cos2 (problem 3.4.1)

Percentage Accurate: 51.5% → 99.8%

Time: 10.3s

Alternatives: 6

Speedup: 17.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (/ (tan (* x 0.5)) x) (/ (sin x) x)))

C

double code(double x) {
	return (tan((x * 0.5)) / x) * (sin(x) / x);
}

Fortran

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

Java

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

Python

def code(x):
	return (math.tan((x * 0.5)) / x) * (math.sin(x) / x)

Julia

function code(x)
	return Float64(Float64(tan(Float64(x * 0.5)) / x) * Float64(sin(x) / x))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.0%

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

   1. flip-- 54.8%

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

   2. clear-num 54.8%

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

   3. metadata-eval 54.8%

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

   4. pow2 54.8%

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

4. Applied egg-rr 54.8%

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

   1. unpow2 54.8%

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

   2. 1-sub-cos 77.6%

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

6. Applied egg-rr 77.6%

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

7. Taylor expanded in x around inf 78.3%

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

   1. unpow2 78.3%

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

   2. associate-*r/ 78.3%

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

   3. hang-0p-tan 78.6%

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

9. Simplified 78.6%

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

    1. *-commutative 78.6%

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

    2. times-frac 99.8%

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

    3. div-inv 99.8%

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

    4. metadata-eval 99.8%

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

11. Applied egg-rr 99.8%

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

Alternative 2: 74.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{1 - \cos x}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0048)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ 1.0 (/ x (/ (- 1.0 (cos x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0048) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = 1.0 / (x / ((1.0 - cos(x)) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0048d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = 1.0d0 / (x / ((1.0d0 - cos(x)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0048) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = 1.0 / (x / ((1.0 - Math.cos(x)) / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0048:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = 1.0 / (x / ((1.0 - math.cos(x)) / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0048)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(1.0 / Float64(x / Float64(Float64(1.0 - cos(x)) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0048)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = 1.0 / (x / ((1.0 - cos(x)) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0048], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x / N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00479999999999999958

   1. Initial program 40.8%

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

   3. Taylor expanded in x around 0 61.4%

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

      1. *-commutative 61.4%

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

   5. Simplified 61.4%

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

   if 0.00479999999999999958 < x

   1. Initial program 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow2 99.5%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. sqr-pow 99.5%

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

      3. metadata-eval 99.5%

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

      4. metadata-eval 99.5%

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

      5. pow-prod-down 99.5%

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

      6. inv-pow 99.5%

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

      7. div-inv 99.5%

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

      8. associate-/r* 99.5%

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

      9. clear-num 99.5%

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

   6. Applied egg-rr 99.5%

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

Alternative 3: 74.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0048)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0048) {
		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 <= 0.0048d0) 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 <= 0.0048) {
		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 <= 0.0048:
		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 <= 0.0048)
		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 <= 0.0048)
		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, 0.0048], 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 < 0.00479999999999999958

   1. Initial program 40.8%

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

   3. Taylor expanded in x around 0 61.4%

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

      1. *-commutative 61.4%

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

   5. Simplified 61.4%

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

   if 0.00479999999999999958 < x

   1. Initial program 99.5%

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

      1. div-sub 99.2%

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

      2. pow2 99.2%

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

      3. pow-flip 99.1%

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

      4. metadata-eval 99.1%

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

      5. div-inv 99.3%

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

      6. pow2 99.3%

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

      7. pow-flip 99.3%

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

      8. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. distribute-rgt-out-- 99.5%

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

      3. *-commutative 99.5%

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

      4. metadata-eval 99.5%

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

      5. pow-flip 99.5%

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

      6. pow2 99.5%

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

      7. div-inv 99.5%

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

      8. associate-/r* 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. Add Preprocessing

Alternative 4: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0048)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0048) {
		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 <= 0.0048d0) 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 <= 0.0048) {
		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 <= 0.0048:
		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 <= 0.0048)
		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 <= 0.0048)
		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, 0.0048], 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 < 0.00479999999999999958

   1. Initial program 40.8%

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

   3. Taylor expanded in x around 0 61.4%

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

      1. *-commutative 61.4%

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

   5. Simplified 61.4%

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

   if 0.00479999999999999958 < x

   1. Initial program 99.5%

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

Alternative 5: 62.9% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 9.5e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 9.5e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	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 = 0.0d0
    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 = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.5e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 9.5e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000003e76

   1. Initial program 45.5%

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

   3. Taylor expanded in x around 0 57.8%

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

   if 9.5000000000000003e76 < x

   1. Initial program 99.7%

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

      1. add-log-exp 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0 69.1%

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

   6. Taylor expanded in x around 0 69.1%

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

Alternative 6: 28.0% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 55.0%

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

   1. add-log-exp 54.8%

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

4. Applied egg-rr 54.8%

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

5. Taylor expanded in x around 0 30.3%

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

6. Taylor expanded in x around 0 30.9%

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))