cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.6%

Time: 10.8s

Alternatives: 10

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: 51.1% 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: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0295:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{x\_m}{1 - \cos x\_m}}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0295)
   (fma
    (* x_m x_m)
    (fma (* x_m x_m) 0.001388888888888889 -0.041666666666666664)
    0.5)
   (/ (/ 1.0 (/ x_m (- 1.0 (cos x_m)))) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0295) {
		tmp = fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 / (x_m / (1.0 - cos(x_m)))) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0295)
		tmp = fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 / Float64(x_m / Float64(1.0 - cos(x_m)))) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0295], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / N[(x$95$m / N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029499999999999998

   1. Initial program 3.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.029499999999999998 < x

   1. Initial program 97.1%

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

   4. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.1

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

   6. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lower--.f64 99.1

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

   8. Applied rewrites 99.1%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0295:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0295)
   (fma
    (* x_m x_m)
    (fma (* x_m x_m) 0.001388888888888889 -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 <= 0.0295) {
		tmp = fma((x_m * x_m), fma((x_m * x_m), 0.001388888888888889, -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 <= 0.0295)
		tmp = fma(Float64(x_m * x_m), fma(Float64(x_m * x_m), 0.001388888888888889, -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, 0.0295], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 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 < 0.029499999999999998

   1. Initial program 2.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.029499999999999998 < x

   1. Initial program 98.2%

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

   4. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. neg-mul-1 N/A

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

      10. sub-neg N/A

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

      11. lift-cos.f64 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. lower--.f64 99.3

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

   6. Applied rewrites 99.3%

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

Reproduce

herbie shell --seed 2024223 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))