ab-angle->ABCF A

Percentage Accurate: 80.7% → 80.7%

Time: 4.9s

Alternatives: 10

Speedup: 1.0×

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

3. Applied rewrites 80.7%

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

Alternative 2: 80.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

3. Applied rewrites 80.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

5. Applied rewrites 80.7%

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

Alternative 3: 80.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

3. Applied rewrites 80.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

5. Applied rewrites 80.7%

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

6. Taylor expanded in angle around inf

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

   1. sqr-cos-a-rev N/A

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

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

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

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

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

   4. sqr-sin-a-rev N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 80.6

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

8. Applied rewrites 80.6%

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

Alternative 4: 80.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

3. Applied rewrites 80.7%

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

4. Taylor expanded in angle around 0

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

6. Applied rewrites 80.7%

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

Alternative 5: 62.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;a \leq 3.7 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - {t\_0}^{2}, b \cdot b, {\left(t\_0 \cdot a\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= a 3.7e+22)
     (* (fma (cos (* (* PI angle) 0.011111111111111112)) 0.5 0.5) (* b b))
     (if (<= a 1.12e+187)
       (fma
        (pow
         (*
          (fma
           0.005555555555555556
           PI
           (* (* (* angle angle) -2.8577960676726107e-8) (* (* PI PI) PI)))
          angle)
         2.0)
        (* a a)
        (* (* 1.0 b) b))
       (fma (- 1.0 (pow t_0 2.0)) (* b b) (pow (* t_0 a) 2.0))))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (a <= 3.7e+22) {
		tmp = fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, 0.5) * (b * b);
	} else if (a <= 1.12e+187) {
		tmp = fma(pow((fma(0.005555555555555556, ((double) M_PI), (((angle * angle) * -2.8577960676726107e-8) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))) * angle), 2.0), (a * a), ((1.0 * b) * b));
	} else {
		tmp = fma((1.0 - pow(t_0, 2.0)), (b * b), pow((t_0 * a), 2.0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (a <= 3.7e+22)
		tmp = Float64(fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, 0.5) * Float64(b * b));
	elseif (a <= 1.12e+187)
		tmp = fma((Float64(fma(0.005555555555555556, pi, Float64(Float64(Float64(angle * angle) * -2.8577960676726107e-8) * Float64(Float64(pi * pi) * pi))) * angle) ^ 2.0), Float64(a * a), Float64(Float64(1.0 * b) * b));
	else
		tmp = fma(Float64(1.0 - (t_0 ^ 2.0)), Float64(b * b), (Float64(t_0 * a) ^ 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[a, 3.7e+22], N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.12e+187], N[(N[Power[N[(N[(0.005555555555555556 * Pi + N[(N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision], 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(1.0 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[Power[N[(t$95$0 * a), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.6999999999999998e22

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in a around 0

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

      1. associate-*l* N/A

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

      2. sqr-cos-a-rev N/A

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

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

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

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

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

      5. sqr-sin-a-rev N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 56.9%

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

   if 3.6999999999999998e22 < a < 1.12000000000000007e187

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{angle}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{angle}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 74.8%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot b\right) \cdot b\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(\frac{1}{180}, \pi, \left(\left(angle \cdot angle\right) \cdot \frac{-1}{34992000}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]
   8. Step-by-step derivation

      1. sqr-cos-a-rev 61.8

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right) \]

      2. sin-+PI/2-rev 61.8

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right) \]

      3. sin-+PI/2-rev 61.8

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right) \]

      4. sqr-sin-a-rev 61.8

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right) \]

   9. Applied rewrites 61.8%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]

   if 1.12000000000000007e187 < a

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. 1-sub-sin-rev N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 58.4

