ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.5%

Time: 4.5s

Alternatives: 11

Speedup: N/A×

• 36.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. pow-prod-down N/A

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

   3. lower-pow.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

4. Applied rewrites 79.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 39.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-to-pow N/A

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

   10. unpow-prod-down N/A

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

   11. pow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

8. Applied rewrites 79.5%

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

Alternative 2: 79.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. pow-prod-down N/A

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

   3. lower-pow.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

4. Applied rewrites 79.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. pow-to-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-*.f64 N/A

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

6. Applied rewrites 39.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. exp-to-pow N/A

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

   10. unpow-prod-down N/A

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

   11. pow2 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

8. Applied rewrites 79.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. sqr-cos-a N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lift-*.f64 79.5

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

10. Applied rewrites 79.5%

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

Alternative 3: 79.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. pow-prod-down N/A

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

   3. lower-pow.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

4. Applied rewrites 79.5%

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

6. Applied rewrites 79.6%

   \[\leadsto \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, \color{blue}{a \cdot a}, {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}\right) \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-cos-a N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-PI.f64 79.6

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

8. Applied rewrites 79.6%

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

Alternative 4: 79.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. pow-prod-down N/A

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

   3. lower-pow.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

4. Applied rewrites 79.5%

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

5. Taylor expanded in angle around 0

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

   1. pow2 N/A

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

   2. lift-*.f64 79.5

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

7. Applied rewrites 79.5%

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

Alternative 5: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{-159}:\\ \;\;\;\;{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b \cdot \left(angle \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.7e-159)
   (pow (* (sin (* (* PI angle) 0.005555555555555556)) b) 2.0)
   (+
    (* a a)
    (* (* (* b angle) PI) (* (* b (* angle PI)) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.7e-159) {
		tmp = pow((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * b), 2.0);
	} else {
		tmp = (a * a) + (((b * angle) * ((double) M_PI)) * ((b * (angle * ((double) M_PI))) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.7e-159) {
		tmp = Math.pow((Math.sin(((Math.PI * angle) * 0.005555555555555556)) * b), 2.0);
	} else {
		tmp = (a * a) + (((b * angle) * Math.PI) * ((b * (angle * Math.PI)) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.7e-159:
		tmp = math.pow((math.sin(((math.pi * angle) * 0.005555555555555556)) * b), 2.0)
	else:
		tmp = (a * a) + (((b * angle) * math.pi) * ((b * (angle * math.pi)) * 3.08641975308642e-5))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.7e-159)
		tmp = Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * b) ^ 2.0;
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(b * angle) * pi) * Float64(Float64(b * Float64(angle * pi)) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.7e-159)
		tmp = (sin(((pi * angle) * 0.005555555555555556)) * b) ^ 2.0;
	else
		tmp = (a * a) + (((b * angle) * pi) * ((b * (angle * pi)) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.7e-159], N[Power[N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(b * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.7e-159

