cos2 (problem 3.4.1)

Percentage Accurate: 51.2% → 99.6%

Time: 13.4s

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: 51.2% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.033:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.033)
   (+
    0.5
    (*
     (pow x_m 2.0)
     (- (* (pow x_m 2.0) 0.001388888888888889) 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 <= 0.033) {
		tmp = 0.5 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.001388888888888889) - 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 <= 0.033d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.001388888888888889d0) - 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 <= 0.033) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.001388888888888889) - 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 <= 0.033:
		tmp = 0.5 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 0.001388888888888889) - 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 <= 0.033)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.001388888888888889) - 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 <= 0.033)
		tmp = 0.5 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 0.001388888888888889) - 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, 0.033], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 0.001388888888888889), $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 < 0.033000000000000002

   1. Initial program 44.5%

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

   3. Taylor expanded in x around 0 59.2%

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

   if 0.033000000000000002 < x

   1. Initial program 97.7%

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

      1. add-sqr-sqrt 97.4%

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

      2. pow2 97.4%

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

      3. div-inv 97.4%

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

      4. sqrt-prod 97.4%

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

      5. unpow-prod-down 97.4%

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

      6. pow2 97.4%

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

      7. add-sqr-sqrt 97.4%

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

      8. pow2 97.4%

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

      9. pow-flip 99.3%

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

      10. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

      2. add-sqr-sqrt 99.5%

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

      3. metadata-eval 99.5%

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

      4. pow-flip 97.6%

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

      5. pow2 97.6%

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

      6. div-inv 97.7%

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

      7. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left({x}^{2} \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(\frac{\frac{\left|\sin x\_m\right|}{\sqrt{1 + \cos x\_m}}}{x\_m}\right)}^{2} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (pow (/ (/ (fabs (sin x_m)) (sqrt (+ 1.0 (cos x_m)))) x_m) 2.0))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((fabs(sin(x_m)) / sqrt((1.0 + cos(x_m)))) / x_m), 2.0);
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(((Math.abs(Math.sin(x_m)) / Math.sqrt((1.0 + Math.cos(x_m)))) / x_m), 2.0);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(((math.fabs(math.sin(x_m)) / math.sqrt((1.0 + math.cos(x_m)))) / x_m), 2.0)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(abs(sin(x_m)) / sqrt(Float64(1.0 + cos(x_m)))) / x_m) ^ 2.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((abs(sin(x_m)) / sqrt((1.0 + cos(x_m)))) / x_m) ^ 2.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[Power[N[(N[(N[Abs[N[Sin[x$95$m], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. add-sqr-sqrt 58.3%

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

   2. pow2 58.3%

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

   3. sqrt-div 58.3%

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

   4. sqrt-prod 27.2%

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

   5. add-sqr-sqrt 59.7%

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

4. Applied egg-rr 59.7%

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

   1. flip-- 59.4%

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

   2. metadata-eval 59.4%

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

   3. metadata-eval 59.4%

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

   4. sqrt-div 59.3%

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

   5. metadata-eval 59.3%

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

   6. pow2 59.3%

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

6. Applied egg-rr 59.3%

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

   1. unpow2 59.3%

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

   2. 1-sub-cos 78.4%

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

   3. rem-sqrt-square 98.3%

      \[\leadsto {\left(\frac{\frac{\color{blue}{\left|\sin x\right|}}{\sqrt{1 + \cos x}}}{x}\right)}^{2} \]

8. Simplified 98.3%

   \[\leadsto {\left(\frac{\color{blue}{\frac{\left|\sin x\right|}{\sqrt{1 + \cos x}}}}{x}\right)}^{2} \]

9. Final simplification 98.3%

   \[\leadsto {\left(\frac{\frac{\left|\sin x\right|}{\sqrt{1 + \cos x}}}{x}\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0057:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0057)
   (+ 0.5 (* (pow x_m 2.0) -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 <= 0.0057) {
		tmp = 0.5 + (pow(x_m, 2.0) * -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 <= 0.0057d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-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 <= 0.0057) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -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 <= 0.0057:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -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 <= 0.0057)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -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 <= 0.0057)
		tmp = 0.5 + ((x_m ^ 2.0) * -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, 0.0057], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $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 < 0.0057000000000000002

   1. Initial program 44.5%

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

   3. Taylor expanded in x around 0 58.6%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0057000000000000002 < x

   1. Initial program 97.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0057:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0057:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0057)
   (+ 0.5 (* (pow x_m 2.0) -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 <= 0.0057) {
		tmp = 0.5 + (pow(x_m, 2.0) * -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 <= 0.0057d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-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 <= 0.0057) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -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 <= 0.0057:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -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 <= 0.0057)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -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 <= 0.0057)
		tmp = 0.5 + ((x_m ^ 2.0) * -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, 0.0057], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $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 < 0.0057000000000000002

   1. Initial program 44.5%

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

   3. Taylor expanded in x around 0 58.6%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0057000000000000002 < x

   1. Initial program 97.7%

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

      1. add-sqr-sqrt 97.4%

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

      2. pow2 97.4%

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

      3. div-inv 97.4%

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

      4. sqrt-prod 97.4%

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

      5. unpow-prod-down 97.4%

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

      6. pow2 97.4%

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

      7. add-sqr-sqrt 97.4%

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

      8. pow2 97.4%

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

      9. pow-flip 99.3%

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

      10. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

      2. add-sqr-sqrt 99.5%

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

      3. metadata-eval 99.5%

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

      4. pow-flip 97.6%

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

      5. pow2 97.6%

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

      6. div-inv 97.7%

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

      7. associate-/r* 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0057:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (/ 0.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = 0.0 / (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 <= 3.5d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = 0.0d0 / (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 <= 3.5) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = 0.0 / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.5:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = 0.0 / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(0.0 / 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 <= 3.5)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = 0.0 / (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, 3.5], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 44.5%

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

   3. Taylor expanded in x around 0 58.6%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 3.5 < x

   1. Initial program 97.7%

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

      1. add-log-exp 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in x around 0 48.8%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.16e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.16e+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 <= 1.16d+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 <= 1.16e+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 <= 1.16e+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 <= 1.16e+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 <= 1.16e+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, 1.16e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1600000000000001e77

   1. Initial program 49.0%

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

   3. Taylor expanded in x around 0 54.5%

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

   if 1.1600000000000001e77 < x

   1. Initial program 97.2%

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

      1. add-log-exp 97.2%

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

   4. Applied egg-rr 97.2%

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

   5. Taylor expanded in x around 0 64.0%

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

   6. Taylor expanded in x around 0 64.0%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.4% 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 58.4%

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

   1. add-log-exp 58.3%

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

4. Applied egg-rr 58.3%

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

5. Taylor expanded in x around 0 25.9%

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

6. Taylor expanded in x around 0 26.6%

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

7. Final simplification 26.6%

   \[\leadsto 0 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024084 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))