cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.3%

Time: 9.8s

Alternatives: 10

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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{x\_m}{2}\right) \cdot \frac{\sin x\_m}{{x\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.6e-8) 0.5 (* (tan (/ x_m 2.0)) (/ (sin x_m) (pow x_m 2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.6e-8) {
		tmp = 0.5;
	} else {
		tmp = tan((x_m / 2.0)) * (sin(x_m) / pow(x_m, 2.0));
	}
	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 <= 2.6d-8) then
        tmp = 0.5d0
    else
        tmp = tan((x_m / 2.0d0)) * (sin(x_m) / (x_m ** 2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.6e-8) {
		tmp = 0.5;
	} else {
		tmp = Math.tan((x_m / 2.0)) * (Math.sin(x_m) / Math.pow(x_m, 2.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.6e-8:
		tmp = 0.5
	else:
		tmp = math.tan((x_m / 2.0)) * (math.sin(x_m) / math.pow(x_m, 2.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.6e-8)
		tmp = 0.5;
	else
		tmp = Float64(tan(Float64(x_m / 2.0)) * Float64(sin(x_m) / (x_m ^ 2.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.6e-8)
		tmp = 0.5;
	else
		tmp = tan((x_m / 2.0)) * (sin(x_m) / (x_m ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.6e-8], 0.5, N[(N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e-8

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 63.6%

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

   if 2.6000000000000001e-8 < x

   1. Initial program 99.3%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.3%

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

      3. flip-- 98.9%

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

      4. associate-/r/ 98.9%

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

      5. unpow-prod-down 98.8%

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

      6. pow2 98.8%

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

      7. metadata-eval 98.8%

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

      8. pow2 98.8%

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

      9. inv-pow 98.8%

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

   4. Applied egg-rr 98.8%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

      3. unpow-1 98.9%

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

   6. Simplified 98.9%

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

      1. unpow2 98.9%

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

      2. 1-sub-cos 99.0%

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

   8. Applied egg-rr 99.0%

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

   9. Taylor expanded in x around inf 98.9%

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

       1. unpow2 98.9%

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

       2. *-commutative 98.9%

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

       3. times-frac 98.9%

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

       4. hang-0p-tan 99.6%

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

   11. Simplified 99.6%

       \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{{x}^{2}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

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

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)))
   (- (pow x_m -2.0) (* (cos x_m) (pow x_m -2.0)))))

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 = pow(x_m, -2.0) - (cos(x_m) * pow(x_m, -2.0));
	}
	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 <= 0.032d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.001388888888888889d0) - 0.041666666666666664d0))
    else
        tmp = (x_m ** (-2.0d0)) - (cos(x_m) * (x_m ** (-2.0d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.032) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = Math.pow(x_m, -2.0) - (Math.cos(x_m) * Math.pow(x_m, -2.0));
	}
	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 = math.pow(x_m, -2.0) - (math.cos(x_m) * math.pow(x_m, -2.0))
	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((x_m ^ -2.0) - Float64(cos(x_m) * (x_m ^ -2.0)));
	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 = (x_m ^ -2.0) - (cos(x_m) * (x_m ^ -2.0));
	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[Power[x$95$m, -2.0], $MachinePrecision] - N[(N[Cos[x$95$m], $MachinePrecision] * N[Power[x$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 63.2%

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

   if 0.032000000000000001 < x

   1. Initial program 99.3%

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

      1. div-sub 99.3%

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

      2. pow2 99.3%

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

      3. pow-flip 99.1%

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

      4. metadata-eval 99.1%

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

      5. div-inv 99.1%

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

      6. pow2 99.1%

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

      7. pow-flip 99.4%

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

      8. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 72.4%

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

Alternative 3: 99.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. clear-num 54.2%

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

   2. inv-pow 54.2%

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

   3. flip-- 54.0%

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

   4. associate-/r/ 54.0%

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

   5. unpow-prod-down 54.0%

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

   6. pow2 54.0%

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

   7. metadata-eval 54.0%

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

   8. pow2 54.0%

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

   9. inv-pow 54.0%

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

4. Applied egg-rr 54.0%

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

   1. associate-*r/ 54.1%

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

   2. *-rgt-identity 54.1%

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

   3. unpow-1 54.1%

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

6. Simplified 54.1%

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

   1. unpow2 54.1%

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

   2. 1-sub-cos 76.1%

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

8. Applied egg-rr 76.1%

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

9. Taylor expanded in x around inf 76.1%

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

    1. unpow2 76.1%

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

    2. unpow2 76.1%

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

    3. times-frac 99.5%

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

    4. unpow2 99.5%

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

11. Simplified 99.5%

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

Alternative 4: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. clear-num 54.2%

