cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.8%

Time: 8.2s

Alternatives: 8

Speedup: 5.2×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \sin x}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (/ (tan (* x 0.5)) x) (sin x)) x))

C

double code(double x) {
	return ((tan((x * 0.5)) / x) * sin(x)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((tan((x * 0.5d0)) / x) * sin(x)) / x
end function

Java

public static double code(double x) {
	return ((Math.tan((x * 0.5)) / x) * Math.sin(x)) / x;
}

Python

def code(x):
	return ((math.tan((x * 0.5)) / x) * math.sin(x)) / x

Julia

function code(x)
	return Float64(Float64(Float64(tan(Float64(x * 0.5)) / x) * sin(x)) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((tan((x * 0.5)) / x) * sin(x)) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \cos x}}{x \cdot x} \]

   3. flip-- N/A

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1} - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   6. lift-cos.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{\cos x} \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{1 - \cos x \cdot \color{blue}{\cos x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   8. 1-sub-cos N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x}} \cdot \frac{\sin x}{1 + \cos x} \]

   12. lower-sin.f64 N/A

       \[\leadsto \frac{\color{blue}{\sin x}}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x} \]

   13. lift-cos.f64 N/A

       \[\leadsto \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \color{blue}{\cos x}} \]

   14. hang-0p-tan N/A

       \[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

   15. lower-tan.f64 N/A

       \[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

   16. lower-/.f64 73.3

       \[\leadsto \frac{\sin x}{x \cdot x} \cdot \tan \color{blue}{\left(\frac{x}{2}\right)} \]

4. Applied rewrites 73.3%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}} \]

   3. lift-/.f64 N/A

      \[\leadsto \tan \left(\frac{x}{2}\right) \cdot \color{blue}{\frac{\sin x}{x \cdot x}} \]

   4. clear-num N/A

      \[\leadsto \tan \left(\frac{x}{2}\right) \cdot \color{blue}{\frac{1}{\frac{x \cdot x}{\sin x}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\sin x}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\sin x}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{\frac{x \cdot x}{\sin x}} \]

   8. clear-num N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{1}{\frac{2}{x}}\right)}}{\frac{x \cdot x}{\sin x}} \]

   9. associate-/r/ N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{\frac{x \cdot x}{\sin x}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\tan \left(\color{blue}{\frac{1}{2}} \cdot x\right)}{\frac{x \cdot x}{\sin x}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{\frac{x \cdot x}{\sin x}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\tan \left(\frac{1}{2} \cdot x\right)}{\frac{\color{blue}{x \cdot x}}{\sin x}} \]

   13. associate-/l* N/A

       \[\leadsto \frac{\tan \left(\frac{1}{2} \cdot x\right)}{\color{blue}{x \cdot \frac{x}{\sin x}}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\tan \left(\frac{1}{2} \cdot x\right)}{\color{blue}{x \cdot \frac{x}{\sin x}}} \]

   15. lower-/.f64 99.3

       \[\leadsto \frac{\tan \left(0.5 \cdot x\right)}{x \cdot \color{blue}{\frac{x}{\sin x}}} \]

6. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\tan \left(0.5 \cdot x\right)}{x \cdot \frac{x}{\sin x}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\tan \left(\frac{1}{2} \cdot x\right)}{x \cdot \frac{x}{\sin x}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\tan \left(\frac{1}{2} \cdot x\right)}{\color{blue}{x \cdot \frac{x}{\sin x}}} \]

   3. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\tan \left(\frac{1}{2} \cdot x\right)}{x}}{\frac{x}{\sin x}}} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\tan \left(\frac{1}{2} \cdot x\right)}{x}}{\color{blue}{\frac{x}{\sin x}}} \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\tan \left(\frac{1}{2} \cdot x\right)}{x}}{x} \cdot \sin x} \]

   6. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\tan \left(\frac{1}{2} \cdot x\right)}{x}\right)}{\mathsf{neg}\left(x\right)}} \cdot \sin x \]

