cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.8%

Time: 8.4s

Alternatives: 6

Speedup: 107.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.1% 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, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.002:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x\_m \cdot {x\_m}^{-2}\right) \cdot \tan \left(\frac{x\_m}{2}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.002)
   (+
    0.5
    (*
     (pow x_m 2.0)
     (- (* (pow x_m 2.0) 0.001388888888888889) 0.041666666666666664)))
   (* (* (sin x_m) (pow x_m -2.0)) (tan (/ x_m 2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.002) {
		tmp = 0.5 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = (sin(x_m) * pow(x_m, -2.0)) * tan((x_m / 2.0));
	}
	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 <= 0.002d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.001388888888888889d0) - 0.041666666666666664d0))
    else
        tmp = (sin(x_m) * (x_m ** (-2.0d0))) * tan((x_m / 2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.002) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = (Math.sin(x_m) * Math.pow(x_m, -2.0)) * Math.tan((x_m / 2.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.002:
		tmp = 0.5 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * 0.001388888888888889) - 0.041666666666666664))
	else:
		tmp = (math.sin(x_m) * math.pow(x_m, -2.0)) * math.tan((x_m / 2.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.002)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * 0.001388888888888889) - 0.041666666666666664)));
	else
		tmp = Float64(Float64(sin(x_m) * (x_m ^ -2.0)) * tan(Float64(x_m / 2.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.002)
		tmp = 0.5 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * 0.001388888888888889) - 0.041666666666666664));
	else
		tmp = (sin(x_m) * (x_m ^ -2.0)) * tan((x_m / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.002], 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[Sin[x$95$m], $MachinePrecision] * N[Power[x$95$m, -2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-3

   1. Initial program 35.0%

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

   3. Taylor expanded in x around 0 67.0%

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

   if 2e-3 < x

   1. Initial program 97.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. frac-times 97.4%

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

      2. unpow2 97.4%

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

      3. *-rgt-identity 97.4%

         \[\leadsto \frac{\color{blue}{1 - \cos x}}{{x}^{2}} \]

      4. add-sqr-sqrt 97.4%

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

      5. sqrt-unprod 68.2%

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

      6. sqr-neg 68.2%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 47.5%

         \[\leadsto \frac{1 - \cos x}{\color{blue}{-{x}^{2}}} \]

      9. un-div-inv 47.5%

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

      10. flip-- 47.5%

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

      11. metadata-eval 47.5%

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

      12. 1-sub-cos 47.5%

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

      13. frac-times 47.5%

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

      14. pow2 47.5%

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

      15. add-sqr-sqrt 0.0%

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

      16. sqrt-unprod 68.2%

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

      17. sqr-neg 68.2%

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

      18. sqrt-unprod 97.2%

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

   6. Applied egg-rr 97.2%

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

      1. *-rgt-identity 97.2%

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

      2. unpow2 97.2%

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

      3. *-commutative 97.2%

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

      4. times-frac 97.1%

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

   8. Simplified 99.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left({x}^{2} \cdot 0.001388888888888889 - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot {x}^{-2}\right) \cdot \tan \left(\frac{x}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

   1. clear-num 48.9%

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

   2. inv-pow 48.9%

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

   3. flip-- 48.7%

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

   4. associate-/r/ 48.7%

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

   5. unpow-prod-down 48.8%

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

   6. pow2 48.8%

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

   7. metadata-eval 48.8%

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

   8. pow2 48.8%

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

   9. inv-pow 48.8%

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

4. Applied egg-rr 48.8%

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

   1. associate-*r/ 48.8%

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

   2. *-rgt-identity 48.8%

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

   3. unpow-1 48.8%

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

6. Simplified 48.8%

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

   1. unpow2 48.8%

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

   2. 1-sub-cos 71.4%

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

8. Applied egg-rr 71.4%

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

9. Taylor expanded in x around inf 71.4%

   \[\leadsto \frac{\color{blue}{\frac{{\sin x}^{2}}{{x}^{2}}}}{1 + \cos x} \]
10. Step-by-step derivation