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

   2. inv-pow 54.2%

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

   3. flip-- 54.0%

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

   4. associate-/r/ 54.0%

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

   5. unpow-prod-down 54.0%

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

   6. pow2 54.0%

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

   7. metadata-eval 54.0%

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

   8. pow2 54.0%

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

   9. inv-pow 54.0%

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

4. Applied egg-rr 54.0%

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

   1. associate-*r/ 54.1%

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

   2. *-rgt-identity 54.1%

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

   3. unpow-1 54.1%

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

6. Simplified 54.1%

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

   1. unpow2 54.1%

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

   2. 1-sub-cos 76.1%

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

8. Applied egg-rr 76.1%

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

   1. frac-2neg 76.1%

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

   2. div-inv 76.1%

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

   3. add-sqr-sqrt 76.0%

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

   4. pow2 76.0%

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

   5. associate-/r/ 74.8%

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

   6. pow-flip 74.8%

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

   7. metadata-eval 74.8%

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

   8. sqrt-prod 74.8%

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

   9. sqrt-pow1 76.3%

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

   10. metadata-eval 76.3%

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

   11. inv-pow 76.3%

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

   12. sqrt-prod 49.0%

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

   13. add-sqr-sqrt 99.3%

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

10. Applied egg-rr 99.3%

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

    1. associate-*r/ 99.3%

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

    2. *-rgt-identity 99.3%

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

    3. associate-*l/ 99.5%

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

    4. *-lft-identity 99.5%

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

    5. distribute-neg-in 99.5%

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

    6. metadata-eval 99.5%

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

    7. unsub-neg 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 5: 99.6% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.031)
   (+
    0.5
    (*
     (pow x_m 2.0)
     (- (* (pow x_m 2.0) 0.001388888888888889) 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 <= 0.031) {
		tmp = 0.5 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.001388888888888889) - 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 <= 0.031d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.001388888888888889d0) - 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 <= 0.031) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.001388888888888889) - 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 <= 0.031:
		tmp = 0.5 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 0.001388888888888889) - 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 <= 0.031)
		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 - 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 <= 0.031)
		tmp = 0.5 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 0.001388888888888889) - 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, 0.031], 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 - 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 < 0.031

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 63.2%

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

   if 0.031 < x

   1. Initial program 99.3%

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

      1. add-cbrt-cube 70.8%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 - \cos x}{x \cdot x} \cdot \frac{1 - \cos x}{x \cdot x}\right) \cdot \frac{1 - \cos x}{x \cdot x}}} \]

      2. pow1/3 69.6%

         \[\leadsto \color{blue}{{\left(\left(\frac{1 - \cos x}{x \cdot x} \cdot \frac{1 - \cos x}{x \cdot x}\right) \cdot \frac{1 - \cos x}{x \cdot x}\right)}^{0.3333333333333333}} \]

      3. pow3 69.5%

         \[\leadsto {\color{blue}{\left({\left(\frac{1 - \cos x}{x \cdot x}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. add-sqr-sqrt 69.5%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{\frac{1 - \cos x}{x \cdot x}} \cdot \sqrt{\frac{1 - \cos x}{x \cdot x}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. pow2 69.5%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{\frac{1 - \cos x}{x \cdot x}}\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 69.5%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{\frac{1 - \cos x}{x \cdot x}}\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. sqrt-div 69.6%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt{1 - \cos x}}{\sqrt{x \cdot x}}\right)}}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      8. sqrt-prod 69.6%

         \[\leadsto {\left({\left(\frac{\sqrt{1 - \cos x}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. add-sqr-sqrt 69.6%

         \[\leadsto {\left({\left(\frac{\sqrt{1 - \cos x}}{\color{blue}{x}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. metadata-eval 69.6%

          \[\leadsto {\left({\left(\frac{\sqrt{1 - \cos x}}{x}\right)}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{1 - \cos x}}{x}\right)}^{6}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.1%

         \[\leadsto \color{blue}{{\left(\frac{\sqrt{1 - \cos x}}{x}\right)}^{\left(6 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.1%

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

      3. pow2 99.1%

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

      4. frac-times 99.1%

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

      5. add-sqr-sqrt 99.3%

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

      6. rem-log-exp 99.1%

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

      7. rem-log-exp 99.3%

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

      8. 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 72.4%

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

Alternative 6: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00014) 0.5 (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00014) {
		tmp = 0.5;
	} 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 <= 0.00014d0) then
        tmp = 0.5d0
    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 <= 0.00014) {
		tmp = 0.5;
	} 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 <= 0.00014:
		tmp = 0.5
	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 <= 0.00014)
		tmp = 0.5;
	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 <= 0.00014)
		tmp = 0.5;
	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, 0.00014], 0.5, 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 < 1.3999999999999999e-4

