cos2 (problem 3.4.1)

Percentage Accurate: 51.0% → 99.8%

Time: 14.5s

Alternatives: 11

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: 51.0% 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, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0001:\\ \;\;\;\;0.5 + {x_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{\frac{x_m}{\sin x_m \cdot \tan \left(\frac{x_m}{2}\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0001)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (/ (/ 1.0 x_m) (/ x_m (* (sin x_m) (tan (/ x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0001) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x_m) / (x_m / (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 <= 0.0001d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 / x_m) / (x_m / (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 <= 0.0001) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 / x_m) / (x_m / (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 <= 0.0001:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = (1.0 / x_m) / (x_m / (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 <= 0.0001)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(x_m / 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 <= 0.0001)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = (1.0 / x_m) / (x_m / (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, 0.0001], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(x$95$m / N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e-4

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.4%

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

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

   if 1.00000000000000005e-4 < x

   1. Initial program 98.9%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. flip-- 99.0%

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

      2. metadata-eval 99.0%

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

      3. 1-sub-cos 98.9%

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

      4. associate-/r/ 98.9%

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

      5. pow2 98.9%

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

   8. Applied egg-rr 98.9%

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

      1. associate-*l/ 98.7%

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

      2. associate-/l* 98.9%

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

      3. unpow2 98.9%

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

      4. associate-*r/ 98.7%

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

      5. hang-0p-tan 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0001:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.03:\\ \;\;\;\;0.5 + \left({x_m}^{2} \cdot -0.041666666666666664 + 0.001388888888888889 \cdot {x_m}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x_m}{x_m}}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.03)
   (+
    0.5
    (+
     (* (pow x_m 2.0) -0.041666666666666664)
     (* 0.001388888888888889 (pow x_m 4.0))))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.03) {
		tmp = 0.5 + ((pow(x_m, 2.0) * -0.041666666666666664) + (0.001388888888888889 * pow(x_m, 4.0)));
	} else {
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.03d0) then
        tmp = 0.5d0 + (((x_m ** 2.0d0) * (-0.041666666666666664d0)) + (0.001388888888888889d0 * (x_m ** 4.0d0)))
    else
        tmp = ((1.0d0 - cos(x_m)) / x_m) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.03) {
		tmp = 0.5 + ((Math.pow(x_m, 2.0) * -0.041666666666666664) + (0.001388888888888889 * Math.pow(x_m, 4.0)));
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.03:
		tmp = 0.5 + ((math.pow(x_m, 2.0) * -0.041666666666666664) + (0.001388888888888889 * math.pow(x_m, 4.0)))
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.03)
		tmp = Float64(0.5 + Float64(Float64((x_m ^ 2.0) * -0.041666666666666664) + Float64(0.001388888888888889 * (x_m ^ 4.0))));
	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 <= 0.03)
		tmp = 0.5 + (((x_m ^ 2.0) * -0.041666666666666664) + (0.001388888888888889 * (x_m ^ 4.0)));
	else
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.03], N[(0.5 + N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x$95$m, 4.0], $MachinePrecision]), $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 < 0.029999999999999999

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.5%

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

   if 0.029999999999999999 < x

   1. Initial program 98.9%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. un-div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;0.5 + \left({x}^{2} \cdot -0.041666666666666664 + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{x_m}}{\frac{1}{\sin x_m} \cdot \frac{x_m}{\tan \left(x_m \cdot 0.5\right)}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (/ 1.0 x_m) (* (/ 1.0 (sin x_m)) (/ x_m (tan (* x_m 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 / x_m) / ((1.0 / sin(x_m)) * (x_m / tan((x_m * 0.5))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (1.0d0 / x_m) / ((1.0d0 / sin(x_m)) * (x_m / tan((x_m * 0.5d0))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 / x_m) / ((1.0 / Math.sin(x_m)) * (x_m / Math.tan((x_m * 0.5))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 / x_m) / ((1.0 / math.sin(x_m)) * (x_m / math.tan((x_m * 0.5))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 / x_m) / Float64(Float64(1.0 / sin(x_m)) * Float64(x_m / tan(Float64(x_m * 0.5)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 / x_m) / ((1.0 / sin(x_m)) * (x_m / tan((x_m * 0.5))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(1.0 / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[Tan[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $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. associate-/r* 55.6%