   7. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{\tan \left(\frac{1}{2} \cdot x\right)}{x}\right)\right) \cdot \sin x}{\mathsf{neg}\left(x\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{\tan \left(\frac{1}{2} \cdot x\right)}{x}\right)\right) \cdot \sin x}{\mathsf{neg}\left(x\right)}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{\tan \left(\frac{1}{2} \cdot x\right)}{x}\right)\right) \cdot \sin x}}{\mathsf{neg}\left(x\right)} \]

   10. distribute-neg-frac2 N/A

       \[\leadsto \frac{\color{blue}{\frac{\tan \left(\frac{1}{2} \cdot x\right)}{\mathsf{neg}\left(x\right)}} \cdot \sin x}{\mathsf{neg}\left(x\right)} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\tan \left(\frac{1}{2} \cdot x\right)}{\mathsf{neg}\left(x\right)}} \cdot \sin x}{\mathsf{neg}\left(x\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\tan \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{\mathsf{neg}\left(x\right)} \cdot \sin x}{\mathsf{neg}\left(x\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{\mathsf{neg}\left(x\right)} \cdot \sin x}{\mathsf{neg}\left(x\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{\mathsf{neg}\left(x\right)} \cdot \sin x}{\mathsf{neg}\left(x\right)} \]

   15. lower-neg.f64 N/A

       \[\leadsto \frac{\frac{\tan \left(x \cdot \frac{1}{2}\right)}{\color{blue}{-x}} \cdot \sin x}{\mathsf{neg}\left(x\right)} \]

   16. lower-neg.f64 99.8

       \[\leadsto \frac{\frac{\tan \left(x \cdot 0.5\right)}{-x} \cdot \sin x}{\color{blue}{-x}} \]

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\tan \left(x \cdot 0.5\right)}{-x} \cdot \sin x}{-x}} \]

9. Final simplification 99.8%

   \[\leadsto \frac{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \sin x}{x} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\tan \left(0.5 \cdot x\right)}{x \cdot \frac{x}{\sin x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (tan (* 0.5 x)) (* x (/ x (sin x)))))

C

double code(double x) {
	return tan((0.5 * x)) / (x * (x / sin(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((0.5d0 * x)) / (x * (x / sin(x)))
end function

Java

public static double code(double x) {
	return Math.tan((0.5 * x)) / (x * (x / Math.sin(x)));
}

Python

def code(x):
	return math.tan((0.5 * x)) / (x * (x / math.sin(x)))

Julia

function code(x)
	return Float64(tan(Float64(0.5 * x)) / Float64(x * Float64(x / sin(x))))
end

MATLAB

function tmp = code(x)
	tmp = tan((0.5 * x)) / (x * (x / sin(x)));
end

Wolfram

code[x_] := N[(N[Tan[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \cos x}}{x \cdot x} \]

   3. flip-- N/A

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1} - \cos x \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   6. lift-cos.f64 N/A

      \[\leadsto \frac{1 - \color{blue}{\cos x} \cdot \cos x}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   7. lift-cos.f64 N/A

      \[\leadsto \frac{1 - \cos x \cdot \color{blue}{\cos x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   8. 1-sub-cos N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot \sin x}}{\left(x \cdot x\right) \cdot \left(1 + \cos x\right)} \]

   9. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x}} \cdot \frac{\sin x}{1 + \cos x} \]

   12. lower-sin.f64 N/A

       \[\leadsto \frac{\color{blue}{\sin x}}{x \cdot x} \cdot \frac{\sin x}{1 + \cos x} \]

   13. lift-cos.f64 N/A

       \[\leadsto \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{1 + \color{blue}{\cos x}} \]

   14. hang-0p-tan N/A

       \[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

   15. lower-tan.f64 N/A

       \[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(\frac{x}{2}\right)} \]

   16. lower-/.f64 73.3

       \[\leadsto \frac{\sin x}{x \cdot x} \cdot \tan \color{blue}{\left(\frac{x}{2}\right)} \]

4. Applied rewrites 73.3%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}} \]

   3. lift-/.f64 N/A

      \[\leadsto \tan \left(\frac{x}{2}\right) \cdot \color{blue}{\frac{\sin x}{x \cdot x}} \]

