cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.6%

Time: 9.6s

Alternatives: 4

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: 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0045)
   (+ 0.5 (* -0.041666666666666664 (* x_m x_m)))
   (* (pow x_m -2.0) (- 1.0 (cos x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} else {
		tmp = pow(x_m, -2.0) * (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 <= 0.0045d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m * x_m))
    else
        tmp = (x_m ** (-2.0d0)) * (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 <= 0.0045) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} else {
		tmp = Math.pow(x_m, -2.0) * (1.0 - Math.cos(x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0045:
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m))
	else:
		tmp = math.pow(x_m, -2.0) * (1.0 - math.cos(x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0045)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x_m * x_m)));
	else
		tmp = Float64((x_m ^ -2.0) * 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 <= 0.0045)
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	else
		tmp = (x_m ^ -2.0) * (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, 0.0045], N[(0.5 + N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 29.4%

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

   3. Taylor expanded in x around 0 71.7%

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

      1. pow2 71.7%

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

   5. Applied egg-rr 71.7%

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

   if 0.00449999999999999966 < 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%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0045)
   (+ 0.5 (* -0.041666666666666664 (* x_m x_m)))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0045d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m * x_m))
    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 <= 0.0045) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0045:
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m))
	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 <= 0.0045)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x_m * x_m)));
	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 <= 0.0045)
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	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, 0.0045], N[(0.5 + N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 0.00449999999999999966

   1. Initial program 29.4%

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

   3. Taylor expanded in x around 0 71.7%

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

      1. pow2 71.7%

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

   5. Applied egg-rr 71.7%

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

   if 0.00449999999999999966 < x

   1. Initial program 99.5%

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

      1. add-cube-cbrt 98.5%

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

      2. pow3 98.5%

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

      3. div-inv 98.5%

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

      4. cbrt-prod 98.5%

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

      5. unpow-prod-down 98.5%

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

      6. pow3 98.4%

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

      7. add-cube-cbrt 99.0%

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

      8. pow2 99.0%

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

      9. pow-flip 99.0%

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

      10. metadata-eval 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. rem-cube-cbrt 99.5%

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

      2. metadata-eval 99.5%

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

      3. pow-flip 99.5%

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

      4. pow2 99.5%

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

      5. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.4%

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

      2. div-inv 99.5%

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

   8. 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 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0045:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0045)
   (+ 0.5 (* -0.041666666666666664 (* x_m x_m)))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0045) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0045d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m * x_m))
    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 <= 0.0045) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0045:
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m))
	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 <= 0.0045)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x_m * x_m)));
	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 <= 0.0045)
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	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, 0.0045], N[(0.5 + N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < 0.00449999999999999966

   1. Initial program 29.4%

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

   3. Taylor expanded in x around 0 71.7%

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

      1. pow2 71.7%

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

   5. Applied egg-rr 71.7%

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

   if 0.00449999999999999966 < 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 4: 50.4% accurate, 15.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 + -0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+ 0.5 (* -0.041666666666666664 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5 + (-0.041666666666666664 * (x_m * x_m));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.5d0 + ((-0.041666666666666664d0) * (x_m * x_m))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5 + (-0.041666666666666664 * (x_m * x_m));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5 + (-0.041666666666666664 * (x_m * x_m))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(0.5 + Float64(-0.041666666666666664 * Float64(x_m * x_m)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(0.5 + N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.1%

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

3. Taylor expanded in x around 0 55.0%

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

   1. pow2 55.0%

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

5. Applied egg-rr 55.0%

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

Reproduce

herbie shell --seed 2024179 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))