cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.8%

Time: 12.4s

Alternatives: 7

Speedup: 107.0×

• could not determine a ground truth

  1. x = -6.528633572252678e+99

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: 51.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. flip-- 51.5%

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

   2. div-inv 51.5%

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

   3. metadata-eval 51.5%

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

   4. pow2 51.5%

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

4. Applied egg-rr 51.5%

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

   1. associate-*r/ 51.5%

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

   2. *-rgt-identity 51.5%

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

6. Simplified 51.5%

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

   1. unpow2 51.5%

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

   2. 1-sub-cos 75.7%

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

8. Applied egg-rr 75.7%

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

   1. associate-/l* 75.7%

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

   2. times-frac 99.7%

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

   3. hang-0p-tan 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 74.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = pow(x, -2.0) * (1.0 - cos(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 = (x ** (-2.0d0)) * (1.0d0 - cos(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 = Math.pow(x, -2.0) * (1.0 - Math.cos(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 = math.pow(x, -2.0) * (1.0 - math.cos(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((x ^ -2.0) * Float64(1.0 - cos(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 = (x ^ -2.0) * (1.0 - cos(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[Power[x, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 39.8%

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

   3. Taylor expanded in x around 0 62.1%

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

      1. *-commutative 62.1%

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

   5. Simplified 62.1%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.3%

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

      1. clear-num 99.1%

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

      2. associate-/r/ 99.3%

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

      3. pow2 99.3%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 69.6%

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

Alternative 3: 74.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((cos(x) + -1.0) * (-1.0 / 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 = ((cos(x) + (-1.0d0)) * ((-1.0d0) / 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 = ((Math.cos(x) + -1.0) * (-1.0 / 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 = ((math.cos(x) + -1.0) * (-1.0 / 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(cos(x) + -1.0) * Float64(-1.0 / 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 = ((cos(x) + -1.0) * (-1.0 / 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[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 39.8%

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

   3. Taylor expanded in x around 0 62.1%

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

      1. *-commutative 62.1%

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

   5. Simplified 62.1%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.3%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. frac-2neg 99.5%

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

      3. associate-*r/ 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 69.5%

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

Alternative 4: 74.6% 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 39.8%

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

   3. Taylor expanded in x around 0 62.1%

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

      1. *-commutative 62.1%

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

   5. Simplified 62.1%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.3%

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

4. Final simplification 69.5%

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

Alternative 5: 74.7% 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 39.8%

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

   3. Taylor expanded in x around 0 62.1%

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

      1. *-commutative 62.1%

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

   5. Simplified 62.1%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.3%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. expm1-log1p-u 99.5%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x} \cdot \frac{1}{x} \]

   6. Applied egg-rr 99.5%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x} \cdot \frac{1}{x} \]
   7. Step-by-step derivation

      1. expm1-undefine 99.3%

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

      2. sub-neg 99.3%

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

      3. log1p-undefine 99.3%

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

      4. rem-exp-log 99.4%

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

      5. associate-+r- 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   8. Simplified 99.4%

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

      1. frac-times 99.1%

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

      2. *-rgt-identity 99.1%

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

      3. +-commutative 99.1%

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

      4. associate-+r- 99.3%

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

      5. metadata-eval 99.3%

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

      6. rem-log-exp 99.2%

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

      7. associate-/r* 99.6%

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

      8. rem-log-exp 99.6%

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

   10. 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 69.5%

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

Alternative 6: 63.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{0}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45e+77) 0.5 (* (/ 1.0 x) (/ 0.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * (0.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d+77) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x) * (0.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * (0.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45e+77:
		tmp = 0.5
	else:
		tmp = (1.0 / x) * (0.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45e+77)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x) * Float64(0.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45e+77)
		tmp = 0.5;
	else
		tmp = (1.0 / x) * (0.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45e+77], 0.5, N[(N[(1.0 / x), $MachinePrecision] * N[(0.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4500000000000001e77

   1. Initial program 43.6%

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

   3. Taylor expanded in x around 0 59.3%

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

   if 1.4500000000000001e77 < x

   1. Initial program 99.3%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. expm1-log1p-u 99.7%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x} \cdot \frac{1}{x} \]

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x} \cdot \frac{1}{x} \]
   7. Step-by-step derivation

      1. expm1-undefine 99.5%

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

      2. sub-neg 99.5%

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

      3. log1p-undefine 99.6%

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

      4. rem-exp-log 99.6%

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

      5. associate-+r- 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 82.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{0}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.3% 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 51.7%

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

3. Taylor expanded in x around 0 51.2%

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

4. Final simplification 51.2%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))