cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.6%

Time: 13.0s

Alternatives: 6

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: 50.9% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.032:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m} \cdot \left(1 - \cos x\_m\right)}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.032)
   (+
    0.5
    (*
     (pow x_m 2.0)
     (- (* (pow x_m 2.0) 0.001388888888888889) 0.041666666666666664)))
   (/ (* (/ 1.0 x_m) (- 1.0 (cos x_m))) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.032) {
		tmp = 0.5 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = ((1.0 / x_m) * (1.0 - cos(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 <= 0.032d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.001388888888888889d0) - 0.041666666666666664d0))
    else
        tmp = ((1.0d0 / x_m) * (1.0d0 - cos(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 <= 0.032) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = ((1.0 / x_m) * (1.0 - Math.cos(x_m))) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.032:
		tmp = 0.5 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 0.001388888888888889) - 0.041666666666666664))
	else:
		tmp = ((1.0 / x_m) * (1.0 - math.cos(x_m))) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.032)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.001388888888888889) - 0.041666666666666664)));
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) * Float64(1.0 - cos(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 <= 0.032)
		tmp = 0.5 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 0.001388888888888889) - 0.041666666666666664));
	else
		tmp = ((1.0 / x_m) * (1.0 - cos(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, 0.032], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.001388888888888889), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 31.5%

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

   3. Taylor expanded in x around 0 71.1%

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

   if 0.032000000000000001 < x

   1. Initial program 98.2%

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

      1. clear-num 98.2%

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

      2. inv-pow 98.2%

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

      3. pow2 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. unpow-1 98.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{2}}{1 - \cos x}}} \]

      2. clear-num 98.2%

         \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^{2}}} \]

      3. unpow2 98.2%

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

      4. associate-/r* 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left({x}^{2} \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} \cdot \left(1 - \cos x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\left(1 + \cos x\_m\right) \cdot {\left(\frac{x\_m}{\sin x\_m}\right)}^{2}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (* (+ 1.0 (cos x_m)) (pow (/ x_m (sin x_m)) 2.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 / ((1.0 + cos(x_m)) * pow((x_m / sin(x_m)), 2.0));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0 / ((1.0d0 + cos(x_m)) * ((x_m / sin(x_m)) ** 2.0d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 / ((1.0 + Math.cos(x_m)) * Math.pow((x_m / Math.sin(x_m)), 2.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 / ((1.0 + math.cos(x_m)) * math.pow((x_m / math.sin(x_m)), 2.0))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / Float64(Float64(1.0 + cos(x_m)) * (Float64(x_m / sin(x_m)) ^ 2.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 / ((1.0 + cos(x_m)) * ((x_m / sin(x_m)) ^ 2.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(N[(1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x$95$m / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.4%

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

   1. clear-num 47.4%

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

   2. inv-pow 47.4%

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

   3. flip-- 47.1%

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

   4. associate-/r/ 47.2%

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

   5. unpow-prod-down 47.2%

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

   6. pow2 47.2%

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

   7. metadata-eval 47.2%

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

   8. pow2 47.2%

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

   9. inv-pow 47.2%

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

4. Applied egg-rr 47.2%

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

   1. unpow-1 47.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{2}}{1 - {\cos x}^{2}}}} \cdot \frac{1}{1 + \cos x} \]

   2. associate-*l/ 47.2%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{1 + \cos x}}{\frac{{x}^{2}}{1 - {\cos x}^{2}}}} \]

   3. *-lft-identity 47.2%

      \[\leadsto \frac{\color{blue}{\frac{1}{1 + \cos x}}}{\frac{{x}^{2}}{1 - {\cos x}^{2}}} \]

6. Simplified 47.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{1 + \cos x}}{\frac{{x}^{2}}{1 - {\cos x}^{2}}}} \]
7. Step-by-step derivation

