ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.8%

Time: 4.9s

Alternatives: 8

Speedup: 1.8×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot \frac{angle}{180}\right)\\ \mathsf{fma}\left(1 - {t\_0}^{2}, a \cdot a, {\left(t\_0 \cdot b\right)}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sin (* PI (/ angle 180.0)))))
   (fma (- 1.0 (pow t_0 2.0)) (* a a) (pow (* t_0 b) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = sin((((double) M_PI) * (angle / 180.0)));
	return fma((1.0 - pow(t_0, 2.0)), (a * a), pow((t_0 * b), 2.0));
}

Julia

function code(a, b, angle)
	t_0 = sin(Float64(pi * Float64(angle / 180.0)))
	return fma(Float64(1.0 - (t_0 ^ 2.0)), Float64(a * a), (Float64(t_0 * b) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[Power[N[(t$95$0 * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 79.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. lift-pow.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

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

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

5. Applied rewrites 79.8%

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

Alternative 2: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. lift-pow.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

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

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 + \cos \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right) \cdot 0.5\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (*
   (+ 0.5 (* (cos (* (* (* 0.005555555555555556 angle) PI) 2.0)) 0.5))
   (* a a))
  (pow (* b (sin (* PI (* 0.005555555555555556 angle)))) 2.0)))

C

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

Java

public static double code(double a, double b, double angle) {
	return ((0.5 + (Math.cos((((0.005555555555555556 * angle) * Math.PI) * 2.0)) * 0.5)) * (a * a)) + Math.pow((b * Math.sin((Math.PI * (0.005555555555555556 * angle)))), 2.0);
}

Python

def code(a, b, angle):
	return ((0.5 + (math.cos((((0.005555555555555556 * angle) * math.pi) * 2.0)) * 0.5)) * (a * a)) + math.pow((b * math.sin((math.pi * (0.005555555555555556 * angle)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(0.5 + Float64(cos(Float64(Float64(Float64(0.005555555555555556 * angle) * pi) * 2.0)) * 0.5)) * Float64(a * a)) + (Float64(b * sin(Float64(pi * Float64(0.005555555555555556 * angle)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((0.5 + (cos((((0.005555555555555556 * angle) * pi) * 2.0)) * 0.5)) * (a * a)) + ((b * sin((pi * (0.005555555555555556 * angle)))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.8

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

4. Applied rewrites 79.8%

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

5. Taylor expanded in angle around 0

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

   1. lower-*.f64 79.8

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 79.8%

    \[\leadsto \color{blue}{\left(0.5 + \cos \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot 2\right) \cdot 0.5\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} \]
11. Add Preprocessing

Alternative 4: 79.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. lift-pow.f64 N/A

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

3. Applied rewrites 79.8%

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

4. Taylor expanded in angle around 0

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

6. Applied rewrites 79.8%

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

Alternative 5: 67.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot b\right)\\ \mathbf{if}\;b \leq 1.1 \cdot 10^{-38}:\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0.5 \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, a \cdot a, t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) (* PI b))))
   (if (<= b 1.1e-38)
     (*
      (-
       0.5
       (*
        0.5
        (cos (* 2.0 (fma (* PI angle) 0.005555555555555556 (* 0.5 PI))))))
      (* a a))
     (fma 1.0 (* a a) (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * angle) * (((double) M_PI) * b);
	double tmp;
	if (b <= 1.1e-38) {
		tmp = (0.5 - (0.5 * cos((2.0 * fma((((double) M_PI) * angle), 0.005555555555555556, (0.5 * ((double) M_PI))))))) * (a * a);
	} else {
		tmp = fma(1.0, (a * a), (t_0 * t_0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * Float64(pi * b))
	tmp = 0.0
	if (b <= 1.1e-38)
		tmp = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * fma(Float64(pi * angle), 0.005555555555555556, Float64(0.5 * pi)))))) * Float64(a * a));
	else
		tmp = fma(1.0, Float64(a * a), Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.1e-38], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(a * a), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.10000000000000004e-38

   1. Initial program 79.8%

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

      1. lift-cos.f64 N/A

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

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

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

      3. sin-sum N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

   3. Applied rewrites 79.8%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 45.6%

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

   7. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

   9. Applied rewrites 56.9%

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

   if 1.10000000000000004e-38 < b

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 74.8

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

   4. Applied rewrites 74.8%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\left(\left(\pi \cdot b\right) \cdot angle\right) \cdot 0.005555555555555556\right)}}^{2} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 74.8%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 74.6%

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

Alternative 6: 67.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot b\right)\\ \mathbf{if}\;b \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\left(0.5 + \cos \left(\left(2 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, a \cdot a, t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) (* PI b))))
   (if (<= b 1.05e-8)
     (*
      (+ 0.5 (* (cos (* (* 2.0 (* PI angle)) 0.005555555555555556)) 0.5))
      (* a a))
     (fma 1.0 (* a a) (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * angle) * (((double) M_PI) * b);
	double tmp;
	if (b <= 1.05e-8) {
		tmp = (0.5 + (cos(((2.0 * (((double) M_PI) * angle)) * 0.005555555555555556)) * 0.5)) * (a * a);
	} else {
		tmp = fma(1.0, (a * a), (t_0 * t_0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * Float64(pi * b))
	tmp = 0.0
	if (b <= 1.05e-8)
		tmp = Float64(Float64(0.5 + Float64(cos(Float64(Float64(2.0 * Float64(pi * angle)) * 0.005555555555555556)) * 0.5)) * Float64(a * a));
	else
		tmp = fma(1.0, Float64(a * a), Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.05e-8], N[(N[(0.5 + N[(N[Cos[N[(N[(2.0 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(a * a), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.04999999999999997e-8

   1. Initial program 79.8%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. lift-pow.f64 N/A

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

   3. Applied rewrites 79.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

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

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\left(0.5 + \cos \left(\left(2 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot \left(a \cdot a\right)} \]

   if 1.04999999999999997e-8 < b

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 74.8

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

   4. Applied rewrites 74.8%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\left(\left(\pi \cdot b\right) \cdot angle\right) \cdot 0.005555555555555556\right)}}^{2} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 74.8%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 74.6%

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

Alternative 7: 67.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot b\right)\\ \mathbf{if}\;b \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, a \cdot a, t\_0 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) (* PI b))))
   (if (<= b 1.3e-6) (* a a) (fma 1.0 (* a a) (* t_0 t_0)))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 * angle) * (((double) M_PI) * b);
	double tmp;
	if (b <= 1.3e-6) {
		tmp = a * a;
	} else {
		tmp = fma(1.0, (a * a), (t_0 * t_0));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * Float64(pi * b))
	tmp = 0.0
	if (b <= 1.3e-6)
		tmp = Float64(a * a);
	else
		tmp = fma(1.0, Float64(a * a), Float64(t_0 * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.3e-6], N[(a * a), $MachinePrecision], N[(1.0 * N[(a * a), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.30000000000000005e-6

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

   if 1.30000000000000005e-6 < b

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-PI.f64 74.8

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

   4. Applied rewrites 74.8%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\color{blue}{\left(\left(\left(\pi \cdot b\right) \cdot angle\right) \cdot 0.005555555555555556\right)}}^{2} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 74.8%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 74.6%

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

Alternative 8: 57.1% accurate, 29.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

2. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 57.1

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

4. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025142 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))