cos2 (problem 3.4.1)

Percentage Accurate: 50.7% → 99.6%

Time: 11.1s

Alternatives: 6

Speedup: 17.8×

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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;\frac{x\_m \cdot \left(0.5 - 0.041666666666666664 \cdot {x\_m}^{2}\right)}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0045)
   (/ (* x_m (- 0.5 (* 0.041666666666666664 (pow x_m 2.0)))) x_m)
   (* (pow x_m -2.0) (- 1.0 (cos x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = (x_m * (0.5 - (0.041666666666666664 * pow(x_m, 2.0)))) / x_m;
	} else {
		tmp = pow(x_m, -2.0) * (1.0 - cos(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 <= 0.0045d0) then
        tmp = (x_m * (0.5d0 - (0.041666666666666664d0 * (x_m ** 2.0d0)))) / x_m
    else
        tmp = (x_m ** (-2.0d0)) * (1.0d0 - cos(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 <= 0.0045) {
		tmp = (x_m * (0.5 - (0.041666666666666664 * Math.pow(x_m, 2.0)))) / x_m;
	} else {
		tmp = Math.pow(x_m, -2.0) * (1.0 - Math.cos(x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0045:
		tmp = (x_m * (0.5 - (0.041666666666666664 * math.pow(x_m, 2.0)))) / x_m
	else:
		tmp = math.pow(x_m, -2.0) * (1.0 - math.cos(x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0045)
		tmp = Float64(Float64(x_m * Float64(0.5 - Float64(0.041666666666666664 * (x_m ^ 2.0)))) / x_m);
	else
		tmp = Float64((x_m ^ -2.0) * Float64(1.0 - cos(x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0045)
		tmp = (x_m * (0.5 - (0.041666666666666664 * (x_m ^ 2.0)))) / x_m;
	else
		tmp = (x_m ^ -2.0) * (1.0 - cos(x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0045], N[(N[(x$95$m * N[(0.5 - N[(0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 36.9%

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

      1. associate-/r* 38.1%

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

      2. div-inv 38.1%

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

   4. Applied egg-rr 38.1%

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

      1. *-commutative 38.1%

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

      2. frac-2neg 38.1%

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

      3. associate-*r/ 38.1%

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

      4. sub-neg 38.1%

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

      5. distribute-neg-in 38.1%

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

      6. metadata-eval 38.1%

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

      7. add-sqr-sqrt 13.8%

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

      8. sqrt-unprod 29.3%

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

      9. sqr-neg 29.3%

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

      10. sqrt-unprod 15.5%

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

      11. add-sqr-sqrt 23.5%

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

      12. add-sqr-sqrt 8.0%

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

      13. sqrt-unprod 32.3%

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

      14. sqr-neg 32.3%

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

      15. sqrt-unprod 24.2%

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

      16. add-sqr-sqrt 38.1%

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

   6. Applied egg-rr 38.1%

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

   7. Taylor expanded in x around 0 65.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.041666666666666664 \cdot {x}^{2} - 0.5\right)}}{-x} \]

   if 0.00449999999999999966 < x

   1. Initial program 99.2%

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

      1. clear-num 99.1%

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

      2. associate-/r/ 99.2%

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

      3. pow2 99.1%

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

      4. pow-flip 99.6%

         \[\leadsto \color{blue}{{x}^{\left(-2\right)}} \cdot \left(1 - \cos x\right) \]

      5. metadata-eval 99.6%

         \[\leadsto {x}^{\color{blue}{-2}} \cdot \left(1 - \cos x\right) \]

   4. Applied egg-rr 99.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;\frac{x \cdot \left(0.5 - 0.041666666666666664 \cdot {x}^{2}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;\frac{x\_m \cdot \left(0.5 - 0.041666666666666664 \cdot {x\_m}^{2}\right)}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0045)
   (/ (* x_m (- 0.5 (* 0.041666666666666664 (pow x_m 2.0)))) 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 <= 0.0045) {
		tmp = (x_m * (0.5 - (0.041666666666666664 * pow(x_m, 2.0)))) / x_m;
	} else {
		tmp = ((1.0 - cos(x_m)) / 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 <= 0.0045d0) then
        tmp = (x_m * (0.5d0 - (0.041666666666666664d0 * (x_m ** 2.0d0)))) / x_m
    else
        tmp = ((1.0d0 - cos(x_m)) / 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 <= 0.0045) {
		tmp = (x_m * (0.5 - (0.041666666666666664 * Math.pow(x_m, 2.0)))) / x_m;
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0045:
		tmp = (x_m * (0.5 - (0.041666666666666664 * math.pow(x_m, 2.0)))) / x_m
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0045)
		tmp = Float64(Float64(x_m * Float64(0.5 - Float64(0.041666666666666664 * (x_m ^ 2.0)))) / x_m);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / 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 <= 0.0045)
		tmp = (x_m * (0.5 - (0.041666666666666664 * (x_m ^ 2.0)))) / x_m;
	else
		tmp = ((1.0 - cos(x_m)) / 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, 0.0045], N[(N[(x$95$m * N[(0.5 - N[(0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $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 < 0.00449999999999999966