   1. unpow2 47.2%

      \[\leadsto \frac{\frac{1}{1 + \cos x}}{\frac{{x}^{2}}{1 - \color{blue}{\cos x \cdot \cos x}}} \]

   2. 1-sub-cos 74.6%

      \[\leadsto \frac{\frac{1}{1 + \cos x}}{\frac{{x}^{2}}{\color{blue}{\sin x \cdot \sin x}}} \]

8. Applied egg-rr 74.6%

   \[\leadsto \frac{\frac{1}{1 + \cos x}}{\frac{{x}^{2}}{\color{blue}{\sin x \cdot \sin x}}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 74.6%

      \[\leadsto \color{blue}{1 \cdot \frac{\frac{1}{1 + \cos x}}{\frac{{x}^{2}}{\sin x \cdot \sin x}}} \]

   2. add-sqr-sqrt 74.6%

      \[\leadsto 1 \cdot \frac{\frac{1}{1 + \cos x}}{\color{blue}{\sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}} \cdot \sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}}}} \]

   3. pow2 74.6%

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

   4. sqrt-div 74.6%

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

   5. sqrt-pow1 74.9%

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

   6. metadata-eval 74.9%

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

   7. pow1 74.9%

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

   8. sqrt-prod 48.7%

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

   9. add-sqr-sqrt 99.2%

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

10. Applied egg-rr 99.2%

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

    1. *-lft-identity 99.2%

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

    2. associate-/r* 99.2%

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

12. Simplified 99.2%

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

13. Final simplification 99.2%

    \[\leadsto \frac{1}{\left(1 + \cos x\right) \cdot {\left(\frac{x}{\sin x}\right)}^{2}} \]
14. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0042:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m} \cdot \left(1 - \cos x\_m\right)}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0042)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (/ (* (/ 1.0 x_m) (- 1.0 (cos x_m))) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0042) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 / x_m) * (1.0 - cos(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 <= 0.0042d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 / x_m) * (1.0d0 - cos(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 <= 0.0042) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 / x_m) * (1.0 - Math.cos(x_m))) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0042:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = ((1.0 / x_m) * (1.0 - math.cos(x_m))) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0042)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) * Float64(1.0 - cos(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 <= 0.0042)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((1.0 / x_m) * (1.0 - cos(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, 0.0042], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 31.5%

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

   3. Taylor expanded in x around 0 70.8%

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

      1. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.2%

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

      1. clear-num 98.2%

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

      2. inv-pow 98.2%

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

      3. pow2 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. unpow-1 98.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{2}}{1 - \cos x}}} \]

      2. clear-num 98.2%

         \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^{2}}} \]

      3. unpow2 98.2%

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

      4. associate-/r* 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} \cdot \left(1 - \cos x\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0042:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0042)
   (+ 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 <= 0.0042) {
		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 <= 0.0042d0) 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 <= 0.0042) {
		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 <= 0.0042:
		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 <= 0.0042)
		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 <= 0.0042)
		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, 0.0042], 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 < 0.00419999999999999974

   1. Initial program 31.5%

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

   3. Taylor expanded in x around 0 70.8%

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

      1. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0042:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0042)
   (+ 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 <= 0.0042) {
		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 <= 0.0042d0) 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 <= 0.0042) {
		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 <= 0.0042:
		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 <= 0.0042)
		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 <= 0.0042)
		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, 0.0042], 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 < 0.00419999999999999974

   1. Initial program 31.5%

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

   3. Taylor expanded in x around 0 70.8%

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

      1. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.2%

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

      1. clear-num 98.2%

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

      2. inv-pow 98.2%

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

      3. pow2 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. unpow-1 98.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{2}}{1 - \cos x}}} \]

      2. clear-num 98.2%

         \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^{2}}} \]

      3. unpow2 98.2%

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

      4. associate-/r* 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.5% 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 47.4%

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

3. Taylor expanded in x around 0 55.3%

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

4. Final simplification 55.3%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))