975308642e-5))))), angle, (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 4.8e+54)
		tmp = fma(Float64(Float64(pi * angle) * Float64(pi * fma(b, Float64(b * 3.08641975308642e-5), Float64(a * Float64(a * -3.08641975308642e-5))))), angle, Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 4.8e+54], N[(N[(N[(Pi * angle), $MachinePrecision] * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision] + N[(a * N[(a * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.79999999999999997e54

   1. Initial program 70.0%

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

   3. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 40.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 44.8%

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

   if 4.79999999999999997e54 < a

   1. Initial program 87.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 84.3

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

   5. Simplified 84.3%

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

Alternative 10: 61.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{+22}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.02e+22)
   (* a a)
   (fma
    (* angle angle)
    (* PI (* PI (* (* b b) 3.08641975308642e-5)))
    (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.02e+22) {
		tmp = a * a;
	} else {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * ((b * b) * 3.08641975308642e-5))), (a * a));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.02e+22)
		tmp = Float64(a * a);
	else
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * Float64(Float64(b * b) * 3.08641975308642e-5))), Float64(a * a));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.02e+22], N[(a * a), $MachinePrecision], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.02e22

   1. Initial program 72.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.6

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

   5. Simplified 60.6%

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

   if 1.02e22 < b

   1. Initial program 79.9%

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

   3. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 37.4%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 63.2

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

   8. Simplified 63.2%

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

Alternative 11: 60.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+151}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4e+151)
   (* a a)
   (* angle (* (* angle (* PI 3.08641975308642e-5)) (* PI (* b b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4e+151) {
		tmp = a * a;
	} else {
		tmp = angle * ((angle * (((double) M_PI) * 3.08641975308642e-5)) * (((double) M_PI) * (b * b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4e+151) {
		tmp = a * a;
	} else {
		tmp = angle * ((angle * (Math.PI * 3.08641975308642e-5)) * (Math.PI * (b * b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4e+151:
		tmp = a * a
	else:
		tmp = angle * ((angle * (math.pi * 3.08641975308642e-5)) * (math.pi * (b * b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4e+151)
		tmp = Float64(a * a);
	else
		tmp = Float64(angle * Float64(Float64(angle * Float64(pi * 3.08641975308642e-5)) * Float64(pi * Float64(b * b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4e+151)
		tmp = a * a;
	else
		tmp = angle * ((angle * (pi * 3.08641975308642e-5)) * (pi * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4e+151], N[(a * a), $MachinePrecision], N[(angle * N[(N[(angle * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000007e151

   1. Initial program 72.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.8

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

   5. Simplified 57.8%

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

   if 4.00000000000000007e151 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 41.9%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. PI-lowering-PI.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. PI-lowering-PI.f64 N/A

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 73.7

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

   8. Simplified 73.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. PI-lowering-PI.f64 N/A

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

      12. *-lowering-*.f64 74.4

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

   10. Applied egg-rr 74.4%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{+151}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.02 \cdot 10^{+152}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.02e+152)
   (* a a)
   (* (* angle angle) (* PI (* 3.08641975308642e-5 (* PI (* b b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.02e+152) {
		tmp = a * a;
	} else {
		tmp = (angle * angle) * (((double) M_PI) * (3.08641975308642e-5 * (((double) M_PI) * (b * b))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.02e+152) {
		tmp = a * a;
	} else {
		tmp = (angle * angle) * (Math.PI * (3.08641975308642e-5 * (Math.PI * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.02e+152:
		tmp = a * a
	else:
		tmp = (angle * angle) * (math.pi * (3.08641975308642e-5 * (math.pi * (b * b))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.02e+152)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle * angle) * Float64(pi * Float64(3.08641975308642e-5 * Float64(pi * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.02e+152)
		tmp = a * a;
	else
		tmp = (angle * angle) * (pi * (3.08641975308642e-5 * (pi * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.02e+152], N[(a * a), $MachinePrecision], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(3.08641975308642e-5 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.01999999999999999e152

   1. Initial program 72.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.8

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

   5. Simplified 57.8%

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

   if 1.01999999999999999e152 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 41.9%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-lowering-*.f64 N/A

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

      14. PI-lowering-PI.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 N/A

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

      18. PI-lowering-PI.f64 N/A

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

      19. unpow2 N/A

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

      20. *-lowering-*.f64 73.7

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

   8. Simplified 73.7%

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

Alternative 13: 56.2% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, angle)
    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 74.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 56.1

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

5. Simplified 56.1%

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

Reproduce

herbie shell --seed 2024197 
(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)))