   1. Initial program 36.9%

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

      1. associate-/r* 38.1%

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

      2. div-inv 38.1%

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

   4. Applied egg-rr 38.1%

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

      1. *-commutative 38.1%

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

      2. frac-2neg 38.1%

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

      3. associate-*r/ 38.1%

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

      4. sub-neg 38.1%

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

      5. distribute-neg-in 38.1%

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

      6. metadata-eval 38.1%

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

      7. add-sqr-sqrt 13.8%

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

      8. sqrt-unprod 29.3%

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

      9. sqr-neg 29.3%

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

      10. sqrt-unprod 15.5%

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

      11. add-sqr-sqrt 23.5%

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

      12. add-sqr-sqrt 8.0%

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

      13. sqrt-unprod 32.3%

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

      14. sqr-neg 32.3%

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

      15. sqrt-unprod 24.2%

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

      16. add-sqr-sqrt 38.1%

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

   6. Applied egg-rr 38.1%

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

   7. Taylor expanded in x around 0 65.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.041666666666666664 \cdot {x}^{2} - 0.5\right)}}{-x} \]

   if 0.00449999999999999966 < x

   1. Initial program 99.2%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. frac-2neg 99.6%

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

      3. associate-*r/ 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. add-sqr-sqrt 52.8%

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

      8. sqrt-unprod 84.0%

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

      9. sqr-neg 84.0%

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

      10. sqrt-unprod 31.2%

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

      11. add-sqr-sqrt 63.0%

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

      12. add-sqr-sqrt 31.8%

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

      13. sqrt-unprod 78.7%

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

      14. sqr-neg 78.7%

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

      15. sqrt-unprod 46.6%

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

      16. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 99.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;\frac{x \cdot \left(0.5 - 0.041666666666666664 \cdot {x}^{2}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0045)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x_m)) / 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 <= 0.0045d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x_m)) / 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 <= 0.0045) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0045:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0045)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / 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 <= 0.0045)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x_m)) / 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, 0.0045], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $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 < 0.00449999999999999966

   1. Initial program 36.9%

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

   3. Taylor expanded in x around 0 65.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.00449999999999999966 < x

   1. Initial program 99.2%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. frac-2neg 99.6%

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

      3. associate-*r/ 99.7%

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

      4. sub-neg 99.7%

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

      5. distribute-neg-in 99.7%

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

      6. metadata-eval 99.7%

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

      7. add-sqr-sqrt 52.8%

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

      8. sqrt-unprod 84.0%

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

      9. sqr-neg 84.0%

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

      10. sqrt-unprod 31.2%

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

      11. add-sqr-sqrt 63.0%

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

      12. add-sqr-sqrt 31.8%

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

      13. sqrt-unprod 78.7%

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

      14. sqr-neg 78.7%

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

      15. sqrt-unprod 46.6%

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

      16. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 99.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0045)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x_m)) / (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 <= 0.0045d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x_m)) / (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 <= 0.0045) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0045:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x_m)) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0045)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / 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 <= 0.0045)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x_m)) / (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, 0.0045], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $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 < 0.00449999999999999966

   1. Initial program 36.9%

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

   3. Taylor expanded in x around 0 65.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.00449999999999999966 < x

   1. Initial program 99.2%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.55e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	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 <= 1.55d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999999e77

   1. Initial program 41.0%

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

   3. Taylor expanded in x around 0 61.7%

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

   if 1.54999999999999999e77 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 70.5%

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

   4. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 27.0% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in x around 0 29.9%

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

4. Taylor expanded in x around 0 30.8%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))