cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.8%

Time: 11.1s

Alternatives: 6

Speedup: 107.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.6% 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

\[\begin{array}{l} \\ \frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (sin x) (/ (tan (/ x 2.0)) x)) x))

C

double code(double x) {
	return (sin(x) * (tan((x / 2.0)) / x)) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sin(x) * (tan((x / 2.0d0)) / x)) / x
end function

Java

public static double code(double x) {
	return (Math.sin(x) * (Math.tan((x / 2.0)) / x)) / x;
}

Python

def code(x):
	return (math.sin(x) * (math.tan((x / 2.0)) / x)) / x

Julia

function code(x)
	return Float64(Float64(sin(x) * Float64(tan(Float64(x / 2.0)) / x)) / x)
end

MATLAB

function tmp = code(x)
	tmp = (sin(x) * (tan((x / 2.0)) / x)) / x;
end

Wolfram

code[x_] := N[(N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 48.1%

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

   1. add-sqr-sqrt 48.0%

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

   2. pow2 48.0%

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

   3. sqrt-div 48.0%

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

   4. sqrt-prod 26.1%

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

   5. add-sqr-sqrt 49.5%

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

4. Applied egg-rr 49.5%

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

   1. unpow2 49.5%

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

   2. frac-times 48.0%

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

   3. add-sqr-sqrt 48.1%

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

   4. flip-- 48.0%

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

   5. metadata-eval 48.0%

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

   6. unpow2 48.0%

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

   7. un-div-inv 48.0%

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

   8. associate-/r* 49.5%

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

   9. un-div-inv 49.5%

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

   10. metadata-eval 49.5%

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

   11. unpow2 49.5%

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

   12. flip-- 49.6%

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

6. Applied egg-rr 49.6%

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

   1. flip-- 49.5%

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

   2. metadata-eval 49.5%

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

   3. 1-sub-cos 75.5%

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

   4. un-div-inv 75.4%

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

   5. associate-*l* 75.5%

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

   6. *-un-lft-identity 75.5%

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

   7. times-frac 99.7%

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

   8. un-div-inv 99.7%

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

8. Applied egg-rr 99.7%

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

   1. /-rgt-identity 99.7%

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

   2. hang-0p-tan 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

    \[\leadsto \frac{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}{x} \]
12. Add Preprocessing

Alternative 2: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0052d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0052)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 31.1%

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

   3. Taylor expanded in x around 0 70.8%

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

      1. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if 0.0051999999999999998 < 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 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0052d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0052)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 31.1%

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

   3. Taylor expanded in x around 0 70.8%

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

      1. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if 0.0051999999999999998 < x

   1. Initial program 98.9%

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

      1. add-sqr-sqrt 98.7%

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

      2. pow2 98.7%

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

      3. sqrt-div 98.7%

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

      4. sqrt-prod 98.7%

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

      5. add-sqr-sqrt 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. frac-times 98.9%

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

      3. add-sqr-sqrt 98.9%

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

      4. flip-- 99.0%

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

      5. metadata-eval 99.0%

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

      6. unpow2 99.0%

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

      7. un-div-inv 98.9%

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

      8. associate-/r* 99.0%

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

      9. un-div-inv 99.1%

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

      10. metadata-eval 99.1%

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

      11. unpow2 99.1%

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

      12. flip-- 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 9.5e+76) 0.5 (/ (+ (/ 1.0 x) (/ -1.0 x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 9.5d+76) then
        tmp = 0.5d0
    else
        tmp = ((1.0d0 / x) + ((-1.0d0) / x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.5e+76:
		tmp = 0.5
	else:
		tmp = ((1.0 / x) + (-1.0 / x)) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9.5e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(-1.0 / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9.5e+76)
		tmp = 0.5;
	else
		tmp = ((1.0 / x) + (-1.0 / x)) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9.5e+76], 0.5, N[(N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000003e76

   1. Initial program 36.8%

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

   3. Taylor expanded in x around 0 65.6%

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

   if 9.5000000000000003e76 < x

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.4%

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

      2. pow2 99.4%

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

      3. sqrt-div 99.5%

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

      4. sqrt-prod 99.5%

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

      5. add-sqr-sqrt 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow2 99.7%

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

      2. frac-times 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. flip-- 99.3%

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

      5. metadata-eval 99.3%

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

      6. unpow2 99.3%

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

      7. un-div-inv 99.3%

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

      8. associate-/r* 99.5%

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

      9. un-div-inv 99.5%

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

      10. metadata-eval 99.5%

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

      11. unpow2 99.5%

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

      12. flip-- 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. div-sub 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around 0 73.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (* x (+ (* x 0.16666666666666666) (* 2.0 (/ 1.0 x))))))

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (x * ((x * 0.16666666666666666) + (2.0 * (1.0 / x))));
}

Python

def code(x):
	return 1.0 / (x * ((x * 0.16666666666666666) + (2.0 * (1.0 / x))))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(2.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.1%

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

   1. add-sqr-sqrt 48.0%

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

   2. pow2 48.0%

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

   3. sqrt-div 48.0%

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

   4. sqrt-prod 26.1%

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

   5. add-sqr-sqrt 49.5%

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

4. Applied egg-rr 49.5%

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

   1. unpow2 49.5%

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

   2. frac-times 48.0%

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

   3. add-sqr-sqrt 48.1%

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

   4. pow2 48.1%

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

   5. div-sub 48.2%

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

   6. pow-flip 47.8%

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

   7. metadata-eval 47.8%

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

6. Applied egg-rr 47.8%

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

   1. metadata-eval 47.8%

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

   2. pow-flip 48.2%

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

   3. div-sub 48.1%

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

   4. pow2 48.1%

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

   5. associate-/l/ 49.6%

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

   6. div-inv 49.5%

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

   7. clear-num 49.5%

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

   8. frac-times 48.8%

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

   9. metadata-eval 48.8%

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

8. Applied egg-rr 48.8%

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

9. Taylor expanded in x around 0 81.4%

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

10. Final simplification 81.4%

    \[\leadsto \frac{1}{x \cdot \left(x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}\right)} \]
11. Add Preprocessing

Alternative 6: 51.9% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 48.1%

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

3. Taylor expanded in x around 0 54.4%

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

4. Final simplification 54.4%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))