cos2 (problem 3.4.1)

Percentage Accurate: 51.4% → 99.6%

Time: 9.0s

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: 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.084:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x\_m}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x\_m}^{6} + 0.001388888888888889 \cdot {x\_m}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x\_m + -1}{x\_m}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.084)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x_m 2.0))
     (+
      (* -2.48015873015873e-5 (pow x_m 6.0))
      (* 0.001388888888888889 (pow x_m 4.0)))))
   (/ (/ (+ (cos x_m) -1.0) x_m) (- x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.084) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x_m, 2.0)) + ((-2.48015873015873e-5 * pow(x_m, 6.0)) + (0.001388888888888889 * pow(x_m, 4.0))));
	} else {
		tmp = ((cos(x_m) + -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 <= 0.084d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x_m ** 2.0d0)) + (((-2.48015873015873d-5) * (x_m ** 6.0d0)) + (0.001388888888888889d0 * (x_m ** 4.0d0))))
    else
        tmp = ((cos(x_m) + (-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 <= 0.084) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x_m, 2.0)) + ((-2.48015873015873e-5 * Math.pow(x_m, 6.0)) + (0.001388888888888889 * Math.pow(x_m, 4.0))));
	} else {
		tmp = ((Math.cos(x_m) + -1.0) / x_m) / -x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.084:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x_m, 2.0)) + ((-2.48015873015873e-5 * math.pow(x_m, 6.0)) + (0.001388888888888889 * math.pow(x_m, 4.0))))
	else:
		tmp = ((math.cos(x_m) + -1.0) / x_m) / -x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.084)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x_m ^ 2.0)) + Float64(Float64(-2.48015873015873e-5 * (x_m ^ 6.0)) + Float64(0.001388888888888889 * (x_m ^ 4.0)))));
	else
		tmp = Float64(Float64(Float64(cos(x_m) + -1.0) / x_m) / Float64(-x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.084)
		tmp = 0.5 + ((-0.041666666666666664 * (x_m ^ 2.0)) + ((-2.48015873015873e-5 * (x_m ^ 6.0)) + (0.001388888888888889 * (x_m ^ 4.0))));
	else
		tmp = ((cos(x_m) + -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, 0.084], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.48015873015873e-5 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0840000000000000052

   1. Initial program 39.9%

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

   3. Taylor expanded in x around 0 61.6%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)} \]

   if 0.0840000000000000052 < x

   1. Initial program 98.0%

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

      1. associate-/r* 99.0%

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

      2. div-inv 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. frac-2neg 98.9%

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

      3. associate-*r/ 99.0%

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

      4. sub-neg 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. add-sqr-sqrt 59.8%

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

      8. sqrt-unprod 81.3%

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

      9. sqr-neg 81.3%

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

      10. sqrt-unprod 21.5%

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

      11. add-sqr-sqrt 51.1%

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

      12. add-sqr-sqrt 29.6%

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

      13. sqrt-unprod 68.7%

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

      14. sqr-neg 68.7%

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

      15. sqrt-unprod 39.0%

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

      16. add-sqr-sqrt 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around inf 99.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.084:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.031:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x\_m}^{2} + 0.001388888888888889 \cdot {x\_m}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x\_m + -1}{x\_m}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.031)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x_m 2.0))
     (* 0.001388888888888889 (pow x_m 4.0))))
   (/ (/ (+ (cos x_m) -1.0) x_m) (- x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.031) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x_m, 2.0)) + (0.001388888888888889 * pow(x_m, 4.0)));
	} else {
		tmp = ((cos(x_m) + -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 <= 0.031d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x_m ** 2.0d0)) + (0.001388888888888889d0 * (x_m ** 4.0d0)))
    else
        tmp = ((cos(x_m) + (-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 <= 0.031) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x_m, 2.0)) + (0.001388888888888889 * Math.pow(x_m, 4.0)));
	} else {
		tmp = ((Math.cos(x_m) + -1.0) / x_m) / -x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.031:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x_m, 2.0)) + (0.001388888888888889 * math.pow(x_m, 4.0)))
	else:
		tmp = ((math.cos(x_m) + -1.0) / x_m) / -x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.031)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x_m ^ 2.0)) + Float64(0.001388888888888889 * (x_m ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(cos(x_m) + -1.0) / x_m) / Float64(-x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.031)
		tmp = 0.5 + ((-0.041666666666666664 * (x_m ^ 2.0)) + (0.001388888888888889 * (x_m ^ 4.0)));
	else
		tmp = ((cos(x_m) + -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, 0.031], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.031