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

   2. div-inv 55.5%

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

4. Applied egg-rr 55.5%

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

   1. *-commutative 55.5%

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

   2. clear-num 55.5%

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

   3. un-div-inv 55.5%

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

6. Applied egg-rr 55.5%

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

   1. flip-- 55.3%

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

   2. metadata-eval 55.3%

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

   3. 1-sub-cos 77.5%

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

   4. associate-/r/ 77.5%

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

   5. pow2 77.5%

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

8. Applied egg-rr 77.5%

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

   1. associate-*l/ 77.5%

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

   2. associate-/l* 77.5%

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

   3. unpow2 77.5%

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

   4. associate-*r/ 77.5%

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

   5. hang-0p-tan 77.8%

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

10. Simplified 77.8%

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

    1. *-un-lft-identity 77.8%

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

    2. times-frac 99.7%

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

    3. div-inv 99.7%

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

    4. metadata-eval 99.7%

       \[\leadsto \frac{\frac{1}{x}}{\frac{1}{\sin x} \cdot \frac{x}{\tan \left(x \cdot \color{blue}{0.5}\right)}} \]

12. Applied egg-rr 99.7%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{1}{\sin x} \cdot \frac{x}{\tan \left(x \cdot 0.5\right)}}} \]

13. Final simplification 99.7%

    \[\leadsto \frac{\frac{1}{x}}{\frac{1}{\sin x} \cdot \frac{x}{\tan \left(x \cdot 0.5\right)}} \]
14. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\sin x_m}{x_m} \cdot \frac{\frac{1}{\frac{1}{\tan \left(x_m \cdot 0.5\right)}}}{x_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (/ (sin x_m) x_m) (/ (/ 1.0 (/ 1.0 (tan (* x_m 0.5)))) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return (sin(x_m) / x_m) * ((1.0 / (1.0 / tan((x_m * 0.5)))) / 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) * ((1.0d0 / (1.0d0 / tan((x_m * 0.5d0)))) / x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (Math.sin(x_m) / x_m) * ((1.0 / (1.0 / Math.tan((x_m * 0.5)))) / x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (math.sin(x_m) / x_m) * ((1.0 / (1.0 / math.tan((x_m * 0.5)))) / x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(sin(x_m) / x_m) * Float64(Float64(1.0 / Float64(1.0 / tan(Float64(x_m * 0.5)))) / x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (sin(x_m) / x_m) * ((1.0 / (1.0 / tan((x_m * 0.5)))) / x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * N[(N[(1.0 / N[(1.0 / N[Tan[N[(x$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

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

   1. add-exp-log 54.2%

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

   2. sub-neg 54.2%

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

   3. log1p-def 54.2%

      \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(-\cos x\right)}}}{x \cdot x} \]

4. Applied egg-rr 54.2%

   \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(-\cos x\right)}}}{x \cdot x} \]
5. Step-by-step derivation

   1. log1p-udef 54.2%

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

   2. add-exp-log 54.2%

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

   3. sub-neg 54.2%

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

   4. flip-- 53.9%

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

   5. metadata-eval 53.9%

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

   6. 1-sub-cos 76.8%

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

   7. associate-/l* 76.7%

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

   8. frac-2neg 76.7%

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

6. Applied egg-rr 76.7%

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

   1. *-lft-identity 76.7%

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

   2. associate-/l* 76.7%

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

   3. hang-0p-tan 77.0%

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

8. Simplified 77.0%

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

   1. div-inv 77.0%

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

   2. times-frac 99.7%

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

   3. add-sqr-sqrt 52.4%

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

   4. sqrt-unprod 53.1%

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

   5. sqr-neg 53.1%

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

   6. sqrt-unprod 10.7%

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

   7. add-sqr-sqrt 24.9%

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

10. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\frac{\sin x}{x} \cdot \frac{\frac{1}{\frac{1}{\tan \left(x \cdot 0.5\right)}}}{x}} \]

11. Final simplification 99.7%

    \[\leadsto \frac{\sin x}{x} \cdot \frac{\frac{1}{\frac{1}{\tan \left(x \cdot 0.5\right)}}}{x} \]
12. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{x_m}}{\frac{\frac{x_m}{\sin x_m}}{\tan \left(x_m \cdot 0.5\right)}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (/ 1.0 x_m) (/ (/ x_m (sin x_m)) (tan (* x_m 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 / x_m) / ((x_m / sin(x_m)) / tan((x_m * 0.5)));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (1.0d0 / x_m) / ((x_m / sin(x_m)) / tan((x_m * 0.5d0)))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 / x_m) / ((x_m / Math.sin(x_m)) / Math.tan((x_m * 0.5)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 / x_m) / ((x_m / math.sin(x_m)) / math.tan((x_m * 0.5)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 / x_m) / Float64(Float64(x_m / sin(x_m)) / tan(Float64(x_m * 0.5))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 / x_m) / ((x_m / sin(x_m)) / tan((x_m * 0.5)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(x$95$m / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / N[Tan[N[(x$95$m * 0.5), $MachinePrecision]], $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. associate-/r* 55.6%