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 63.6%

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

   if 1.3999999999999999e-4 < x

   1. Initial program 99.3%

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

      1. add-cbrt-cube 70.8%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1 - \cos x}{x \cdot x} \cdot \frac{1 - \cos x}{x \cdot x}\right) \cdot \frac{1 - \cos x}{x \cdot x}}} \]

      2. pow1/3 69.6%

         \[\leadsto \color{blue}{{\left(\left(\frac{1 - \cos x}{x \cdot x} \cdot \frac{1 - \cos x}{x \cdot x}\right) \cdot \frac{1 - \cos x}{x \cdot x}\right)}^{0.3333333333333333}} \]

      3. pow3 69.5%

         \[\leadsto {\color{blue}{\left({\left(\frac{1 - \cos x}{x \cdot x}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. add-sqr-sqrt 69.5%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{\frac{1 - \cos x}{x \cdot x}} \cdot \sqrt{\frac{1 - \cos x}{x \cdot x}}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. pow2 69.5%

         \[\leadsto {\left({\color{blue}{\left({\left(\sqrt{\frac{1 - \cos x}{x \cdot x}}\right)}^{2}\right)}}^{3}\right)}^{0.3333333333333333} \]

      6. pow-pow 69.5%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{\frac{1 - \cos x}{x \cdot x}}\right)}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} \]

      7. sqrt-div 69.6%

         \[\leadsto {\left({\color{blue}{\left(\frac{\sqrt{1 - \cos x}}{\sqrt{x \cdot x}}\right)}}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      8. sqrt-prod 69.6%

         \[\leadsto {\left({\left(\frac{\sqrt{1 - \cos x}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      9. add-sqr-sqrt 69.6%

         \[\leadsto {\left({\left(\frac{\sqrt{1 - \cos x}}{\color{blue}{x}}\right)}^{\left(2 \cdot 3\right)}\right)}^{0.3333333333333333} \]

      10. metadata-eval 69.6%

          \[\leadsto {\left({\left(\frac{\sqrt{1 - \cos x}}{x}\right)}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   4. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{{\left({\left(\frac{\sqrt{1 - \cos x}}{x}\right)}^{6}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow-pow 99.1%

         \[\leadsto \color{blue}{{\left(\frac{\sqrt{1 - \cos x}}{x}\right)}^{\left(6 \cdot 0.3333333333333333\right)}} \]

      2. metadata-eval 99.1%

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

      3. pow2 99.1%

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

      4. frac-times 99.1%

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

      5. add-sqr-sqrt 99.3%

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

      6. rem-log-exp 99.1%

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

      7. rem-log-exp 99.3%

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

      8. 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. Add Preprocessing

Alternative 7: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00014) 0.5 (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00014) {
		tmp = 0.5;
	} 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 <= 0.00014d0) then
        tmp = 0.5d0
    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 <= 0.00014) {
		tmp = 0.5;
	} 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 <= 0.00014:
		tmp = 0.5
	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 <= 0.00014)
		tmp = 0.5;
	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 <= 0.00014)
		tmp = 0.5;
	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, 0.00014], 0.5, 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 < 1.3999999999999999e-4

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 63.6%

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

   if 1.3999999999999999e-4 < x

   1. Initial program 99.3%

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

Alternative 8: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.35:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-4} \cdot 120\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.35)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (* (pow x_m -4.0) 120.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.35) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = pow(x_m, -4.0) * 120.0;
	}
	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 <= 3.35d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (x_m ** (-4.0d0)) * 120.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 3.35) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = Math.pow(x_m, -4.0) * 120.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.35:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = math.pow(x_m, -4.0) * 120.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.35)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64((x_m ^ -4.0) * 120.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 3.35)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = (x_m ^ -4.0) * 120.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.35], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, -4.0], $MachinePrecision] * 120.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.35000000000000009

   1. Initial program 39.2%

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

   3. Taylor expanded in x around 0 62.6%

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

   if 3.35000000000000009 < x

   1. Initial program 99.3%

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

      1. associate-/r* 99.3%

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

      2. clear-num 99.3%

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

      3. inv-pow 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0 49.5%

      \[\leadsto {\color{blue}{\left(2 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)\right)}}^{-1} \]
   6. Step-by-step derivation

      1. *-commutative 49.5%

         \[\leadsto {\left(2 + {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{{x}^{2} \cdot 0.008333333333333333}\right)\right)}^{-1} \]

   7. Simplified 49.5%

      \[\leadsto {\color{blue}{\left(2 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot 0.008333333333333333\right)\right)}}^{-1} \]

