cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.6%

Time: 9.7s

Alternatives: 7

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: 51.3% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.1:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.001388888888888889 + {x\_m}^{2} \cdot -2.48015873015873 \cdot 10^{-5}\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\mathsf{log1p}\left(-\cos x\_m\right)}}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.1)
   (+
    0.5
    (*
     (pow x_m 2.0)
     (-
      (*
       (pow x_m 2.0)
       (+ 0.001388888888888889 (* (pow x_m 2.0) -2.48015873015873e-5)))
      0.041666666666666664)))
   (/ (/ (exp (log1p (- (cos x_m)))) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.1) {
		tmp = 0.5 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * (0.001388888888888889 + (pow(x_m, 2.0) * -2.48015873015873e-5))) - 0.041666666666666664));
	} else {
		tmp = (exp(log1p(-cos(x_m))) / x_m) / x_m;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.1) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * (0.001388888888888889 + (Math.pow(x_m, 2.0) * -2.48015873015873e-5))) - 0.041666666666666664));
	} else {
		tmp = (Math.exp(Math.log1p(-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 <= 0.1:
		tmp = 0.5 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * (0.001388888888888889 + (math.pow(x_m, 2.0) * -2.48015873015873e-5))) - 0.041666666666666664))
	else:
		tmp = (math.exp(math.log1p(-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 <= 0.1)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.001388888888888889 + Float64((x_m ^ 2.0) * -2.48015873015873e-5))) - 0.041666666666666664)));
	else
		tmp = Float64(Float64(exp(log1p(Float64(-cos(x_m)))) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.1], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.001388888888888889 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[Log[1 + (-N[Cos[x$95$m], $MachinePrecision])], $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 37.8%

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

   3. Taylor expanded in x around 0 63.8%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.001388888888888889 + -2.48015873015873 \cdot 10^{-5} \cdot {x}^{2}\right) - 0.041666666666666664\right)} \]

   if 0.10000000000000001 < x

   1. Initial program 98.3%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.7%

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

      3. div-inv 97.5%

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

      4. pow2 97.5%

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

      5. pow-flip 98.9%

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

      6. metadata-eval 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. rem-cube-cbrt 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow-sqr 99.5%

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

      4. inv-pow 99.5%

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

      5. inv-pow 99.5%

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

      6. un-div-inv 99.6%

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

      7. associate-/l* 99.5%

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

      8. div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. add-exp-log 99.5%

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

      2. sub-neg 99.5%

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

      3. log1p-define 99.6%

         \[\leadsto \frac{\frac{e^{\color{blue}{\mathsf{log1p}\left(-\cos x\right)}}}{x}}{x} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(-\cos x\right)}}}{x}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.1:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.001388888888888889 + {x}^{2} \cdot -2.48015873015873 \cdot 10^{-5}\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\mathsf{log1p}\left(-\cos x\right)}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.031:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\mathsf{log1p}\left(-\cos x\_m\right)}}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.031)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* 0.001388888888888889 (* x_m x_m)) 0.041666666666666664)))
   (/ (/ (exp (log1p (- (cos x_m)))) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.031) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} else {
		tmp = (exp(log1p(-cos(x_m))) / x_m) / x_m;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.031) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} else {
		tmp = (Math.exp(Math.log1p(-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 <= 0.031:
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664))
	else:
		tmp = (math.exp(math.log1p(-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 <= 0.031)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(0.001388888888888889 * Float64(x_m * x_m)) - 0.041666666666666664)));
	else
		tmp = Float64(Float64(exp(log1p(Float64(-cos(x_m)))) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.031], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[Log[1 + (-N[Cos[x$95$m], $MachinePrecision])], $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.031

   1. Initial program 37.8%

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

   3. Taylor expanded in x around 0 64.1%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 64.1%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   5. Applied egg-rr 64.1%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. pow2 64.1%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   7. Applied egg-rr 64.1%

      \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664\right) \]

   if 0.031 < x

   1. Initial program 98.3%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.7%

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

      3. div-inv 97.5%

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

      4. pow2 97.5%

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

      5. pow-flip 98.9%

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

      6. metadata-eval 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. rem-cube-cbrt 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow-sqr 99.5%

