ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.2%

Time: 4.9s

Alternatives: 12

Speedup: 1.0×

• 63.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* (/ 0.005555555555555556 (/ 1.0 angle)) PI)))
  (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 / (1.0 / angle)) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (0.005555555555555556 / (1.0 / angle)) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (0.005555555555555556 / (1.0 / angle)) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(0.005555555555555556 / Float64(1.0 / angle)) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (0.005555555555555556 / (1.0 / angle)) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(0.005555555555555556 / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 80.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 80.2%

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

3. Applied rewrites 80.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 80.2%

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

5. Applied rewrites 80.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. mult-flip N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 80.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)}^{2} \]

7. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{0.005555555555555556}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. mult-flip N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 80.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} \]

9. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{0.005555555555555556}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 80.2% accurate, 1.2× speedup

Math

\[{\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{0.6666666666666666}{angle} + \frac{0.3333333333333333}{angle}} \cdot \pi\right)\right)}^{2} + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556 + 0.5\right)\right)\right)\right) \cdot b\right) \cdot b \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (pow
  (*
   a
   (sin
    (*
     (/
      0.005555555555555556
      (+ (/ 0.6666666666666666 angle) (/ 0.3333333333333333 angle)))
     PI)))
  2.0)
 (*
  (*
   (-
    0.5
    (*
     0.5
     (cos (* 2.0 (* PI (+ (* angle 0.005555555555555556) 0.5))))))
   b)
  b)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((0.005555555555555556 / ((0.6666666666666666 / angle) + (0.3333333333333333 / angle))) * ((double) M_PI)))), 2.0) + (((0.5 - (0.5 * cos((2.0 * (((double) M_PI) * ((angle * 0.005555555555555556) + 0.5)))))) * b) * b);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((0.005555555555555556 / ((0.6666666666666666 / angle) + (0.3333333333333333 / angle))) * Math.PI))), 2.0) + (((0.5 - (0.5 * Math.cos((2.0 * (Math.PI * ((angle * 0.005555555555555556) + 0.5)))))) * b) * b);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((0.005555555555555556 / ((0.6666666666666666 / angle) + (0.3333333333333333 / angle))) * math.pi))), 2.0) + (((0.5 - (0.5 * math.cos((2.0 * (math.pi * ((angle * 0.005555555555555556) + 0.5)))))) * b) * b)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(0.005555555555555556 / Float64(Float64(0.6666666666666666 / angle) + Float64(0.3333333333333333 / angle))) * pi))) ^ 2.0) + Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(Float64(angle * 0.005555555555555556) + 0.5)))))) * b) * b))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((0.005555555555555556 / ((0.6666666666666666 / angle) + (0.3333333333333333 / angle))) * pi))) ^ 2.0) + (((0.5 - (0.5 * cos((2.0 * (pi * ((angle * 0.005555555555555556) + 0.5)))))) * b) * b);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(0.005555555555555556 / N[(N[(0.6666666666666666 / angle), $MachinePrecision] + N[(0.3333333333333333 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(N[(angle * 0.005555555555555556), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 80.2%

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

3. Applied rewrites 80.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 80.2%

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

5. Applied rewrites 80.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. mult-flip N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 80.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)}^{2} \]

7. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{0.005555555555555556}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. mult-flip N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 80.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} \]

9. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{0.005555555555555556}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. metadata-eval N/A

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

    3. div-add N/A

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

    4. lower-+.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 80.2%

       \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{0.6666666666666666}{angle} + \color{blue}{\frac{0.3333333333333333}{angle}}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} \]

11. Applied rewrites 80.2%

    \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{0.6666666666666666}{angle} + \frac{0.3333333333333333}{angle}}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} \]
12. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. metadata-eval N/A

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

    3. div-add N/A

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

    4. lower-+.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. lower-/.f64 80.2%

       \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{0.6666666666666666}{angle} + \frac{0.3333333333333333}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{0.005555555555555556}{\frac{0.6666666666666666}{angle} + \color{blue}{\frac{0.3333333333333333}{angle}}} \cdot \pi\right)\right)}^{2} \]