   8. Taylor expanded in x around inf 49.5%

      \[\leadsto \color{blue}{\frac{120}{{x}^{4}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 49.5%

         \[\leadsto \color{blue}{1 \cdot \frac{120}{{x}^{4}}} \]

      2. pow1 49.5%

         \[\leadsto 1 \cdot \color{blue}{{\left(\frac{120}{{x}^{4}}\right)}^{1}} \]

      3. pow1 49.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{120}{{x}^{4}}} \]

      4. div-inv 49.5%

         \[\leadsto 1 \cdot \color{blue}{\left(120 \cdot \frac{1}{{x}^{4}}\right)} \]

      5. pow-flip 49.5%

         \[\leadsto 1 \cdot \left(120 \cdot \color{blue}{{x}^{\left(-4\right)}}\right) \]

      6. metadata-eval 49.5%

         \[\leadsto 1 \cdot \left(120 \cdot {x}^{\color{blue}{-4}}\right) \]

   10. Applied egg-rr 49.5%

       \[\leadsto \color{blue}{1 \cdot \left(120 \cdot {x}^{-4}\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 49.5%

          \[\leadsto \color{blue}{120 \cdot {x}^{-4}} \]

       2. *-commutative 49.5%

          \[\leadsto \color{blue}{{x}^{-4} \cdot 120} \]

   12. Simplified 49.5%

       \[\leadsto \color{blue}{{x}^{-4} \cdot 120} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.35:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x}^{-4} \cdot 120\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-4} \cdot 120\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.0) 0.5 (* (pow x_m -4.0) 120.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.0) {
		tmp = 0.5;
	} else {
		tmp = pow(x_m, -4.0) * 120.0;
	}
	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 <= 4.0d0) then
        tmp = 0.5d0
    else
        tmp = (x_m ** (-4.0d0)) * 120.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 4.0) {
		tmp = 0.5;
	} else {
		tmp = Math.pow(x_m, -4.0) * 120.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 4.0:
		tmp = 0.5
	else:
		tmp = math.pow(x_m, -4.0) * 120.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.0)
		tmp = 0.5;
	else
		tmp = Float64((x_m ^ -4.0) * 120.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 4.0)
		tmp = 0.5;
	else
		tmp = (x_m ^ -4.0) * 120.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 4.0], 0.5, N[(N[Power[x$95$m, -4.0], $MachinePrecision] * 120.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 39.2%

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

   3. Taylor expanded in x around 0 63.4%

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

   if 4 < x

   1. Initial program 99.3%

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

      1. associate-/r* 99.3%

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

      2. clear-num 99.3%

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

      3. inv-pow 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0 49.5%

      \[\leadsto {\color{blue}{\left(2 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)\right)}}^{-1} \]
   6. Step-by-step derivation

      1. *-commutative 49.5%

         \[\leadsto {\left(2 + {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{{x}^{2} \cdot 0.008333333333333333}\right)\right)}^{-1} \]

   7. Simplified 49.5%

      \[\leadsto {\color{blue}{\left(2 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot 0.008333333333333333\right)\right)}}^{-1} \]

   8. Taylor expanded in x around inf 49.5%

      \[\leadsto \color{blue}{\frac{120}{{x}^{4}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 49.5%

         \[\leadsto \color{blue}{1 \cdot \frac{120}{{x}^{4}}} \]

      2. pow1 49.5%

         \[\leadsto 1 \cdot \color{blue}{{\left(\frac{120}{{x}^{4}}\right)}^{1}} \]

      3. pow1 49.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{120}{{x}^{4}}} \]

      4. div-inv 49.5%

         \[\leadsto 1 \cdot \color{blue}{\left(120 \cdot \frac{1}{{x}^{4}}\right)} \]

      5. pow-flip 49.5%

         \[\leadsto 1 \cdot \left(120 \cdot \color{blue}{{x}^{\left(-4\right)}}\right) \]

      6. metadata-eval 49.5%

         \[\leadsto 1 \cdot \left(120 \cdot {x}^{\color{blue}{-4}}\right) \]

   10. Applied egg-rr 49.5%

       \[\leadsto \color{blue}{1 \cdot \left(120 \cdot {x}^{-4}\right)} \]
   11. Step-by-step derivation

       1. *-lft-identity 49.5%

          \[\leadsto \color{blue}{120 \cdot {x}^{-4}} \]

       2. *-commutative 49.5%

          \[\leadsto \color{blue}{{x}^{-4} \cdot 120} \]

   12. Simplified 49.5%

       \[\leadsto \color{blue}{{x}^{-4} \cdot 120} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.8% 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 54.2%

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

3. Taylor expanded in x around 0 48.7%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))