cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.8%

Time: 10.8s

Alternatives: 10

Speedup: 34.9×

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: 50.9% 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

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

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(Float64(tan(Float64(x * 0.5)) / x) * 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[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

   1. flip-- 49.1%

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

   2. div-inv 49.1%

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

   3. metadata-eval 49.1%

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

   4. 1-sub-cos 74.0%

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

   5. pow2 74.0%

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

3. Applied egg-rr 74.0%

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

   1. unpow2 74.0%

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

   2. associate-*l* 74.0%

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

   3. associate-*r/ 74.0%

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

   4. *-rgt-identity 74.0%

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

   5. hang-0p-tan 74.3%

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

5. Simplified 74.3%

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

   1. *-commutative 74.3%

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

   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%

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

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. flip-- 49.1%

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

   2. div-inv 49.1%

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

   3. metadata-eval 49.1%

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

   4. 1-sub-cos 74.0%

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

   5. pow2 74.0%

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

3. Applied egg-rr 74.0%

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

   1. unpow2 74.0%

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

   2. associate-*l* 74.0%

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

   3. associate-*r/ 74.0%

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

   4. *-rgt-identity 74.0%

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

   5. hang-0p-tan 74.3%

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

5. Simplified 74.3%

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

   1. *-commutative 74.3%

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

   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%

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

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]
8. Step-by-step derivation

   1. associate-*r/ 99.8%

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

9. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \sin x}{x}} \]
10. Step-by-step derivation

    1. expm1-log1p-u 99.8%

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

    2. expm1-udef 78.0%

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

    3. associate-/l* 78.0%

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

11. Applied egg-rr 78.0%

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

    1. expm1-def 99.8%

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

    2. expm1-log1p 99.8%

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

    3. associate-/l/ 99.0%

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

    4. associate-/r* 99.8%

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

    5. associate-/l* 75.4%

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

    6. associate-*r/ 99.8%

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

    7. *-rgt-identity 99.8%

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

    8. associate-*r/ 99.7%

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

    9. associate-*l* 99.7%

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

    10. *-commutative 99.7%

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

    11. associate-*r/ 99.7%

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

    12. *-rgt-identity 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 3: 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 49.4%

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

   1. flip-- 49.1%

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

   2. div-inv 49.1%

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

   3. metadata-eval 49.1%

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

   4. 1-sub-cos 74.0%

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

   5. pow2 74.0%

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

3. Applied egg-rr 74.0%

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

   1. unpow2 74.0%

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

   2. associate-*l* 74.0%

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

   3. associate-*r/ 74.0%

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

   4. *-rgt-identity 74.0%

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

   5. hang-0p-tan 74.3%

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

5. Simplified 74.3%

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

   1. *-commutative 74.3%

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

   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%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. flip-- 49.1%

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

   2. div-inv 49.1%

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

   3. metadata-eval 49.1%

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

   4. 1-sub-cos 74.0%

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

   5. pow2 74.0%

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

3. Applied egg-rr 74.0%

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

   1. unpow2 74.0%

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

   2. associate-*l* 74.0%

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

   3. associate-*r/ 74.0%

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

   4. *-rgt-identity 74.0%

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

   5. hang-0p-tan 74.3%

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

5. Simplified 74.3%

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

   1. *-commutative 74.3%

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

   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%

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

7. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]
8. Step-by-step derivation

   1. associate-*l/ 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;\frac{x \cdot 0.5 + -0.041666666666666664 \cdot {x}^{3}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (/ (+ (* x 0.5) (* -0.041666666666666664 (pow x 3.0))) x)
   (* (/ (- 1.0 (cos x)) x) (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = ((x * 0.5) + (-0.041666666666666664 * pow(x, 3.0))) / x;
	} else {
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0052d0) then
        tmp = ((x * 0.5d0) + ((-0.041666666666666664d0) * (x ** 3.0d0))) / x
    else
        tmp = ((1.0d0 - cos(x)) / x) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = ((x * 0.5) + (-0.041666666666666664 * Math.pow(x, 3.0))) / x;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = ((x * 0.5) + (-0.041666666666666664 * math.pow(x, 3.0))) / x
	else:
		tmp = ((1.0 - math.cos(x)) / x) * (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = Float64(Float64(Float64(x * 0.5) + Float64(-0.041666666666666664 * (x ^ 3.0))) / x);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0052)
		tmp = ((x * 0.5) + (-0.041666666666666664 * (x ^ 3.0))) / x;
	else
		tmp = ((1.0 - cos(x)) / x) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(N[(N[(x * 0.5), $MachinePrecision] + N[(-0.041666666666666664 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 33.7%