13. Applied rewrites 80.2%

    \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{0.6666666666666666}{angle} + \frac{0.3333333333333333}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{0.005555555555555556}{\color{blue}{\frac{0.6666666666666666}{angle} + \frac{0.3333333333333333}{angle}}} \cdot \pi\right)\right)}^{2} \]

15. Applied rewrites 80.2%

    \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{0.6666666666666666}{angle} + \frac{0.3333333333333333}{angle}} \cdot \pi\right)\right)}^{2} + \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556 + 0.5\right)\right)\right)\right) \cdot b\right) \cdot b} \]
16. Add Preprocessing

Alternative 3: 80.2% accurate, 1.2× speedup

Math

\[\left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b + \left(-\sin \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot a\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right) \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (*
  (* (+ 0.5 (* 0.5 (cos (* (* PI angle) 0.011111111111111112)))) b)
  b)
 (*
  (- (* (sin (* (* -0.005555555555555556 angle) PI)) a))
  (* (sin (* PI (* 0.005555555555555556 angle))) a))))

C

double code(double a, double b, double angle) {
	return (((0.5 + (0.5 * cos(((((double) M_PI) * angle) * 0.011111111111111112)))) * b) * b) + (-(sin(((-0.005555555555555556 * angle) * ((double) M_PI))) * a) * (sin((((double) M_PI) * (0.005555555555555556 * angle))) * a));
}

Java

public static double code(double a, double b, double angle) {
	return (((0.5 + (0.5 * Math.cos(((Math.PI * angle) * 0.011111111111111112)))) * b) * b) + (-(Math.sin(((-0.005555555555555556 * angle) * Math.PI)) * a) * (Math.sin((Math.PI * (0.005555555555555556 * angle))) * a));
}

Python

def code(a, b, angle):
	return (((0.5 + (0.5 * math.cos(((math.pi * angle) * 0.011111111111111112)))) * b) * b) + (-(math.sin(((-0.005555555555555556 * angle) * math.pi)) * a) * (math.sin((math.pi * (0.005555555555555556 * angle))) * a))

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(Float64(pi * angle) * 0.011111111111111112)))) * b) * b) + Float64(Float64(-Float64(sin(Float64(Float64(-0.005555555555555556 * angle) * pi)) * a)) * Float64(sin(Float64(pi * Float64(0.005555555555555556 * angle))) * a)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (((0.5 + (0.5 * cos(((pi * angle) * 0.011111111111111112)))) * b) * b) + (-(sin(((-0.005555555555555556 * angle) * pi)) * a) * (sin((pi * (0.005555555555555556 * angle))) * a));
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] + N[((-N[(N[Sin[N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]) * N[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Applied rewrites 80.2%

   \[\leadsto \color{blue}{\left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b + \left(-\sin \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot a\right) \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)} \]
4. Add Preprocessing

Alternative 4: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + ((0.5 + (0.5 * Math.cos(((Math.PI * angle) * 0.011111111111111112)))) * (b * b));
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((angle / 180.0) * math.pi))), 2.0) + ((0.5 + (0.5 * math.cos(((math.pi * angle) * 0.011111111111111112)))) * (b * b))

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((0.5 + (0.5 * cos(((pi * angle) * 0.011111111111111112)))) * (b * b));
end

Wolfram

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

Derivation

1. Initial program 80.2%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. swap-sqr N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

3. Applied rewrites 80.2%

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

Alternative 5: 80.1% accurate, 1.3× speedup

Math

\[{\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (pow (* a (sin (* (/ 0.005555555555555556 (/ 1.0 angle)) PI))) 2.0)
 (pow (* b 1.0) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((0.005555555555555556 / (1.0 / angle)) * ((double) M_PI)))), 2.0) + pow((b * 1.0), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((0.005555555555555556 / (1.0 / angle)) * Math.PI))), 2.0) + Math.pow((b * 1.0), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((0.005555555555555556 / (1.0 / angle)) * math.pi))), 2.0) + math.pow((b * 1.0), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(0.005555555555555556 / Float64(1.0 / angle)) * pi))) ^ 2.0) + (Float64(b * 1.0) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((0.005555555555555556 / (1.0 / angle)) * pi))) ^ 2.0) + ((b * 1.0) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(0.005555555555555556 / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 80.2%