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

      4. inv-pow 99.5%

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

      5. inv-pow 99.5%

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

      6. un-div-inv 99.6%

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

      7. associate-/l* 99.5%

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

      8. div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. add-exp-log 99.5%

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

      2. sub-neg 99.5%

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

      3. log1p-define 99.6%

         \[\leadsto \frac{\frac{e^{\color{blue}{\mathsf{log1p}\left(-\cos x\right)}}}{x}}{x} \]

   8. Applied egg-rr 99.6%

      \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(-\cos x\right)}}}{x}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.031:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.031)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* 0.001388888888888889 (* x_m x_m)) 0.041666666666666664)))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.031) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} 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 <= 0.031d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((0.001388888888888889d0 * (x_m * x_m)) - 0.041666666666666664d0))
    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 <= 0.031) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} 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 <= 0.031:
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664))
	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 <= 0.031)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(0.001388888888888889 * Float64(x_m * x_m)) - 0.041666666666666664)));
	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 <= 0.031)
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	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, 0.031], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $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 < 0.031

   1. Initial program 37.8%

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

   3. Taylor expanded in x around 0 64.1%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 64.1%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   5. Applied egg-rr 64.1%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. pow2 64.1%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   7. Applied egg-rr 64.1%

      \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664\right) \]

   if 0.031 < x

   1. Initial program 98.3%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.7%

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

      3. div-inv 97.5%

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

      4. pow2 97.5%

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

      5. pow-flip 98.9%

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

      6. metadata-eval 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. rem-cube-cbrt 99.6%

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

      2. metadata-eval 99.6%

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

      3. pow-sqr 99.5%

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

      4. inv-pow 99.5%

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

      5. inv-pow 99.5%

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

      6. un-div-inv 99.6%

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

      7. associate-/l* 99.5%

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

      8. div-inv 99.6%

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

   6. Applied egg-rr 99.6%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.031:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.031)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* 0.001388888888888889 (* x_m x_m)) 0.041666666666666664)))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.031) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} 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 <= 0.031d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((0.001388888888888889d0 * (x_m * x_m)) - 0.041666666666666664d0))
    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 <= 0.031) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} 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 <= 0.031:
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664))
	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 <= 0.031)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(0.001388888888888889 * Float64(x_m * x_m)) - 0.041666666666666664)));
	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 <= 0.031)
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	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, 0.031], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $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 < 0.031

   1. Initial program 37.8%

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

   3. Taylor expanded in x around 0 64.1%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 64.1%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   5. Applied egg-rr 64.1%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. pow2 64.1%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   7. Applied egg-rr 64.1%

      \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664\right) \]

   if 0.031 < x

   1. Initial program 98.3%

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

Alternative 5: 75.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.8 \cdot 10^{+38}:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.8e+38)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* 0.001388888888888889 (* x_m x_m)) 0.041666666666666664)))
   0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.8e+38) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} 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 <= 4.8d+38) then
        tmp = 0.5d0 + ((x_m * x_m) * ((0.001388888888888889d0 * (x_m * x_m)) - 0.041666666666666664d0))
    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 <= 4.8e+38) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 4.8e+38:
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664))
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.8e+38)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(0.001388888888888889 * Float64(x_m * x_m)) - 0.041666666666666664)));
	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 <= 4.8e+38)
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	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, 4.8e+38], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.80000000000000035e38

   1. Initial program 42.0%

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

   3. Taylor expanded in x around 0 60.0%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 60.0%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   5. Applied egg-rr 60.0%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. pow2 60.0%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   7. Applied egg-rr 60.0%

      \[\leadsto 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664\right) \]

   if 4.80000000000000035e38 < x

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0 62.6%

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

   4. Taylor expanded in x around 0 62.6%

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

Alternative 6: 75.4% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.05e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.05e+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 <= 1.05d+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 <= 1.05e+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 <= 1.05e+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 <= 1.05e+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 <= 1.05e+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, 1.05e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.0499999999999999e77

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0 58.2%

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

   if 1.0499999999999999e77 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 70.4%

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

   4. Taylor expanded in x around 0 70.4%

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

Alternative 7: 27.5% 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 56.9%

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

3. Taylor expanded in x around 0 30.0%

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

4. Taylor expanded in x around 0 30.7%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))