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

   8. Applied rewrites 58.4%

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

   9. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 56.8

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

   11. Applied rewrites 56.8%

       \[\leadsto \mathsf{fma}\left(1 - {\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}, b \cdot b, {\left(\color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot a\right)}^{2}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.7e+22)
   (* (fma (cos (* (* PI angle) 0.011111111111111112)) 0.5 0.5) (* b b))
   (fma
    (pow
     (*
      (fma
       0.005555555555555556
       PI
       (* (* (* angle angle) -2.8577960676726107e-8) (* (* PI PI) PI)))
      angle)
     2.0)
    (* a a)
    (* (* 1.0 b) b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.7e+22) {
		tmp = fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, 0.5) * (b * b);
	} else {
		tmp = fma(pow((fma(0.005555555555555556, ((double) M_PI), (((angle * angle) * -2.8577960676726107e-8) * ((((double) M_PI) * ((double) M_PI)) * ((double) M_PI)))) * angle), 2.0), (a * a), ((1.0 * b) * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.7e+22)
		tmp = Float64(fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, 0.5) * Float64(b * b));
	else
		tmp = fma((Float64(fma(0.005555555555555556, pi, Float64(Float64(Float64(angle * angle) * -2.8577960676726107e-8) * Float64(Float64(pi * pi) * pi))) * angle) ^ 2.0), Float64(a * a), Float64(Float64(1.0 * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.7e+22], N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(0.005555555555555556 * Pi + N[(N[(N[(angle * angle), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle), $MachinePrecision], 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(1.0 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.6999999999999998e22

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in a around 0

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

      1. associate-*l* N/A

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

      2. sqr-cos-a-rev N/A

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

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

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

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

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

      5. sqr-sin-a-rev N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 56.9%

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

   if 3.6999999999999998e22 < a

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

      \[\leadsto {\left(a \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{angle}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

      2. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \left(\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{angle}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied rewrites 74.8%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(-2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot b\right) \cdot b\right)} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(\frac{1}{180}, \pi, \left(\left(angle \cdot angle\right) \cdot \frac{-1}{34992000}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]
   8. Step-by-step derivation

      1. sqr-cos-a-rev 61.8

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right) \]

      2. sin-+PI/2-rev 61.8

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right) \]

      3. sin-+PI/2-rev 61.8

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right) \]

      4. sqr-sin-a-rev 61.8

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(1 \cdot b\right) \cdot b\right) \]

   9. Applied rewrites 61.8%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(0.005555555555555556, \pi, \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot angle\right)}^{2}, a \cdot a, \left(\color{blue}{1} \cdot b\right) \cdot b\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 57.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \mathsf{fma}\left(\pi \cdot a, \pi \cdot a, \left(\left(\pi \cdot \pi\right) \cdot -1\right) \cdot \left(b \cdot b\right)\right), angle, \left(0 \cdot \left(b \cdot b\right)\right) \cdot 0.005555555555555556\right), angle, 1 \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3e+151)
   (fma
    (fma
     (*
      3.08641975308642e-5
      (fma (* PI a) (* PI a) (* (* (* PI PI) -1.0) (* b b))))
     angle
     (* (* 0.0 (* b b)) 0.005555555555555556))
    angle
    (* 1.0 (* b b)))
   (* (fma (cos (* (* PI angle) 0.011111111111111112)) 0.5 0.5) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3e+151) {
		tmp = fma(fma((3.08641975308642e-5 * fma((((double) M_PI) * a), (((double) M_PI) * a), (((((double) M_PI) * ((double) M_PI)) * -1.0) * (b * b)))), angle, ((0.0 * (b * b)) * 0.005555555555555556)), angle, (1.0 * (b * b)));
	} else {
		tmp = fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, 0.5) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3e+151)
		tmp = fma(fma(Float64(3.08641975308642e-5 * fma(Float64(pi * a), Float64(pi * a), Float64(Float64(Float64(pi * pi) * -1.0) * Float64(b * b)))), angle, Float64(Float64(0.0 * Float64(b * b)) * 0.005555555555555556)), angle, Float64(1.0 * Float64(b * b)));
	else
		tmp = Float64(fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, 0.5) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3e+151], N[(N[(N[(3.08641975308642e-5 * N[(N[(Pi * a), $MachinePrecision] * N[(Pi * a), $MachinePrecision] + N[(N[(N[(Pi * Pi), $MachinePrecision] * -1.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(N[(0.0 * N[(b * b), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * angle + N[(1.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9999999999999999e151