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

3. Applied rewrites 80.2%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 80.2%

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

5. Applied rewrites 80.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. mult-flip N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 80.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)}^{2} \]

7. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{0.005555555555555556}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. associate-/r/ N/A

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

   5. mult-flip N/A

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

   6. associate-/r* N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 80.2%

      \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{0.005555555555555556}{\color{blue}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} \]

9. Applied rewrites 80.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{0.005555555555555556}{\frac{1}{angle}}} \cdot \pi\right)\right)}^{2} \]

10. Taylor expanded in angle around 0

    \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
11. Step-by-step derivation

12. Applied rewrites 80.1%

    \[\leadsto {\left(a \cdot \sin \left(\frac{0.005555555555555556}{\frac{1}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
13. Add Preprocessing

Alternative 6: 80.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sin(((Math.PI * 0.005555555555555556) * angle)) * a;
	return (b * b) - (-t_0 * t_0);
}

Python

def code(a, b, angle):
	t_0 = math.sin(((math.pi * 0.005555555555555556) * angle)) * a
	return (b * b) - (-t_0 * t_0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	t_0 = sin(((pi * 0.005555555555555556) * angle)) * a;
	tmp = (b * b) - (-t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 80.1%

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

4. Applied rewrites 80.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. associate-/r/ N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. unpow2 N/A

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

6. Applied rewrites 80.1%

   \[\leadsto \color{blue}{b \cdot b - \left(-\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right) \cdot \left(\sin \left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right) \cdot a\right)} \]
7. Add Preprocessing

Alternative 7: 80.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 80.1%

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

4. Applied rewrites 80.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 80.1%

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

6. Applied rewrites 80.1%

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

Alternative 8: 79.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 0.00076:\\ \;\;\;\;\left(\left(\left(\left(-3.175328964080679 \cdot 10^{-10} \cdot \left(\left(\left|angle\right| \cdot \left|angle\right|\right) \cdot a\right)\right) \cdot {\pi}^{4} - -3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot a\right)\right) \cdot \left|angle\right|\right) \cdot \left|angle\right|\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right)\right) \cdot \left(a \cdot a\right) + b \cdot b\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<= (fabs angle) 0.00076)
  (+
   (*
    (*
     (*
      (-
       (*
        (* -3.175328964080679e-10 (* (* (fabs angle) (fabs angle)) a))
        (pow PI 4.0))
       (* -3.08641975308642e-5 (* (* PI PI) a)))
      (fabs angle))
     (fabs angle))
    a)
   (* (* (+ 0.5 0.5) b) b))
  (+
   (*
    (-
     0.5
     (*
      0.5
      (cos (* 2.0 (* (* PI 0.005555555555555556) (fabs angle))))))
    (* a a))
   (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 0.00076) {
		tmp = ((((((-3.175328964080679e-10 * ((fabs(angle) * fabs(angle)) * a)) * pow(((double) M_PI), 4.0)) - (-3.08641975308642e-5 * ((((double) M_PI) * ((double) M_PI)) * a))) * fabs(angle)) * fabs(angle)) * a) + (((0.5 + 0.5) * b) * b);
	} else {
		tmp = ((0.5 - (0.5 * cos((2.0 * ((((double) M_PI) * 0.005555555555555556) * fabs(angle)))))) * (a * a)) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 0.00076) {
		tmp = ((((((-3.175328964080679e-10 * ((Math.abs(angle) * Math.abs(angle)) * a)) * Math.pow(Math.PI, 4.0)) - (-3.08641975308642e-5 * ((Math.PI * Math.PI) * a))) * Math.abs(angle)) * Math.abs(angle)) * a) + (((0.5 + 0.5) * b) * b);
	} else {
		tmp = ((0.5 - (0.5 * Math.cos((2.0 * ((Math.PI * 0.005555555555555556) * Math.abs(angle)))))) * (a * a)) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 0.00076:
		tmp = ((((((-3.175328964080679e-10 * ((math.fabs(angle) * math.fabs(angle)) * a)) * math.pow(math.pi, 4.0)) - (-3.08641975308642e-5 * ((math.pi * math.pi) * a))) * math.fabs(angle)) * math.fabs(angle)) * a) + (((0.5 + 0.5) * b) * b)
	else:
		tmp = ((0.5 - (0.5 * math.cos((2.0 * ((math.pi * 0.005555555555555556) * math.fabs(angle)))))) * (a * a)) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 0.00076)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-3.175328964080679e-10 * Float64(Float64(abs(angle) * abs(angle)) * a)) * (pi ^ 4.0)) - Float64(-3.08641975308642e-5 * Float64(Float64(pi * pi) * a))) * abs(angle)) * abs(angle)) * a) + Float64(Float64(Float64(0.5 + 0.5) * b) * b));
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(pi * 0.005555555555555556) * abs(angle)))))) * Float64(a * a)) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 0.00076)
		tmp = ((((((-3.175328964080679e-10 * ((abs(angle) * abs(angle)) * a)) * (pi ^ 4.0)) - (-3.08641975308642e-5 * ((pi * pi) * a))) * abs(angle)) * abs(angle)) * a) + (((0.5 + 0.5) * b) * b);
	else
		tmp = ((0.5 - (0.5 * cos((2.0 * ((pi * 0.005555555555555556) * abs(angle)))))) * (a * a)) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[angle], $MachinePrecision], 0.00076], N[(N[(N[(N[(N[(N[(N[(-3.175328964080679e-10 * N[(N[(N[Abs[angle], $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] - N[(-3.08641975308642e-5 * N[(N[(Pi * Pi), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(0.5 + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 7.6000000000000004e-4