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

   2. div-inv 55.5%

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

4. Applied egg-rr 55.5%

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

   1. *-commutative 55.5%

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

   2. clear-num 55.5%

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

   3. un-div-inv 55.5%

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

6. Applied egg-rr 55.5%

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

   1. flip-- 55.3%

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

   2. metadata-eval 55.3%

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

   3. 1-sub-cos 77.5%

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

   4. associate-/r/ 77.5%

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

   5. pow2 77.5%

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

8. Applied egg-rr 77.5%

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

   1. associate-*l/ 77.5%

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

   2. associate-/l* 77.5%

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

   3. unpow2 77.5%

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

   4. associate-*r/ 77.5%

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

   5. hang-0p-tan 77.8%

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

10. Simplified 77.8%

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

    1. expm1-log1p-u 40.8%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sin x \cdot \tan \left(\frac{x}{2}\right)}\right)\right)}} \]

    2. expm1-udef 40.8%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\sin x \cdot \tan \left(\frac{x}{2}\right)}\right)} - 1}} \]

    3. *-commutative 40.8%

       \[\leadsto \frac{\frac{1}{x}}{e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\tan \left(\frac{x}{2}\right) \cdot \sin x}}\right)} - 1} \]

    4. div-inv 40.8%

       \[\leadsto \frac{\frac{1}{x}}{e^{\mathsf{log1p}\left(\frac{x}{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \sin x}\right)} - 1} \]

    5. metadata-eval 40.8%

       \[\leadsto \frac{\frac{1}{x}}{e^{\mathsf{log1p}\left(\frac{x}{\tan \left(x \cdot \color{blue}{0.5}\right) \cdot \sin x}\right)} - 1} \]

12. Applied egg-rr 40.8%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\tan \left(x \cdot 0.5\right) \cdot \sin x}\right)} - 1}} \]
13. Step-by-step derivation

    1. expm1-def 40.8%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan \left(x \cdot 0.5\right) \cdot \sin x}\right)\right)}} \]

    2. expm1-log1p 77.8%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{x}{\tan \left(x \cdot 0.5\right) \cdot \sin x}}} \]

    3. *-lft-identity 77.8%

       \[\leadsto \frac{\frac{1}{x}}{\frac{\color{blue}{1 \cdot x}}{\tan \left(x \cdot 0.5\right) \cdot \sin x}} \]

    4. times-frac 99.7%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{1}{\tan \left(x \cdot 0.5\right)} \cdot \frac{x}{\sin x}}} \]

    5. associate-*l/ 99.7%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{1 \cdot \frac{x}{\sin x}}{\tan \left(x \cdot 0.5\right)}}} \]

    6. *-lft-identity 99.7%

       \[\leadsto \frac{\frac{1}{x}}{\frac{\color{blue}{\frac{x}{\sin x}}}{\tan \left(x \cdot 0.5\right)}} \]

14. Simplified 99.7%

    \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{\frac{x}{\sin x}}{\tan \left(x \cdot 0.5\right)}}} \]

15. Final simplification 99.7%

    \[\leadsto \frac{\frac{1}{x}}{\frac{\frac{x}{\sin x}}{\tan \left(x \cdot 0.5\right)}} \]
16. Add Preprocessing

Alternative 6: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.03:\\ \;\;\;\;\frac{\frac{1}{x_m}}{0.008333333333333333 \cdot {x_m}^{3} + \left(x_m \cdot 0.16666666666666666 + \frac{1}{x_m} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x_m}{x_m}}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.03)
   (/
    (/ 1.0 x_m)
    (+
     (* 0.008333333333333333 (pow x_m 3.0))
     (+ (* x_m 0.16666666666666666) (* (/ 1.0 x_m) 2.0))))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.03) {
		tmp = (1.0 / x_m) / ((0.008333333333333333 * pow(x_m, 3.0)) + ((x_m * 0.16666666666666666) + ((1.0 / x_m) * 2.0)));
	} else {
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.03d0) then
        tmp = (1.0d0 / x_m) / ((0.008333333333333333d0 * (x_m ** 3.0d0)) + ((x_m * 0.16666666666666666d0) + ((1.0d0 / x_m) * 2.0d0)))
    else
        tmp = ((1.0d0 - cos(x_m)) / x_m) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.03) {
		tmp = (1.0 / x_m) / ((0.008333333333333333 * Math.pow(x_m, 3.0)) + ((x_m * 0.16666666666666666) + ((1.0 / x_m) * 2.0)));
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.03:
		tmp = (1.0 / x_m) / ((0.008333333333333333 * math.pow(x_m, 3.0)) + ((x_m * 0.16666666666666666) + ((1.0 / x_m) * 2.0)))
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.03)
		tmp = Float64(Float64(1.0 / x_m) / Float64(Float64(0.008333333333333333 * (x_m ^ 3.0)) + Float64(Float64(x_m * 0.16666666666666666) + Float64(Float64(1.0 / x_m) * 2.0))));
	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 <= 0.03)
		tmp = (1.0 / x_m) / ((0.008333333333333333 * (x_m ^ 3.0)) + ((x_m * 0.16666666666666666) + ((1.0 / x_m) * 2.0)));
	else
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.03], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(0.008333333333333333 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / x$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $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 < 0.029999999999999999