   4. clear-num N/A

      \[\leadsto \tan \left(\frac{x}{2}\right) \cdot \color{blue}{\frac{1}{\frac{x \cdot x}{\sin x}}} \]

   5. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\sin x}}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{\frac{x \cdot x}{\sin x}}} \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{\frac{x \cdot x}{\sin x}} \]

   8. clear-num N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{1}{\frac{2}{x}}\right)}}{\frac{x \cdot x}{\sin x}} \]

   9. associate-/r/ N/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{\frac{x \cdot x}{\sin x}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\tan \left(\color{blue}{\frac{1}{2}} \cdot x\right)}{\frac{x \cdot x}{\sin x}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\tan \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{\frac{x \cdot x}{\sin x}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\tan \left(\frac{1}{2} \cdot x\right)}{\frac{\color{blue}{x \cdot x}}{\sin x}} \]

   13. associate-/l* N/A

       \[\leadsto \frac{\tan \left(\frac{1}{2} \cdot x\right)}{\color{blue}{x \cdot \frac{x}{\sin x}}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\tan \left(\frac{1}{2} \cdot x\right)}{\color{blue}{x \cdot \frac{x}{\sin x}}} \]

   15. lower-/.f64 99.3

       \[\leadsto \frac{\tan \left(0.5 \cdot x\right)}{x \cdot \color{blue}{\frac{x}{\sin x}}} \]

6. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{\tan \left(0.5 \cdot x\right)}{x \cdot \frac{x}{\sin x}}} \]
7. Add Preprocessing

Alternative 3: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (* (pow x -2.0) (- 1.0 (cos x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.095) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = pow(x, -2.0) * (1.0 - cos(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.095)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64((x ^ -2.0) * Float64(1.0 - cos(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.095], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Power[x, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 32.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right), {x}^{2}, \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{40320} \cdot {x}^{2} + \frac{1}{720}}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{40320}, {x}^{2}, \frac{1}{720}\right)}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      15. lower-*.f64 69.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.095000000000000001 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{{x}^{-2} \cdot \left(1 - \cos x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.095) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.095)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.095], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 32.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right), {x}^{2}, \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{40320} \cdot {x}^{2} + \frac{1}{720}}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{40320}, {x}^{2}, \frac{1}{720}\right)}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      15. lower-*.f64 69.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.095000000000000001 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.095) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.095)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.095], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 32.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right), {x}^{2}, \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{40320} \cdot {x}^{2} + \frac{1}{720}}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{40320}, {x}^{2}, \frac{1}{720}\right)}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      15. lower-*.f64 69.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.095000000000000001 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 62.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.85e+76) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.85e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.85d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.85e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.85e+76:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.85e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.85e+76)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.85e+76], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.85000000000000002e76

   1. Initial program 37.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.8%

      \[\leadsto \color{blue}{0.5} \]

   if 2.85000000000000002e76 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
   4. Step-by-step derivation

   5. Applied rewrites 61.7%

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(-0.16666666666666666, x \cdot x, -2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (fma -0.16666666666666666 (* x x) -2.0)))

C

double code(double x) {
	return -1.0 / fma(-0.16666666666666666, (x * x), -2.0);
}

Julia

function code(x)
	return Float64(-1.0 / fma(-0.16666666666666666, Float64(x * x), -2.0))
end

Wolfram

code[x_] := N[(-1.0 / N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

4. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\frac{-1}{\frac{x \cdot x}{\cos x - 1}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{6} \cdot {x}^{2} - 2}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{6} \cdot {x}^{2} + \left(\mathsf{neg}\left(2\right)\right)}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{2}, \mathsf{neg}\left(2\right)\right)}} \]

   3. unpow2 N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, \mathsf{neg}\left(2\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, \mathsf{neg}\left(2\right)\right)} \]

   5. metadata-eval 76.7

      \[\leadsto \frac{-1}{\mathsf{fma}\left(-0.16666666666666666, x \cdot x, \color{blue}{-2}\right)} \]

7. Applied rewrites 76.7%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot x, -2\right)}} \]
8. Add Preprocessing

Alternative 8: 51.1% accurate, 120.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 50.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 52.3%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024324 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))