   1. Initial program 80.2%

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

   3. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(0.5 - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.3%

      \[\leadsto \left(\left(0.5 - \color{blue}{0.5}\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b \]
   8. Step-by-step derivation

   9. Applied rewrites 57.4%

      \[\leadsto \left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + \color{blue}{0.5}\right) \cdot b\right) \cdot b \]

   10. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\pi}^{2}\right)\right)\right)} \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       2. lower-pow.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \left(\color{blue}{\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)} + \frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       3. lower-+.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \color{blue}{\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       4. lower-*.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \color{blue}{\frac{1}{32400}} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       5. lower-*.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       6. lower-*.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       7. lower-pow.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       8. lower-pow.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       9. lower-PI.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       10. lower-*.f64 N/A

           \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \color{blue}{\left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       11. lower-*.f64 N/A

           \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       12. lower-pow.f64 N/A

           \[\leadsto \left({angle}^{2} \cdot \left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       13. lower-PI.f64 50.9%

           \[\leadsto \left({angle}^{2} \cdot \left(-3.175328964080679 \cdot 10^{-10} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot {\pi}^{2}\right)\right)\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]

   12. Applied rewrites 50.9%

       \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left(-3.175328964080679 \cdot 10^{-10} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot {\pi}^{2}\right)\right)\right)} \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left({angle}^{2} \cdot \color{blue}{\left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\pi}^{2}\right)\right)}\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       2. *-commutative N/A

          \[\leadsto \left(\left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\pi}^{2}\right)\right) \cdot \color{blue}{{angle}^{2}}\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       3. lift-pow.f64 N/A

          \[\leadsto \left(\left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\pi}^{2}\right)\right) \cdot {angle}^{\color{blue}{2}}\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       4. unpow2 N/A

          \[\leadsto \left(\left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\pi}^{2}\right)\right) \cdot \left(angle \cdot \color{blue}{angle}\right)\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       5. associate-*r* N/A

          \[\leadsto \left(\left(\left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\pi}^{2}\right)\right) \cdot angle\right) \cdot \color{blue}{angle}\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\frac{-1}{3149280000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right) + \frac{1}{32400} \cdot \left(a \cdot {\pi}^{2}\right)\right) \cdot angle\right) \cdot \color{blue}{angle}\right) \cdot a + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot b\right) \cdot b \]

