cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.6%

Time: 13.1s

Alternatives: 5

Speedup: 107.0×

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: 50.5% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.105:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.001388888888888889 + {x\_m}^{2} \cdot -2.48015873015873 \cdot 10^{-5}\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.105)
   (+
    0.5
    (*
     (pow x_m 2.0)
     (-
      (*
       (pow x_m 2.0)
       (+ 0.001388888888888889 (* (pow x_m 2.0) -2.48015873015873e-5)))
      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 <= 0.105) {
		tmp = 0.5 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * (0.001388888888888889 + (pow(x_m, 2.0) * -2.48015873015873e-5))) - 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 <= 0.105d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * (0.001388888888888889d0 + ((x_m ** 2.0d0) * (-2.48015873015873d-5)))) - 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 <= 0.105) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * (0.001388888888888889 + (Math.pow(x_m, 2.0) * -2.48015873015873e-5))) - 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 <= 0.105:
		tmp = 0.5 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * (0.001388888888888889 + (math.pow(x_m, 2.0) * -2.48015873015873e-5))) - 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 <= 0.105)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.001388888888888889 + Float64((x_m ^ 2.0) * -2.48015873015873e-5))) - 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 <= 0.105)
		tmp = 0.5 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * (0.001388888888888889 + ((x_m ^ 2.0) * -2.48015873015873e-5))) - 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, 0.105], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.001388888888888889 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $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 < 0.104999999999999996

   1. Initial program 32.9%

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

   3. Taylor expanded in x around 0 68.8%

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

   if 0.104999999999999996 < x

   1. Initial program 98.4%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. add-sqr-sqrt 98.2%

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

      4. times-frac 98.1%

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

      5. unpow-prod-down 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. pow-sqr 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

      \[\leadsto \color{blue}{{\left(\frac{x}{\sqrt{1 - \cos x}}\right)}^{-2}} \]
   7. Step-by-step derivation

      1. sqr-pow 99.2%

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

      2. pow2 99.2%

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

      3. metadata-eval 99.2%

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

      4. unpow-1 99.2%

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

      5. clear-num 99.3%

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

      6. add-sqr-sqrt 99.0%

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

      7. sqrt-prod 98.3%

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

      8. sqrt-div 98.3%

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

      9. pow2 98.3%

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

      10. add-sqr-sqrt 98.4%

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

      11. associate-/r* 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.105:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.001388888888888889 + {x}^{2} \cdot -2.48015873015873 \cdot 10^{-5}\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0055:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0055)
   (+ 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 <= 0.0055) {
		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 <= 0.0055d0) 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 <= 0.0055) {
		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 <= 0.0055:
		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 <= 0.0055)
		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 <= 0.0055)
		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, 0.0055], 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 < 0.0054999999999999997

   1. Initial program 32.9%

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

   3. Taylor expanded in x around 0 68.7%

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

   if 0.0054999999999999997 < x

   1. Initial program 98.4%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0055:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0055:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0055)
   (+ 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 <= 0.0055) {
		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 <= 0.0055d0) 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 <= 0.0055) {
		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 <= 0.0055:
		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 <= 0.0055)
		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 <= 0.0055)
		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, 0.0055], 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 < 0.0054999999999999997

   1. Initial program 32.9%

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

   3. Taylor expanded in x around 0 68.7%

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

   if 0.0054999999999999997 < x

   1. Initial program 98.4%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. add-sqr-sqrt 98.2%

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

      4. times-frac 98.1%

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

      5. unpow-prod-down 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. pow-sqr 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

      \[\leadsto \color{blue}{{\left(\frac{x}{\sqrt{1 - \cos x}}\right)}^{-2}} \]
   7. Step-by-step derivation

      1. sqr-pow 99.2%

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

      2. pow2 99.2%

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

      3. metadata-eval 99.2%

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

      4. unpow-1 99.2%

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

      5. clear-num 99.3%

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

      6. add-sqr-sqrt 99.0%

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

      7. sqrt-prod 98.3%

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

      8. sqrt-div 98.3%

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

      9. pow2 98.3%

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

      10. add-sqr-sqrt 98.4%

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

      11. associate-/r* 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0055:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.4e+77) 0.5 (/ (/ 0.0 x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.4e+77) {
		tmp = 0.5;
	} else {
		tmp = (0.0 / 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 <= 1.4d+77) then
        tmp = 0.5d0
    else
        tmp = (0.0d0 / 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 <= 1.4e+77) {
		tmp = 0.5;
	} else {
		tmp = (0.0 / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.4e+77:
		tmp = 0.5
	else:
		tmp = (0.0 / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.4e+77)
		tmp = 0.5;
	else
		tmp = Float64(Float64(0.0 / 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 <= 1.4e+77)
		tmp = 0.5;
	else
		tmp = (0.0 / 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, 1.4e+77], 0.5, N[(N[(0.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e77

   1. Initial program 38.1%

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

   3. Taylor expanded in x around 0 64.5%

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

   if 1.4e77 < x

   1. Initial program 98.2%

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

      1. clear-num 98.1%

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

      2. inv-pow 98.1%

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

      3. add-sqr-sqrt 98.0%

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

      4. times-frac 97.9%

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

      5. unpow-prod-down 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. pow-sqr 99.3%

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

      2. metadata-eval 99.3%

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

   6. Simplified 99.3%

      \[\leadsto \color{blue}{{\left(\frac{x}{\sqrt{1 - \cos x}}\right)}^{-2}} \]
   7. Step-by-step derivation

      1. sqr-pow 99.3%

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

      2. pow2 99.3%

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

      3. metadata-eval 99.3%

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

      4. unpow-1 99.3%

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

      5. clear-num 99.4%

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

      6. add-sqr-sqrt 99.2%

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

      7. sqrt-prod 98.1%

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

      8. sqrt-div 98.1%

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

      9. pow2 98.1%

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

      10. add-sqr-sqrt 98.2%

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

      11. associate-/r* 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. add-log-exp 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in x around 0 63.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.0% accurate, 107.0× 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 49.8%

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

3. Taylor expanded in x around 0 52.5%

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

4. Final simplification 52.5%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))