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

      1. flip-- 33.6%

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

      2. div-inv 33.6%

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

      3. metadata-eval 33.6%

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

      4. 1-sub-cos 66.3%

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

      5. pow2 66.3%

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

   3. Applied egg-rr 66.3%

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

      1. unpow2 66.3%

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

      2. associate-*l* 66.3%

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

      3. associate-*r/ 66.3%

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

      4. *-rgt-identity 66.3%

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

      5. hang-0p-tan 66.4%

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

   5. Simplified 66.4%

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

      1. *-commutative 66.4%

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

      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%

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

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0 67.2%

       \[\leadsto \frac{\color{blue}{-0.041666666666666664 \cdot {x}^{3} + 0.5 \cdot x}}{x} \]

   if 0.0051999999999999998 < x

   1. Initial program 99.4%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;\frac{x \cdot 0.5 + -0.041666666666666664 \cdot {x}^{3}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 6: 75.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = ((1.0 - cos(x)) / x) * (1.0 / 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 + ((-0.041666666666666664d0) * (x * x))
    else
        tmp = ((1.0d0 - cos(x)) / x) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 33.7%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 67.2%

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

      1. unpow2 67.2%

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

   4. Simplified 67.2%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.4%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 74.9%

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

Alternative 7: 75.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * (x * x));
	} 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 + ((-0.041666666666666664d0) * (x * x))
    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 + (-0.041666666666666664 * (x * x));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0052:
		tmp = 0.5 + (-0.041666666666666664 * (x * x))
	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(-0.041666666666666664 * Float64(x * x)));
	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 + (-0.041666666666666664 * (x * x));
	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[(-0.041666666666666664 * N[(x * x), $MachinePrecision]), $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 33.7%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 67.2%

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

      1. unpow2 67.2%

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

   4. Simplified 67.2%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.4%

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

4. Final simplification 74.9%

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

Alternative 8: 78.1% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.4%

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

   1. clear-num 49.4%

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

   2. inv-pow 49.4%

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

   3. associate-/l* 50.1%

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

3. Applied egg-rr 50.1%

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

   1. unpow-1 50.1%

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

   2. associate-/r/ 50.1%

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

5. Applied egg-rr 50.1%

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

6. Taylor expanded in x around 0 79.6%

   \[\leadsto \frac{1}{\color{blue}{2 + 0.16666666666666666 \cdot {x}^{2}}} \]
7. Step-by-step derivation

   1. *-commutative 79.6%

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

   2. unpow2 79.6%

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

8. Simplified 79.6%

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

9. Final simplification 79.6%

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

Alternative 9: 63.6% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 6.2e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.2e+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 <= 6.2d+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 <= 6.2e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 6.2e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.2e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 6.2e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 6.2e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000023e76

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]

   2. Taylor expanded in x around 0 64.8%

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

   if 6.20000000000000023e76 < x

   1. Initial program 99.5%

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

      1. div-sub 99.5%

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

      2. pow2 99.5%

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

      3. pow-flip 99.3%

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

      4. metadata-eval 99.3%

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

      5. div-inv 99.4%

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

      6. pow2 99.4%

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

      7. pow-flip 99.5%

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

      8. metadata-eval 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in x around 0 67.2%

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

      1. unpow2 67.2%

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

   6. Simplified 67.2%

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

   7. Taylor expanded in x around 0 67.3%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 27.2% 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 49.4%

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

   1. div-sub 49.4%

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

   2. pow2 49.4%

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

   3. pow-flip 49.5%

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

   4. metadata-eval 49.5%

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

   5. div-inv 49.5%

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

   6. pow2 49.5%

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

   7. pow-flip 50.2%

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

   8. metadata-eval 50.2%

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

3. Applied egg-rr 50.2%

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

4. Taylor expanded in x around 0 27.3%

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

   1. unpow2 27.3%

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

6. Simplified 27.3%

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

7. Taylor expanded in x around 0 28.0%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 28.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))