   1. Initial program 34.6%

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

      1. associate-/r* 36.3%

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

      2. div-inv 36.3%

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

   4. Applied egg-rr 36.3%

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

      1. *-commutative 36.3%

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

      2. clear-num 36.3%

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

      3. un-div-inv 36.3%

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

   6. Applied egg-rr 36.3%

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

   7. Taylor expanded in x around 0 81.5%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{0.008333333333333333 \cdot {x}^{3} + \left(0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)}} \]

   if 0.029999999999999999 < x

   1. Initial program 98.9%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. un-div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\frac{\frac{1}{x}}{0.008333333333333333 \cdot {x}^{3} + \left(x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 34.6%

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

   3. Taylor expanded in x around 0 67.4%

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

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.9%

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

4. Final simplification 77.0%

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

Alternative 8: 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 34.6%

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

   3. Taylor expanded in x around 0 67.4%

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

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.9%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. un-div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 77.1%

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

Alternative 9: 78.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{x_m}}{x_m \cdot 0.16666666666666666 + \frac{1}{x_m} \cdot 2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (/ 1.0 x_m) (+ (* x_m 0.16666666666666666) (* (/ 1.0 x_m) 2.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 / x_m) / ((x_m * 0.16666666666666666) + ((1.0 / x_m) * 2.0));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (1.0d0 / x_m) / ((x_m * 0.16666666666666666d0) + ((1.0d0 / x_m) * 2.0d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (1.0 / x_m) / ((x_m * 0.16666666666666666) + ((1.0 / x_m) * 2.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (1.0 / x_m) / ((x_m * 0.16666666666666666) + ((1.0 / x_m) * 2.0))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 / x_m) / Float64(Float64(x_m * 0.16666666666666666) + Float64(Float64(1.0 / x_m) * 2.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (1.0 / x_m) / ((x_m * 0.16666666666666666) + ((1.0 / x_m) * 2.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(x$95$m * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / x$95$m), $MachinePrecision] * 2.0), $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. associate-/r* 55.6%

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

   2. div-inv 55.5%

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

4. Applied egg-rr 55.5%

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

   1. *-commutative 55.5%

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

   2. clear-num 55.5%

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

   3. un-div-inv 55.5%

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

6. Applied egg-rr 55.5%

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

7. Taylor expanded in x around 0 74.3%

   \[\leadsto \frac{\frac{1}{x}}{\color{blue}{0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}}} \]

8. Final simplification 74.3%

   \[\leadsto \frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2} \]
9. Add Preprocessing

Alternative 10: 78.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{x_m \cdot 0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5) 0.5 (/ (/ 1.0 x_m) (* x_m 0.16666666666666666))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) / (x_m * 0.16666666666666666);
	}
	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 <= 3.5d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x_m) / (x_m * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) / (x_m * 0.16666666666666666);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.5:
		tmp = 0.5
	else:
		tmp = (1.0 / x_m) / (x_m * 0.16666666666666666)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(x_m * 0.16666666666666666));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 3.5)
		tmp = 0.5;
	else
		tmp = (1.0 / x_m) / (x_m * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.5], 0.5, N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(x$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.6%

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

   if 3.5 < x

   1. Initial program 98.9%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in x around 0 54.4%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}}} \]

   8. Taylor expanded in x around inf 54.4%

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{0.16666666666666666 \cdot x}} \]
   9. Step-by-step derivation

      1. *-commutative 54.4%

         \[\leadsto \frac{\frac{1}{x}}{\color{blue}{x \cdot 0.16666666666666666}} \]

   10. Simplified 54.4%

       \[\leadsto \frac{\frac{1}{x}}{\color{blue}{x \cdot 0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot 0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.4% 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 48.3%

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

4. Final simplification 48.3%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))