    1. *-rgt-identity 71.4%

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

    2. associate-/l* 70.2%

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

    3. unpow2 70.2%

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

    4. associate-/r* 70.9%

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

    5. *-rgt-identity 70.9%

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

    6. associate-*r/ 70.8%

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

    7. *-commutative 70.8%

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

    8. unpow2 70.8%

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

    9. swap-sqr 99.3%

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

    10. associate-/r/ 99.4%

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

    11. unpow-1 99.4%

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

    12. associate-/r/ 99.5%

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

    13. unpow-1 99.5%

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

    14. pow-sqr 99.5%

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

    15. metadata-eval 99.5%

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

11. Simplified 99.5%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

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)))
   (/ (/ (+ (cos x_m) -1.0) 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 = ((cos(x_m) + -1.0) / 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 <= 0.033d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * 0.001388888888888889d0) - 0.041666666666666664d0))
    else
        tmp = ((cos(x_m) + (-1.0d0)) / 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 <= 0.033) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * 0.001388888888888889) - 0.041666666666666664));
	} else {
		tmp = ((Math.cos(x_m) + -1.0) / 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 = ((math.cos(x_m) + -1.0) / 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(cos(x_m) + -1.0) / x_m) / Float64(-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 = ((cos(x_m) + -1.0) / 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, 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[(N[Cos[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.033000000000000002

   1. Initial program 35.0%

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

   3. Taylor expanded in x around 0 67.0%

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

   if 0.033000000000000002 < x

   1. Initial program 97.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. un-div-inv 99.4%

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

      2. frac-2neg 99.4%

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

      3. frac-2neg 99.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 47.4%

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

      6. sqr-neg 47.4%

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

      7. sqrt-unprod 47.4%

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

      8. add-sqr-sqrt 47.4%

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

      9. distribute-frac-neg 47.4%

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

      10. frac-2neg 47.4%

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

      11. add-sqr-sqrt 47.4%

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

      12. sqrt-unprod 47.4%

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

      13. sqr-neg 47.4%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 99.4%

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

      16. neg-sub0 99.4%

          \[\leadsto \frac{\frac{\color{blue}{0 - \left(1 - \cos x\right)}}{x}}{-x} \]

      17. associate--r- 99.4%

          \[\leadsto \frac{\frac{\color{blue}{\left(0 - 1\right) + \cos x}}{x}}{-x} \]

      18. metadata-eval 99.4%

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

   6. 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 74.2%

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

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0048:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x\_m + -1}{x\_m}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0048)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (/ (/ (+ (cos x_m) -1.0) x_m) (- x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0048) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((cos(x_m) + -1.0) / 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 <= 0.0048d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = ((cos(x_m) + (-1.0d0)) / 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 <= 0.0048) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((Math.cos(x_m) + -1.0) / x_m) / -x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0048:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = ((math.cos(x_m) + -1.0) / x_m) / -x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0048)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(cos(x_m) + -1.0) / x_m) / Float64(-x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0048)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((cos(x_m) + -1.0) / 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, 0.0048], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision] / x$95$m), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00479999999999999958

   1. Initial program 35.0%

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

   3. Taylor expanded in x around 0 66.6%

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

   if 0.00479999999999999958 < x

   1. Initial program 97.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. un-div-inv 99.4%

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

      2. frac-2neg 99.4%

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

      3. frac-2neg 99.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 47.4%

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

      6. sqr-neg 47.4%

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

      7. sqrt-unprod 47.4%

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

      8. add-sqr-sqrt 47.4%

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

      9. distribute-frac-neg 47.4%

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

      10. frac-2neg 47.4%

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

      11. add-sqr-sqrt 47.4%

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

      12. sqrt-unprod 47.4%

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

      13. sqr-neg 47.4%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 99.4%

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

      16. neg-sub0 99.4%

          \[\leadsto \frac{\frac{\color{blue}{0 - \left(1 - \cos x\right)}}{x}}{-x} \]

      17. associate--r- 99.4%

          \[\leadsto \frac{\frac{\color{blue}{\left(0 - 1\right) + \cos x}}{x}}{-x} \]

      18. metadata-eval 99.4%

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

   6. 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 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0048:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0048)
   (+ 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 <= 0.0048) {
		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 <= 0.0048d0) 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 <= 0.0048) {
		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 <= 0.0048:
		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 <= 0.0048)
		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 <= 0.0048)
		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, 0.0048], 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 < 0.00479999999999999958

   1. Initial program 35.0%

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

   3. Taylor expanded in x around 0 66.6%

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

   if 0.00479999999999999958 < x

   1. Initial program 97.4%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.9%

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

3. Taylor expanded in x around 0 53.0%

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

Reproduce

herbie shell --seed 2024108 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))