cos2 (problem 3.4.1)

Percentage Accurate: 51.4% → 99.4%

Time: 8.9s

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: 51.4% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m}{x\_m \cdot x\_m} \cdot \tan \left(0.5 \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0002)
   (fma (* x_m x_m) -0.041666666666666664 0.5)
   (* (/ (sin x_m) (* x_m x_m)) (tan (* 0.5 x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0002) {
		tmp = fma((x_m * x_m), -0.041666666666666664, 0.5);
	} else {
		tmp = (sin(x_m) / (x_m * x_m)) * tan((0.5 * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0002)
		tmp = fma(Float64(x_m * x_m), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(sin(x_m) / Float64(x_m * x_m)) * tan(Float64(0.5 * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0002], N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[(N[Sin[x$95$m], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{-1}{24}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 2.0000000000000001e-4 < x

   1. Initial program 99.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 99.6

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

   6. Applied rewrites 99.6%

      \[\leadsto \frac{\sin x}{x \cdot x} \cdot \color{blue}{\tan \left(0.5 \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 78.4% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/
  (pow x_m -1.0)
  (pow (/ x_m (fma (* x_m x_m) 0.16666666666666666 2.0)) -1.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[x$95$m, -1.0], $MachinePrecision] / N[Power[N[(x$95$m / N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

4. Applied rewrites 51.8%

   \[\leadsto \color{blue}{\frac{{x}^{-1}}{\frac{x}{1 - \cos x}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow-1 N/A

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

   3. lower-/.f64 51.8

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

6. Applied rewrites 51.8%

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

7. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2 + \frac{1}{6} \cdot {x}^{2}}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{x}}{\frac{\color{blue}{\frac{1}{6} \cdot {x}^{2} + 2}}{x}} \]

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 78.4

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

9. Applied rewrites 78.4%

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 2\right)}{x}}} \]
10. Step-by-step derivation

11. Applied rewrites 78.4%

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

12. Final simplification 78.4%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (tan (* 0.5 x_m)) (* x_m (/ x_m (sin x_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Tan[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(x$95$m / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 76.1

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

4. Applied rewrites 76.1%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
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 99.5

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

6. Applied rewrites 99.5%

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

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (fma (* x_m x_m) -0.041666666666666664 0.5)
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0056) {
		tmp = fma((x_m * x_m), -0.041666666666666664, 0.5);
	} else {
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0056)
		tmp = fma(Float64(x_m * x_m), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0056], N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00559999999999999994

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{-1}{24}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 0.00559999999999999994 < x

   1. Initial program 99.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (fma (* x_m x_m) -0.041666666666666664 0.5)
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0056) {
		tmp = fma((x_m * x_m), -0.041666666666666664, 0.5);
	} else {
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0056)
		tmp = fma(Float64(x_m * x_m), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0056], N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00559999999999999994

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{-1}{24}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 0.00559999999999999994 < x

   1. Initial program 99.3%

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

Alternative 6: 75.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 7.6e+76) 0.5 (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 7.6e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 7.6d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 7.6e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 7.6e+76:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 7.6e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 7.6e+76)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 7.6e+76], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.60000000000000049e76

   1. Initial program 39.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.1%

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

   if 7.60000000000000049e76 < x

   1. Initial program 99.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.6%

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

Alternative 7: 78.4% accurate, 5.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ -1.0 (fma -0.16666666666666666 (* x_m x_m) -2.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(-1.0 / N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.1%

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

4. Applied rewrites 50.8%

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

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. metadata-eval N/A

      \[\leadsto \frac{-1}{\frac{-1}{6} \cdot {x}^{2} + \color{blue}{-2}} \]

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 78.4

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

7. Applied rewrites 78.4%

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

Alternative 8: 51.1% accurate, 120.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 51.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.7%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))