cos2 (problem 3.4.1)

Percentage Accurate: 50.4% → 99.6%

Time: 11.9s

Alternatives: 8

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: 50.4% 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: 99.6% accurate, 0.9× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.1)
   (/
    (*
     x_m
     (fma
      x_m
      (*
       x_m
       (fma
        x_m
        (* x_m (fma (* x_m x_m) -2.48015873015873e-5 0.001388888888888889))
        -0.041666666666666664))
      0.5))
    x_m)
   (/ (/ (- 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 = (x_m * fma(x_m, (x_m * fma(x_m, (x_m * fma((x_m * x_m), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664)), 0.5)) / x_m;
	} 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 = Float64(Float64(x_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664)), 0.5)) / x_m);
	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[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / x$95$m), $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 3.6%

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

   4. Applied rewrites 3.6%

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      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 5.3

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

   6. Applied rewrites 5.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

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

   if 0.10000000000000001 < x

   1. Initial program 98.5%

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

   4. Applied rewrites 98.5%

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      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 99.2

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

   6. Applied rewrites 99.2%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.1)
   (/
    (*
     x_m
     (fma
      x_m
      (*
       x_m
       (fma
        x_m
        (* x_m (fma (* x_m x_m) -2.48015873015873e-5 0.001388888888888889))
        -0.041666666666666664))
      0.5))
    x_m)
   (/ (- 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 = (x_m * fma(x_m, (x_m * fma(x_m, (x_m * fma((x_m * x_m), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664)), 0.5)) / x_m;
	} 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 = Float64(Float64(x_m * fma(x_m, Float64(x_m * fma(x_m, Float64(x_m * fma(Float64(x_m * x_m), -2.48015873015873e-5, 0.001388888888888889)), -0.041666666666666664)), 0.5)) / x_m);
	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[(x$95$m * N[(x$95$m * N[(x$95$m * N[(x$95$m * N[(N[(x$95$m * x$95$m), $MachinePrecision] * -2.48015873015873e-5 + 0.001388888888888889), $MachinePrecision]), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / x$95$m), $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 2.7%

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

   4. Applied rewrites 2.6%

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

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      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 4.1

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

   6. Applied rewrites 4.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

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

   if 0.10000000000000001 < x

   1. Initial program 98.3%

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))