cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.8%

Time: 9.7s

Alternatives: 8

Speedup: 17.8×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 50.3% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-7}:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(\sin x\_m \cdot \tan \left(\frac{x\_m}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1e-7)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* (* x_m x_m) 0.001388888888888889) 0.041666666666666664)))
   (* (pow x_m -2.0) (* (sin x_m) (tan (/ x_m 2.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1e-7) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = pow(x_m, -2.0) * (sin(x_m) * tan((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 <= 1d-7) then
        tmp = 0.5d0 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889d0) - 0.041666666666666664d0))
    else
        tmp = (x_m ** (-2.0d0)) * (sin(x_m) * tan((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 <= 1e-7) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = Math.pow(x_m, -2.0) * (Math.sin(x_m) * Math.tan((x_m / 2.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1e-7:
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889) - 0.041666666666666664))
	else:
		tmp = math.pow(x_m, -2.0) * (math.sin(x_m) * math.tan((x_m / 2.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1e-7)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * 0.001388888888888889) - 0.041666666666666664)));
	else
		tmp = Float64((x_m ^ -2.0) * Float64(sin(x_m) * tan(Float64(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 <= 1e-7)
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889) - 0.041666666666666664));
	else
		tmp = (x_m ^ -2.0) * (sin(x_m) * tan((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, 1e-7], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999995e-8

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 65.3%

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

      1. unpow2 65.3%

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

   5. Applied egg-rr 65.3%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. unpow2 65.3%

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

   7. Applied egg-rr 65.3%

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

   if 9.9999999999999995e-8 < x

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. clear-num 99.5%

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

      3. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-/r/ 99.5%

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

      2. div-inv 99.3%

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

      3. frac-times 99.5%

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

      4. metadata-eval 99.5%

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

      5. unpow2 99.5%

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

      6. flip-- 99.3%

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

      7. metadata-eval 99.3%

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

      8. 1-sub-cos 99.3%

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

      9. associate-/l* 99.2%

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

      10. *-commutative 99.2%

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

      11. 1-sub-cos 99.3%

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

      12. unpow2 99.3%

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

      13. *-un-lft-identity 99.3%

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

      14. times-frac 99.3%

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

      15. unpow2 99.3%

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

      16. 1-sub-cos 99.3%

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

      17. pow2 99.3%

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

   8. Applied egg-rr 99.4%

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

      1. /-rgt-identity 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*l/ 99.3%

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

      4. unpow2 99.3%

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

      5. associate-*l* 99.3%

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

      6. associate-*r/ 99.3%

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

      7. associate-*l* 99.4%

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

      8. hang-0p-tan 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-7}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.038:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} - \cos x\_m \cdot {x\_m}^{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.038)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* (* x_m x_m) 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 <= 0.038) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.038d0) then
        tmp = 0.5d0 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.038) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.038:
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.038)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * 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 <= 0.038)
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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, 0.038], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $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 < 0.0379999999999999991

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 65.3%

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

      1. unpow2 65.3%

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

   5. Applied egg-rr 65.3%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. unpow2 65.3%

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

   7. Applied egg-rr 65.3%

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

   if 0.0379999999999999991 < x

   1. Initial program 99.6%

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

      1. div-sub 99.6%

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

      2. pow2 99.6%

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

      3. pow-flip 99.7%

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

      4. metadata-eval 99.7%

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

      5. div-inv 99.6%

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

      6. pow2 99.6%

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

      7. pow-flip 99.7%

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

      8. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.038:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} - \cos x \cdot {x}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((sin(x_m) / x_m) ^ 2.0) / (1.0 + cos(x_m));
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[(1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. clear-num 50.9%

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

   2. inv-pow 50.9%

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

   3. flip-- 50.8%

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

   4. associate-/r/ 50.8%

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

   5. unpow-prod-down 50.8%

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

   6. pow2 50.8%

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

   7. metadata-eval 50.8%

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

   8. pow2 50.8%

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

   9. inv-pow 50.8%

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

4. Applied egg-rr 50.8%

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

   1. associate-*r/ 50.8%

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

   2. *-rgt-identity 50.8%

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

   3. unpow-1 50.8%

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

   4. associate-/r/ 50.8%

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

6. Simplified 50.8%

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

   1. unpow2 50.8%

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

   2. 1-sub-cos 72.6%

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

8. Applied egg-rr 72.6%

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

9. Taylor expanded in x around inf 74.2%

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

    1. unpow2 74.2%

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

    2. unpow2 74.2%

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

    3. times-frac 99.6%

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

    4. unpow2 99.6%

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

11. Simplified 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.034:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.034)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* (* x_m x_m) 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 <= 0.034) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.034d0) then
        tmp = 0.5d0 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.034) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.034:
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.034)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * 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 <= 0.034)
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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, 0.034], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $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 < 0.034000000000000002

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 65.3%

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

      1. unpow2 65.3%

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

   5. Applied egg-rr 65.3%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. unpow2 65.3%

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

   7. Applied egg-rr 65.3%

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

   if 0.034000000000000002 < x

   1. Initial program 99.6%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. un-div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.034:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.034:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.034)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* (* x_m x_m) 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 <= 0.034) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.034d0) then
        tmp = 0.5d0 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.034) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.034:
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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 <= 0.034)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * 0.001388888888888889) - 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 <= 0.034)
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 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, 0.034], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.034000000000000002

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 65.3%

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

      1. unpow2 65.3%

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

   5. Applied egg-rr 65.3%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. unpow2 65.3%

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

   7. Applied egg-rr 65.3%

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

   if 0.034000000000000002 < x

   1. Initial program 99.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.034:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 6.8 \cdot 10^{+38}:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(\left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 6.8e+38)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* (* x_m x_m) 0.001388888888888889) 0.041666666666666664)))
   0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 6.8e+38) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 6.8d+38) then
        tmp = 0.5d0 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889d0) - 0.041666666666666664d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 6.8e+38) {
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 6.8e+38:
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889) - 0.041666666666666664))
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 6.8e+38)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(Float64(x_m * x_m) * 0.001388888888888889) - 0.041666666666666664)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 6.8e+38)
		tmp = 0.5 + ((x_m * x_m) * (((x_m * x_m) * 0.001388888888888889) - 0.041666666666666664));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 6.8e+38], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.79999999999999992e38

   1. Initial program 38.8%

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

   3. Taylor expanded in x around 0 63.4%

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

      1. unpow2 63.4%

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

   5. Applied egg-rr 63.4%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. unpow2 63.4%

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

   7. Applied egg-rr 63.4%

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

   if 6.79999999999999992e38 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 63.4%

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

   4. Taylor expanded in x around 0 63.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+38}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 17.8× speedup

Math

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

FPCore

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

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;
	}
	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
    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;
	}
	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
	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 = 0.0;
	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;
	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, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e77

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0 62.7%

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

   if 1.4e77 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 68.4%

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

   4. Taylor expanded in x around 0 68.4%

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

Alternative 8: 27.4% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

3. Taylor expanded in x around 0 29.7%

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

4. Taylor expanded in x around 0 30.5%

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

Reproduce

herbie shell --seed 2024121 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))