   1. Initial program 39.9%

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

   3. Taylor expanded in x around 0 62.0%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)} \]

   if 0.031 < x

   1. Initial program 98.0%

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

      1. associate-/r* 99.0%

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

      2. div-inv 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. frac-2neg 98.9%

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

      3. associate-*r/ 99.0%

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

      4. sub-neg 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. add-sqr-sqrt 59.8%

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

      8. sqrt-unprod 81.3%

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

      9. sqr-neg 81.3%

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

      10. sqrt-unprod 21.5%

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

      11. add-sqr-sqrt 51.1%

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

      12. add-sqr-sqrt 29.6%

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

      13. sqrt-unprod 68.7%

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

      14. sqr-neg 68.7%

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

      15. sqrt-unprod 39.0%

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

      16. add-sqr-sqrt 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around inf 99.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.031:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = ((cos(x_m) + -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 <= 0.0045d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = ((cos(x_m) + (-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 <= 0.0045) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = ((Math.cos(x_m) + -1.0) / 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 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = ((math.cos(x_m) + -1.0) / 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(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(cos(x_m) + -1.0) / x_m) / Float64(-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 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = ((cos(x_m) + -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, 0.0045], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 39.9%

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

   3. Taylor expanded in x around 0 61.9%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 61.9%

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

   5. Simplified 61.9%

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

   if 0.00449999999999999966 < x

   1. Initial program 98.0%

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

      1. associate-/r* 99.0%

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

      2. div-inv 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. frac-2neg 98.9%

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

      3. associate-*r/ 99.0%

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

      4. sub-neg 99.0%

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

      5. distribute-neg-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. add-sqr-sqrt 59.8%

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

      8. sqrt-unprod 81.3%

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

      9. sqr-neg 81.3%

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

      10. sqrt-unprod 21.5%

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

      11. add-sqr-sqrt 51.1%

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

      12. add-sqr-sqrt 29.6%

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

      13. sqrt-unprod 68.7%

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

      14. sqr-neg 68.7%

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

      15. sqrt-unprod 39.0%

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

      16. add-sqr-sqrt 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around inf 99.0%

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

4. Final simplification 72.7%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0045)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (- 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 + (-0.041666666666666664 * pow(x_m, 2.0));
	} 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 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    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 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} 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 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	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(-0.041666666666666664 * (x_m ^ 2.0)));
	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 + (-0.041666666666666664 * (x_m ^ 2.0));
	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[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $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 39.9%

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

   3. Taylor expanded in x around 0 61.9%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 61.9%

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

   5. Simplified 61.9%

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

   if 0.00449999999999999966 < x

   1. Initial program 98.0%

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

4. Final simplification 72.5%

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

Alternative 5: 75.5% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8.6 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 8.6e+76) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 8.6e+76) {
		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 <= 8.6d+76) 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 <= 8.6e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 8.6e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 8.6e+76)
		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 <= 8.6e+76)
		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, 8.6e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.59999999999999957e76

   1. Initial program 47.2%

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

   3. Taylor expanded in x around 0 55.8%

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

   if 8.59999999999999957e76 < x

   1. Initial program 97.9%

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

      1. add-log-exp 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in x around 0 66.1%

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

   6. Taylor expanded in x around 0 66.1%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.8% 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 56.9%

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

   1. add-log-exp 56.8%

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

4. Applied egg-rr 56.8%

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

5. Taylor expanded in x around 0 28.3%

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

6. Taylor expanded in x around 0 28.9%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 28.9%

   \[\leadsto 0 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))