   1. Initial program 78.8%

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

   2. Taylor expanded in a around 0

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 47.2

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

   4. Applied rewrites 47.2%

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

   if 2.7e-159 < a

   1. Initial program 80.9%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 78.5

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in angle around 0

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

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

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

      2. pow2 N/A

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

      3. lift-*.f64 78.2

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

   7. Applied rewrites 78.2%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 78.3

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 78.3

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

   9. Applied rewrites 78.3%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 78.3

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

   11. Applied rewrites 78.3%

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

Alternative 6: 66.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{-86}:\\ \;\;\;\;{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b \cdot \left(angle \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 4.6e-86)
   (pow (* (cos (* (* PI angle) 0.005555555555555556)) a) 2.0)
   (+
    (* a a)
    (* (* (* b angle) PI) (* (* b (* angle PI)) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.6e-86) {
		tmp = pow((cos(((((double) M_PI) * angle) * 0.005555555555555556)) * a), 2.0);
	} else {
		tmp = (a * a) + (((b * angle) * ((double) M_PI)) * ((b * (angle * ((double) M_PI))) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 4.6e-86) {
		tmp = Math.pow((Math.cos(((Math.PI * angle) * 0.005555555555555556)) * a), 2.0);
	} else {
		tmp = (a * a) + (((b * angle) * Math.PI) * ((b * (angle * Math.PI)) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 4.6e-86:
		tmp = math.pow((math.cos(((math.pi * angle) * 0.005555555555555556)) * a), 2.0)
	else:
		tmp = (a * a) + (((b * angle) * math.pi) * ((b * (angle * math.pi)) * 3.08641975308642e-5))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 4.6e-86)
		tmp = Float64(cos(Float64(Float64(pi * angle) * 0.005555555555555556)) * a) ^ 2.0;
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(b * angle) * pi) * Float64(Float64(b * Float64(angle * pi)) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 4.6e-86)
		tmp = (cos(((pi * angle) * 0.005555555555555556)) * a) ^ 2.0;
	else
		tmp = (a * a) + (((b * angle) * pi) * ((b * (angle * pi)) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 4.6e-86], N[Power[N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(b * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.59999999999999992e-86

   1. Initial program 78.9%

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

   2. Taylor expanded in a around inf

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

      1. pow-prod-down N/A

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

      2. lower-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 61.2

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

   4. Applied rewrites 61.2%

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

   if 4.59999999999999992e-86 < b

   1. Initial program 81.0%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in angle around 0

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

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

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

      2. pow2 N/A

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

      3. lift-*.f64 77.4

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

   7. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 77.5

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.5

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

   9. Applied rewrites 77.5%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 77.6

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

   11. Applied rewrites 77.6%

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

Alternative 7: 66.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-97}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b \cdot \left(angle \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.45e-97)
   (* a a)
   (+
    (* a a)
    (* (* (* b angle) PI) (* (* b (* angle PI)) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-97) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((b * angle) * ((double) M_PI)) * ((b * (angle * ((double) M_PI))) * 3.08641975308642e-5));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-97) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((b * angle) * Math.PI) * ((b * (angle * Math.PI)) * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.45e-97:
		tmp = a * a
	else:
		tmp = (a * a) + (((b * angle) * math.pi) * ((b * (angle * math.pi)) * 3.08641975308642e-5))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.45e-97)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(b * angle) * pi) * Float64(Float64(b * Float64(angle * pi)) * 3.08641975308642e-5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.45e-97)
		tmp = a * a;
	else
		tmp = (a * a) + (((b * angle) * pi) * ((b * (angle * pi)) * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.45e-97], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(b * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45e-97

   1. Initial program 79.1%

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

   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 61.4

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

   4. Applied rewrites 61.4%

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

   if 1.45e-97 < b

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 77.3

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in angle around 0

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

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

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

      2. pow2 N/A

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

      3. lift-*.f64 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 77.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.2

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

   9. Applied rewrites 77.2%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 77.2

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

   11. Applied rewrites 77.2%

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

Alternative 8: 66.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-97}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot angle\right) \cdot \pi\right) \cdot \left(\left(b \cdot angle\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.45e-97)
   (* a a)
   (+
    (* a a)
    (* (* (* b angle) PI) (* (* b angle) (* PI 3.08641975308642e-5))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-97) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((b * angle) * ((double) M_PI)) * ((b * angle) * (((double) M_PI) * 3.08641975308642e-5)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-97) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((b * angle) * Math.PI) * ((b * angle) * (Math.PI * 3.08641975308642e-5)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.45e-97:
		tmp = a * a
	else:
		tmp = (a * a) + (((b * angle) * math.pi) * ((b * angle) * (math.pi * 3.08641975308642e-5)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.45e-97)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(b * angle) * pi) * Float64(Float64(b * angle) * Float64(pi * 3.08641975308642e-5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.45e-97)
		tmp = a * a;
	else
		tmp = (a * a) + (((b * angle) * pi) * ((b * angle) * (pi * 3.08641975308642e-5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.45e-97], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(b * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(b * angle), $MachinePrecision] * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45e-97