   14. Applied rewrites 53.8%

       \[\leadsto \left(\left(\left(\left(-3.175328964080679 \cdot 10^{-10} \cdot \left(\left(angle \cdot angle\right) \cdot a\right)\right) \cdot {\pi}^{4} - -3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \pi\right) \cdot a\right)\right) \cdot angle\right) \cdot \color{blue}{angle}\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]

   if 7.6000000000000004e-4 < angle

   1. Initial program 80.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 80.1%

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

   4. Applied rewrites 80.1%

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

   6. Applied rewrites 62.8%

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

Alternative 9: 68.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|angle\right| \leq 3.15 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left|angle\right|\right)\right)\right) \cdot \left(a \cdot a\right) + b \cdot b\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<= (fabs angle) 3.15e-8)
  (+ (* (* (- 0.5 0.5) a) a) (* (* (+ 0.5 0.5) b) b))
  (+
   (*
    (-
     0.5
     (*
      0.5
      (cos (* 2.0 (* (* PI 0.005555555555555556) (fabs angle))))))
    (* a a))
   (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(angle) <= 3.15e-8) {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
	} else {
		tmp = ((0.5 - (0.5 * cos((2.0 * ((((double) M_PI) * 0.005555555555555556) * fabs(angle)))))) * (a * a)) + (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(angle) <= 3.15e-8) {
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
	} else {
		tmp = ((0.5 - (0.5 * Math.cos((2.0 * ((Math.PI * 0.005555555555555556) * Math.abs(angle)))))) * (a * a)) + (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(angle) <= 3.15e-8:
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b)
	else:
		tmp = ((0.5 - (0.5 * math.cos((2.0 * ((math.pi * 0.005555555555555556) * math.fabs(angle)))))) * (a * a)) + (b * b)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(angle) <= 3.15e-8)
		tmp = Float64(Float64(Float64(Float64(0.5 - 0.5) * a) * a) + Float64(Float64(Float64(0.5 + 0.5) * b) * b));
	else
		tmp = Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(pi * 0.005555555555555556) * abs(angle)))))) * Float64(a * a)) + Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(angle) <= 3.15e-8)
		tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
	else
		tmp = ((0.5 - (0.5 * cos((2.0 * ((pi * 0.005555555555555556) * abs(angle)))))) * (a * a)) + (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[angle], $MachinePrecision], 3.15e-8], N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(0.5 + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.1499999999999998e-8

   1. Initial program 80.2%

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

   3. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(0.5 - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b \]
   5. Step-by-step derivation

   6. Applied rewrites 57.3%

      \[\leadsto \left(\left(0.5 - \color{blue}{0.5}\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b \]
   8. Step-by-step derivation

   9. Applied rewrites 57.4%

      \[\leadsto \left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + \color{blue}{0.5}\right) \cdot b\right) \cdot b \]

   if 3.1499999999999998e-8 < angle

   1. Initial program 80.2%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 80.1%

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

   4. Applied rewrites 80.1%

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

   6. Applied rewrites 62.8%

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

Alternative 10: 68.2% accurate, 3.1× speedup

Math

\[\left(\left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (*
  (* (- 0.5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))) a)
  a)
 (* (* (+ 0.5 0.5) b) b)))

C

double code(double a, double b, double angle) {
	return (((0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) * a) * a) + (((0.5 + 0.5) * b) * b);
}

Java

public static double code(double a, double b, double angle) {
	return (((0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) * a) * a) + (((0.5 + 0.5) * b) * b);
}

Python

def code(a, b, angle):
	return (((0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) * a) * a) + (((0.5 + 0.5) * b) * b)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) * a) * a) + Float64(Float64(Float64(0.5 + 0.5) * b) * b))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (((0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))) * a) * a) + (((0.5 + 0.5) * b) * b);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(0.5 + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Applied rewrites 68.3%

   \[\leadsto \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b} \]