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

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

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-PI.f64 N/A

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

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

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

      15. sqr-sin-a N/A

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

      16. lower--.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 44.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5} \cdot \mathsf{fma}\left(\pi \cdot a, \pi \cdot a, \left(\left(\pi \cdot \pi\right) \cdot -1\right) \cdot \left(b \cdot b\right)\right), angle, \left(0 \cdot \left(b \cdot b\right)\right) \cdot 0.005555555555555556\right), angle, 1 \cdot \left(b \cdot b\right)\right)} \]

   if 2.9999999999999999e151 < b

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in a around 0

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

      1. associate-*l* N/A

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

      2. sqr-cos-a-rev N/A

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

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

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

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

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

      5. sqr-sin-a-rev N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 56.9%

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

Alternative 8: 56.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b, b, \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right), \pi \cdot \pi, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5, 0.5\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3e+151)
   (fma
    b
    b
    (*
     (* angle angle)
     (fma
      (* -3.08641975308642e-5 (* b b))
      (* PI PI)
      (* (* 3.08641975308642e-5 (* a a)) (* PI PI)))))
   (* (fma (cos (* (* PI angle) 0.011111111111111112)) 0.5 0.5) (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3e+151) {
		tmp = fma(b, b, ((angle * angle) * fma((-3.08641975308642e-5 * (b * b)), (((double) M_PI) * ((double) M_PI)), ((3.08641975308642e-5 * (a * a)) * (((double) M_PI) * ((double) M_PI))))));
	} else {
		tmp = fma(cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5, 0.5) * (b * b);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3e+151)
		tmp = fma(b, b, Float64(Float64(angle * angle) * fma(Float64(-3.08641975308642e-5 * Float64(b * b)), Float64(pi * pi), Float64(Float64(3.08641975308642e-5 * Float64(a * a)) * Float64(pi * pi)))));
	else
		tmp = Float64(fma(cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5, 0.5) * Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3e+151], N[(b * b + N[(N[(angle * angle), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9999999999999999e151

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 42.0%

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

   if 2.9999999999999999e151 < b

   1. Initial program 80.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

   3. Applied rewrites 80.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in a around 0

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

      1. associate-*l* N/A

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

      2. sqr-cos-a-rev N/A

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

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

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

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

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

      5. sqr-sin-a-rev N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 56.9%

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

Alternative 9: 54.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(b, b, \left(angle \cdot angle\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right), \pi \cdot \pi, \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 3e+151)
   (fma
    b
    b
    (*
     (* angle angle)
     (fma
      (* -3.08641975308642e-5 (* b b))
      (* PI PI)
      (* (* 3.08641975308642e-5 (* a a)) (* PI PI)))))
   (* b b)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 3e+151) {
		tmp = fma(b, b, ((angle * angle) * fma((-3.08641975308642e-5 * (b * b)), (((double) M_PI) * ((double) M_PI)), ((3.08641975308642e-5 * (a * a)) * (((double) M_PI) * ((double) M_PI))))));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 3e+151)
		tmp = fma(b, b, Float64(Float64(angle * angle) * fma(Float64(-3.08641975308642e-5 * Float64(b * b)), Float64(pi * pi), Float64(Float64(3.08641975308642e-5 * Float64(a * a)) * Float64(pi * pi)))));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 3e+151], N[(b * b + N[(N[(angle * angle), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9999999999999999e151

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 42.0%

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

   if 2.9999999999999999e151 < b

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

Alternative 10: 54.3% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.1

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

4. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025139 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))