cos2 (problem 3.4.1)

Percentage Accurate: 50.7% → 99.4%

Time: 16.8s

Alternatives: 6

Speedup: 120.0×

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: 50.7% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \left(x\_m \cdot x\_m\right) \cdot -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, -0.041666666666666664\right)\right), x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m}{x\_m \cdot x\_m} \cdot \tan \left(\frac{x\_m}{2}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.05)
   (fma
    (*
     x_m
     (fma
      (* x_m x_m)
      (* (* x_m x_m) -2.48015873015873e-5)
      (fma (* 0.001388888888888889 x_m) x_m -0.041666666666666664)))
    x_m
    0.5)
   (* (/ (sin x_m) (* x_m x_m)) (tan (/ x_m 2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.05) {
		tmp = fma((x_m * fma((x_m * x_m), ((x_m * x_m) * -2.48015873015873e-5), fma((0.001388888888888889 * x_m), x_m, -0.041666666666666664))), x_m, 0.5);
	} else {
		tmp = (sin(x_m) / (x_m * x_m)) * tan((x_m / 2.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.05)
		tmp = fma(Float64(x_m * fma(Float64(x_m * x_m), Float64(Float64(x_m * x_m) * -2.48015873015873e-5), fma(Float64(0.001388888888888889 * x_m), x_m, -0.041666666666666664))), x_m, 0.5);
	else
		tmp = Float64(Float64(sin(x_m) / Float64(x_m * x_m)) * tan(Float64(x_m / 2.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.05], N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision] + N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision], N[(N[(N[Sin[x$95$m], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.050000000000000003

   1. Initial program 37.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

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

   if 0.050000000000000003 < x

   1. Initial program 99.1%

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

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

      5. metadata-eval N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. 1-sub-cos N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 98.8

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

   4. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(\left(\cos x + 1\right) \cdot x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-cos.f64 N/A

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

      16. hang-0p-tan N/A

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

      17. lower-tan.f64 N/A

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

      18. lower-/.f64 99.5

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

   6. Applied rewrites 99.5%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \left(x\_m \cdot x\_m\right) \cdot -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, -0.041666666666666664\right)\right), x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.1)
   (fma
    (*
     x_m
     (fma
      (* x_m x_m)
      (* (* x_m x_m) -2.48015873015873e-5)
      (fma (* 0.001388888888888889 x_m) x_m -0.041666666666666664)))
    x_m
    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 <= 0.1) {
		tmp = fma((x_m * fma((x_m * x_m), ((x_m * x_m) * -2.48015873015873e-5), fma((0.001388888888888889 * x_m), x_m, -0.041666666666666664))), x_m, 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 <= 0.1)
		tmp = fma(Float64(x_m * fma(Float64(x_m * x_m), Float64(Float64(x_m * x_m) * -2.48015873015873e-5), fma(Float64(0.001388888888888889 * x_m), x_m, -0.041666666666666664))), x_m, 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, 0.1], N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision] + N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m + 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 < 0.10000000000000001

   1. Initial program 37.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

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

   if 0.10000000000000001 < x

   1. Initial program 99.1%

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

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.9

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

   4. Applied rewrites 98.9%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot \mathsf{fma}\left(x\_m \cdot x\_m, \left(x\_m \cdot x\_m\right) \cdot -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, -0.041666666666666664\right)\right), x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.1)
   (fma
    (*
     x_m
     (fma
      (* x_m x_m)
      (* (* x_m x_m) -2.48015873015873e-5)
      (fma (* 0.001388888888888889 x_m) x_m -0.041666666666666664)))
    x_m
    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 <= 0.1) {
		tmp = fma((x_m * fma((x_m * x_m), ((x_m * x_m) * -2.48015873015873e-5), fma((0.001388888888888889 * x_m), x_m, -0.041666666666666664))), x_m, 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 <= 0.1)
		tmp = fma(Float64(x_m * fma(Float64(x_m * x_m), Float64(Float64(x_m * x_m) * -2.48015873015873e-5), fma(Float64(0.001388888888888889 * x_m), x_m, -0.041666666666666664))), x_m, 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, 0.1], N[(N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision] + N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m + 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 < 0.10000000000000001

   1. Initial program 37.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

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

   if 0.10000000000000001 < x

   1. Initial program 99.1%

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

Alternative 4: 76.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x\_m \cdot \frac{1}{x\_m}}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.8e+14)
   (fma
    (* x_m x_m)
    (fma (* 0.001388888888888889 x_m) x_m -0.041666666666666664)
    0.5)
   (/ (- 1.0 (* x_m (/ 1.0 x_m))) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.8e+14) {
		tmp = fma((x_m * x_m), fma((0.001388888888888889 * x_m), x_m, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - (x_m * (1.0 / x_m))) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.8e+14)
		tmp = fma(Float64(x_m * x_m), fma(Float64(0.001388888888888889 * x_m), x_m, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - Float64(x_m * Float64(1.0 / 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, 4.8e+14], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[(x$95$m * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.8e14

   1. Initial program 39.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.9%

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

   if 4.8e14 < x

   1. Initial program 99.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 40.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-div N/A

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

      7. frac-sub N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{{x}^{\color{blue}{0}} - x \cdot \frac{1}{x}}{x \cdot x} \]

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 41.5

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

   7. Applied rewrites 41.5%

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

Alternative 5: 76.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.15e+77) 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 <= 1.15e+77) {
		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 <= 1.15d+77) 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 <= 1.15e+77) {
		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 <= 1.15e+77:
		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 <= 1.15e+77)
		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 <= 1.15e+77)
		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, 1.15e+77], 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 < 1.14999999999999997e77

   1. Initial program 43.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.8%

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

   if 1.14999999999999997e77 < x

   1. Initial program 99.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 55.5%

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

Alternative 6: 51.8% 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 52.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.7%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))