4. Taylor expanded in angle around 0

   \[\leadsto \left(\left(0.5 - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b \]
5. Step-by-step derivation

6. Applied rewrites 57.3%

   \[\leadsto \left(\left(0.5 - \color{blue}{0.5}\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b \]

7. Taylor expanded in angle around 0

   \[\leadsto \left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b \]
8. Step-by-step derivation

9. Applied rewrites 57.4%

   \[\leadsto \left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + \color{blue}{0.5}\right) \cdot b\right) \cdot b \]

10. Taylor expanded in angle around inf

    \[\leadsto \left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot a\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]
11. Step-by-step derivation

    1. lower--.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-cos.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-PI.f64 68.2%

       \[\leadsto \left(\left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]

12. Applied rewrites 68.2%

    \[\leadsto \left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot a\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]
13. Add Preprocessing

Alternative 11: 57.4% accurate, 3.3× speedup

Math

\[1 \cdot \left(\left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right) \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (*
 1.0
 (*
  (* (+ 0.5 (* 0.5 (cos (* (* PI angle) 0.011111111111111112)))) b)
  b)))

C

double code(double a, double b, double angle) {
	return 1.0 * (((0.5 + (0.5 * cos(((((double) M_PI) * angle) * 0.011111111111111112)))) * b) * b);
}

Java

public static double code(double a, double b, double angle) {
	return 1.0 * (((0.5 + (0.5 * Math.cos(((Math.PI * angle) * 0.011111111111111112)))) * b) * b);
}

Python

def code(a, b, angle):
	return 1.0 * (((0.5 + (0.5 * math.cos(((math.pi * angle) * 0.011111111111111112)))) * b) * b)

Julia

function code(a, b, angle)
	return Float64(1.0 * Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(Float64(pi * angle) * 0.011111111111111112)))) * b) * b))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = 1.0 * (((0.5 + (0.5 * cos(((pi * angle) * 0.011111111111111112)))) * b) * b);
end

Wolfram

code[a_, b_, angle_] := N[(1.0 * N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right) \cdot a}{\left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b}\right) \cdot \left(\left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right)} \]

4. Taylor expanded in a around 0

   \[\leadsto \color{blue}{1} \cdot \left(\left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right) \]
5. Step-by-step derivation

6. Applied rewrites 57.3%

   \[\leadsto \color{blue}{1} \cdot \left(\left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b\right) \]
7. Add Preprocessing

Alternative 12: 57.3% accurate, 14.9× speedup

Math

\[\left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5\right) \cdot b\right) \cdot b \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+ (* (* (- 0.5 0.5) a) a) (* (* (+ 0.5 0.5) b) b)))

C

double code(double a, double b, double angle) {
	return (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = (((0.5d0 - 0.5d0) * a) * a) + (((0.5d0 + 0.5d0) * b) * b)
end function

Java

public static double code(double a, double b, double angle) {
	return (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
}

Python

def code(a, b, angle):
	return (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(Float64(0.5 - 0.5) * a) * a) + Float64(Float64(Float64(0.5 + 0.5) * b) * b))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (((0.5 - 0.5) * a) * a) + (((0.5 + 0.5) * b) * b);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision] + N[(N[(N[(0.5 + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.2%

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

3. Applied rewrites 68.3%

   \[\leadsto \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b} \]

4. Taylor expanded in angle around 0

   \[\leadsto \left(\left(0.5 - \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b \]
5. Step-by-step derivation

6. Applied rewrites 57.3%

   \[\leadsto \left(\left(0.5 - \color{blue}{0.5}\right) \cdot a\right) \cdot a + \left(\left(0.5 + 0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot b\right) \cdot b \]

7. Taylor expanded in angle around 0

   \[\leadsto \left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b \]
8. Step-by-step derivation

9. Applied rewrites 57.4%

   \[\leadsto \left(\left(0.5 - 0.5\right) \cdot a\right) \cdot a + \left(\left(0.5 + \color{blue}{0.5}\right) \cdot b\right) \cdot b \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))