cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.4%

Time: 12.2s

Alternatives: 6

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: 50.8% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x_m}}{\frac{x_m}{1 - \cos x_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000118) 0.5 (/ (/ 1.0 x_m) (/ x_m (- 1.0 (cos x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) / (x_m / (1.0 - cos(x_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000118d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x_m) / (x_m / (1.0d0 - cos(x_m)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) / (x_m / (1.0 - Math.cos(x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.000118:
		tmp = 0.5
	else:
		tmp = (1.0 / x_m) / (x_m / (1.0 - math.cos(x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000118)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(x_m / Float64(1.0 - cos(x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.000118)
		tmp = 0.5;
	else
		tmp = (1.0 / x_m) / (x_m / (1.0 - cos(x_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000118], 0.5, N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(x$95$m / N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e-4

   1. Initial program 35.4%

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

   2. Taylor expanded in x around 0 66.7%

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

   if 1.18e-4 < x

   1. Initial program 98.5%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. add-cube-cbrt 98.5%

         \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}}}{x} \]

      3. unpow3 98.6%

         \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3}}}{x} \]

      4. clear-num 98.6%

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{1}{\frac{x}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3}}}} \]

      5. un-div-inv 98.6%

         \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{x}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3}}}} \]

      6. unpow3 98.5%

         \[\leadsto \frac{\frac{1}{x}}{\frac{x}{\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}}}} \]

      7. add-cube-cbrt 99.3%

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

   5. Applied egg-rr 99.3%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{x}{1 - \cos x}}\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x_m}{x_m \cdot x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000118) 0.5 (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000118d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - cos(x_m)) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.000118:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x_m)) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000118)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.000118)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000118], 0.5, N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e-4

   1. Initial program 35.4%

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

   2. Taylor expanded in x around 0 66.7%

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

   if 1.18e-4 < x

   1. Initial program 98.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x_m}{x_m}}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000118) 0.5 (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000118d0) then
        tmp = 0.5d0
    else
        tmp = ((1.0d0 - cos(x_m)) / x_m) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.000118:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000118)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / x_m) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.000118)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000118], 0.5, N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e-4

   1. Initial program 35.4%

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

   2. Taylor expanded in x around 0 66.7%

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

   if 1.18e-4 < x

   1. Initial program 98.5%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. un-div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 4: 78.9% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(x$95$m * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / x$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. associate-/r* 52.0%

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

   2. div-inv 52.0%

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

3. Applied egg-rr 52.0%

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

   1. *-commutative 52.0%

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

   2. add-cube-cbrt 51.7%

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}}}{x} \]

   3. unpow3 51.8%

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3}}}{x} \]

   4. clear-num 51.7%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{1}{\frac{x}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3}}}} \]

   5. un-div-inv 51.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{x}{{\left(\sqrt[3]{1 - \cos x}\right)}^{3}}}} \]

   6. unpow3 51.7%

      \[\leadsto \frac{\frac{1}{x}}{\frac{x}{\color{blue}{\left(\sqrt[3]{1 - \cos x} \cdot \sqrt[3]{1 - \cos x}\right) \cdot \sqrt[3]{1 - \cos x}}}} \]

   7. add-cube-cbrt 52.0%

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

5. Applied egg-rr 52.0%

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

6. Taylor expanded in x around 0 78.9%

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

7. Final simplification 78.9%

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

Alternative 5: 76.4% accurate, 34.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.3e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.3e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.3d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.3e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.3e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.3e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.3e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.3e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.3000000000000001e77

   1. Initial program 39.0%

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

   2. Taylor expanded in x around 0 63.2%

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

   if 1.3000000000000001e77 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in x around 0 61.4%

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

   3. Taylor expanded in x around 0 61.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 28.1% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 50.9%

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

2. Taylor expanded in x around 0 28.1%

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

3. Taylor expanded in x around 0 28.8%

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

4. Final simplification 28.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023339 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))