cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.8%

Time: 13.9s

Alternatives: 5

Speedup: 107.0×

• could not determine a ground truth

  1. x = 3821643566.863322

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: 50.3% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-13}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m \cdot \tan \left(\frac{x\_m}{2}\right)}{x\_m} \cdot \frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1e-13)
   0.5
   (* (/ (* (sin x_m) (tan (/ x_m 2.0))) x_m) (/ 1.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1e-13) {
		tmp = 0.5;
	} else {
		tmp = ((sin(x_m) * tan((x_m / 2.0))) / x_m) * (1.0 / 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 <= 1d-13) then
        tmp = 0.5d0
    else
        tmp = ((sin(x_m) * tan((x_m / 2.0d0))) / x_m) * (1.0d0 / 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 <= 1e-13) {
		tmp = 0.5;
	} else {
		tmp = ((Math.sin(x_m) * Math.tan((x_m / 2.0))) / x_m) * (1.0 / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1e-13:
		tmp = 0.5
	else:
		tmp = ((math.sin(x_m) * math.tan((x_m / 2.0))) / x_m) * (1.0 / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1e-13)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(sin(x_m) * tan(Float64(x_m / 2.0))) / x_m) * Float64(1.0 / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1e-13)
		tmp = 0.5;
	else
		tmp = ((sin(x_m) * tan((x_m / 2.0))) / x_m) * (1.0 / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1e-13], 0.5, N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-13

   1. Initial program 29.4%

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

   3. Taylor expanded in x around 0 72.6%

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

   if 1e-13 < x

   1. Initial program 93.7%

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

      1. associate-/r* 94.9%

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

      2. div-inv 94.8%

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

   4. Applied egg-rr 94.8%

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

      1. frac-2neg 94.8%

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

      2. div-inv 94.7%

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

      3. sub-neg 94.7%

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

      4. distribute-neg-in 94.7%

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

      5. add-sqr-sqrt 48.2%

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

      6. sqrt-unprod 76.3%

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

      7. sqr-neg 76.3%

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

      8. sqrt-prod 28.1%

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

      9. add-sqr-sqrt 53.2%

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

      10. distribute-neg-in 53.2%

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

      11. +-commutative 53.2%

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

      12. distribute-neg-in 53.2%

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

      13. add-sqr-sqrt 25.1%

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

      14. sqrt-unprod 71.6%

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

      15. sqr-neg 71.6%

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

      16. sqrt-prod 46.4%

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

      17. add-sqr-sqrt 94.7%

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

      18. metadata-eval 94.7%

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

   6. Applied egg-rr 94.7%

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

      1. flip-+ 94.6%

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

      2. *-un-lft-identity 94.6%

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

      3. fma-neg 94.6%

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

      4. metadata-eval 94.6%

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

      5. fma-define 94.6%

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

      6. *-un-lft-identity 94.6%

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

      7. +-commutative 94.6%

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

      8. frac-2neg 94.6%

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

      9. metadata-eval 94.6%

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

      10. frac-times 94.6%

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

      11. metadata-eval 94.6%

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

      12. sub-1-cos 99.3%

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

      13. pow2 99.3%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 46.3%

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

      16. sqr-neg 46.3%

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

   8. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-mul-1 99.3%

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

      3. remove-double-neg 99.3%

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

      4. associate-/r* 99.2%

         \[\leadsto \color{blue}{\frac{\frac{{\sin x}^{2}}{1 + \cos x}}{x}} \cdot \frac{1}{x} \]

      5. *-lft-identity 99.2%

         \[\leadsto \frac{\frac{\color{blue}{1 \cdot {\sin x}^{2}}}{1 + \cos x}}{x} \cdot \frac{1}{x} \]

      6. associate-*l/ 99.2%

         \[\leadsto \frac{\color{blue}{\frac{1}{1 + \cos x} \cdot {\sin x}^{2}}}{x} \cdot \frac{1}{x} \]

      7. *-commutative 99.2%

         \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \frac{1}{1 + \cos x}}}{x} \cdot \frac{1}{x} \]

      8. unpow2 99.2%

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

      9. associate-*r* 99.2%

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

      10. associate-*r/ 99.1%

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

      11. *-rgt-identity 99.1%

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

      12. hang-0p-tan 99.6%

          \[\leadsto \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x} \cdot \frac{1}{x} \]

   10. Simplified 99.6%

       \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}} \cdot \frac{1}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-13}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000165:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000165) 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 <= 0.000165) {
		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 <= 0.000165d0) 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 <= 0.000165) {
		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 <= 0.000165:
		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 <= 0.000165)
		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 <= 0.000165)
		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, 0.000165], 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 < 1.65e-4

   1. Initial program 29.0%

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

   3. Taylor expanded in x around 0 73.0%

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

   if 1.65e-4 < x

   1. Initial program 98.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000165:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000165:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000165) 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 <= 0.000165) {
		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 <= 0.000165d0) 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 <= 0.000165) {
		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 <= 0.000165:
		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 <= 0.000165)
		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 <= 0.000165)
		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, 0.000165], 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 < 1.65e-4

   1. Initial program 29.0%

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

   3. Taylor expanded in x around 0 73.0%

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

   if 1.65e-4 < x

   1. Initial program 98.1%

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

      1. associate-/r* 99.4%

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

      2. div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. add-log-exp 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. un-div-inv 99.3%

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

      2. rem-log-exp 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000165:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 7.4 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 7.4e+76) 0.5 (/ 0.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 7.4e+76) {
		tmp = 0.5;
	} 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 <= 7.4d+76) then
        tmp = 0.5d0
    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 <= 7.4e+76) {
		tmp = 0.5;
	} 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 <= 7.4e+76:
		tmp = 0.5
	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 <= 7.4e+76)
		tmp = 0.5;
	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 <= 7.4e+76)
		tmp = 0.5;
	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, 7.4e+76], 0.5, N[(0.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.3999999999999999e76

   1. Initial program 32.4%

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

   3. Taylor expanded in x around 0 69.7%

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

   if 7.3999999999999999e76 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 58.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.2% 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 45.5%

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

3. Taylor expanded in x around 0 56.5%

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

4. Final simplification 56.5%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024045 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))