ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.1%

Time: 15.2s

Alternatives: 15

Speedup: 2.0×

• 37.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.1% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (* a (sin (pow (pow (* PI (* angle_m 0.005555555555555556)) 0.5) 2.0)))
   2.0)
  (* b b)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[Power[N[Power[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 78.5%

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

6. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 78.5

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

8. Applied rewrites 78.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 78.5

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

10. Applied rewrites 78.5%

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

    1. lift-/.f64 N/A

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

    2. clear-num N/A

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

    3. lift-/.f64 N/A

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

    4. associate-/r* N/A

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

    5. lift-*.f64 N/A

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

    6. lift-/.f64 N/A

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

    7. inv-pow N/A

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

    8. sqr-pow N/A

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

    9. pow2 N/A

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

    10. lower-pow.f64 N/A

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

12. Applied rewrites 78.5%

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

Alternative 2: 80.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b + {\left(a \cdot \sin \left(\pi \cdot \frac{-1}{\frac{-180}{angle\_m}}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (* b b) (pow (* a (sin (* PI (/ -1.0 (/ -180.0 angle_m))))) 2.0)))

C

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

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return (b * b) + Math.pow((a * Math.sin((Math.PI * (-1.0 / (-180.0 / angle_m))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return (b * b) + math.pow((a * math.sin((math.pi * (-1.0 / (-180.0 / angle_m))))), 2.0)

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (b * b) + ((a * sin((pi * (-1.0 / (-180.0 / angle_m))))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[Sin[N[(Pi * N[(-1.0 / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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. Add Preprocessing

3. Taylor expanded in angle around 0

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

5. Applied rewrites 80.2%

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

6. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.2

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

8. Applied rewrites 80.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-*l/ N/A

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

   5. *-lft-identity N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 80.2

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

10. Applied rewrites 80.2%

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

    1. lift-/.f64 N/A

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

    2. frac-2neg N/A

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

    3. div-inv N/A

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

    4. lower-*.f64 N/A

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

    5. lower-neg.f64 N/A

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

    6. lower-/.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. distribute-neg-frac N/A

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

    9. lower-/.f64 N/A

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

    10. metadata-eval 80.2

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

12. Applied rewrites 80.2%

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

13. Final simplification 80.2%

    \[\leadsto b \cdot b + {\left(a \cdot \sin \left(\pi \cdot \frac{-1}{\frac{-180}{angle}}\right)\right)}^{2} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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)))