cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.4%

Time: 11.6s

Alternatives: 3

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000115:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000115) 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 <= 0.000115) {
		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 <= 0.000115d0) 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 <= 0.000115) {
		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 <= 0.000115:
		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 <= 0.000115)
		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 <= 0.000115)
		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, 0.000115], 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 < 1.15e-4

   1. Initial program 35.5%

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

   3. Taylor expanded in x around 0 66.8%

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

   if 1.15e-4 < x

   1. Initial program 98.8%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.0%

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

      3. div-inv 98.0%

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

      4. cbrt-prod 97.8%

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

      5. unpow-prod-down 97.8%

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

      6. pow3 97.9%

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

      7. add-cube-cbrt 97.9%

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

      8. pow2 97.9%

         \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{\frac{1}{\color{blue}{{x}^{2}}}}\right)}^{3} \]

      9. pow-flip 98.7%

         \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{\color{blue}{{x}^{\left(-2\right)}}}\right)}^{3} \]

      10. metadata-eval 98.7%

          \[\leadsto \left(1 - \cos x\right) \cdot {\left(\sqrt[3]{{x}^{\color{blue}{-2}}}\right)}^{3} \]

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\left(1 - \cos x\right) \cdot {\left(\sqrt[3]{{x}^{-2}}\right)}^{3}} \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 99.3%

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

      2. metadata-eval 99.3%

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

      3. pow-prod-up 99.1%

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

      4. inv-pow 99.1%

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

      5. inv-pow 99.1%

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

      6. frac-2neg 99.1%

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

      7. metadata-eval 99.1%

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

      8. frac-2neg 99.1%

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

      9. metadata-eval 99.1%

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

      10. frac-times 98.7%

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

      11. metadata-eval 98.7%

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

   6. Applied egg-rr 98.7%

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

      1. div-inv 98.8%

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

      2. sqr-neg 98.8%

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

      3. associate-/r* 99.4%

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

   8. Applied egg-rr 99.4%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000115:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000115) 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 <= 0.000115) {
		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 <= 0.000115d0) 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 <= 0.000115) {
		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 <= 0.000115:
		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 <= 0.000115)
		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 <= 0.000115)
		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, 0.000115], 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 < 1.15e-4

   1. Initial program 35.5%

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

   3. Taylor expanded in x around 0 66.8%

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

   if 1.15e-4 < x

   1. Initial program 98.8%

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

Alternative 3: 51.2% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 52.6%

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

3. Taylor expanded in x around 0 50.0%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))