   1. Initial program 79.1%

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

   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 61.4

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

   4. Applied rewrites 61.4%

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

   if 1.45e-97 < b

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 77.3

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in angle around 0

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

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

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

      2. pow2 N/A

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

      3. lift-*.f64 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 77.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.2

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

   9. Applied rewrites 77.2%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 77.2

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

   11. Applied rewrites 77.2%

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

Alternative 9: 66.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-97}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(\left(b \cdot angle\right) \cdot \pi\right) \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.45e-97)
   (* a a)
   (+
    (* a a)
    (* (* (* b angle) PI) (* (* 3.08641975308642e-5 angle) (* b PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-97) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((b * angle) * ((double) M_PI)) * ((3.08641975308642e-5 * angle) * (b * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-97) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (((b * angle) * Math.PI) * ((3.08641975308642e-5 * angle) * (b * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.45e-97:
		tmp = a * a
	else:
		tmp = (a * a) + (((b * angle) * math.pi) * ((3.08641975308642e-5 * angle) * (b * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.45e-97)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(b * angle) * pi) * Float64(Float64(3.08641975308642e-5 * angle) * Float64(b * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.45e-97)
		tmp = a * a;
	else
		tmp = (a * a) + (((b * angle) * pi) * ((3.08641975308642e-5 * angle) * (b * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.45e-97], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(b * angle), $MachinePrecision] * Pi), $MachinePrecision] * N[(N[(3.08641975308642e-5 * angle), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45e-97

   1. Initial program 79.1%

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

   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 61.4

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

   4. Applied rewrites 61.4%

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

   if 1.45e-97 < b

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 77.3

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in angle around 0

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

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

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

      2. pow2 N/A

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

      3. lift-*.f64 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 77.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.2

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

   9. Applied rewrites 77.2%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lift-PI.f64 77.2

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

   11. Applied rewrites 77.2%

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

Alternative 10: 66.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{-97}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot angle\right) \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\right) \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.45e-97)
   (* a a)
   (+
    (* a a)
    (* (* b angle) (* PI (* 3.08641975308642e-5 (* (* b angle) PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-97) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((b * angle) * (((double) M_PI) * (3.08641975308642e-5 * ((b * angle) * ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e-97) {
		tmp = a * a;
	} else {
		tmp = (a * a) + ((b * angle) * (Math.PI * (3.08641975308642e-5 * ((b * angle) * Math.PI))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.45e-97:
		tmp = a * a
	else:
		tmp = (a * a) + ((b * angle) * (math.pi * (3.08641975308642e-5 * ((b * angle) * math.pi))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.45e-97)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(b * angle) * Float64(pi * Float64(3.08641975308642e-5 * Float64(Float64(b * angle) * pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.45e-97)
		tmp = a * a;
	else
		tmp = (a * a) + ((b * angle) * (pi * (3.08641975308642e-5 * ((b * angle) * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.45e-97], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(b * angle), $MachinePrecision] * N[(Pi * N[(3.08641975308642e-5 * N[(N[(b * angle), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45e-97

   1. Initial program 79.1%

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

   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 61.4

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

   4. Applied rewrites 61.4%

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

   if 1.45e-97 < b

   1. Initial program 80.7%

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

   2. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 77.3

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in angle around 0

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

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

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

      2. pow2 N/A

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

      3. lift-*.f64 77.1

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

   7. Applied rewrites 77.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 77.2

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 77.2

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

   9. Applied rewrites 77.2%

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

       1. lift-*.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-PI.f64 77.2

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 77.2

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

   11. Applied rewrites 77.2%

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

Alternative 11: 56.3% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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 79.6%

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

2. 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 56.3

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

4. Applied rewrites 56.3%

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

Reproduce

herbie shell --seed 2025093 
(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)))