cos2 (problem 3.4.1)

Percentage Accurate: 49.8% → 99.8%

Time: 10.9s

Alternatives: 7

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: 49.8% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(\sin x\_m \cdot \tan \left(\frac{x\_m}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5e-5)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (* (pow x_m -2.0) (* (sin x_m) (tan (/ x_m 2.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5e-5) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = pow(x_m, -2.0) * (sin(x_m) * 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 <= 5d-5) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (x_m ** (-2.0d0)) * (sin(x_m) * 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 <= 5e-5) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = Math.pow(x_m, -2.0) * (Math.sin(x_m) * Math.tan((x_m / 2.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 5e-5:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = math.pow(x_m, -2.0) * (math.sin(x_m) * math.tan((x_m / 2.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5e-5)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64((x_m ^ -2.0) * Float64(sin(x_m) * 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 <= 5e-5)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = (x_m ^ -2.0) * (sin(x_m) * 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, 5e-5], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000024e-5

   1. Initial program 30.7%

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

   3. Taylor expanded in x around 0 71.1%

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

      1. *-commutative 71.1%

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

   5. Simplified 71.1%

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

   if 5.00000000000000024e-5 < x

   1. Initial program 99.2%

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

      1. flip-- 99.0%

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

      2. div-inv 98.9%

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

      3. metadata-eval 98.9%

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

      4. pow2 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. unpow2 98.9%

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

      2. 1-sub-cos 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. div-inv 99.1%

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

      2. associate-*l* 99.3%

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

      3. pow2 99.3%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*l* 99.4%

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

      7. un-div-inv 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*r* 99.5%

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

      4. *-commutative 99.5%

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

      5. hang-0p-tan 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(\sin x \cdot \tan \left(\frac{x}{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0052:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x\_m + -1\right) \cdot \frac{-1}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (/ (* (+ (cos x_m) -1.0) (/ -1.0 x_m)) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((cos(x_m) + -1.0) * (-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 <= 0.0052d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = ((cos(x_m) + (-1.0d0)) * ((-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 <= 0.0052) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((Math.cos(x_m) + -1.0) * (-1.0 / x_m)) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0052:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = ((math.cos(x_m) + -1.0) * (-1.0 / x_m)) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0052)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(cos(x_m) + -1.0) * Float64(-1.0 / 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 <= 0.0052)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((cos(x_m) + -1.0) * (-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, 0.0052], 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] * N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 30.7%

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

   3. Taylor expanded in x around 0 71.1%

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

      1. *-commutative 71.1%

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

   5. Simplified 71.1%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.2%

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

      1. clear-num 99.2%

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

      2. associate-/r/ 99.2%

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

      3. pow2 99.2%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. metadata-eval 99.5%

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

      2. pow-prod-up 99.5%

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

      3. inv-pow 99.5%

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

      4. inv-pow 99.5%

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

      5. associate-*l* 99.3%

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

      6. associate-/r/ 99.5%

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

      7. clear-num 99.5%

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

      8. frac-2neg 99.5%

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

      9. associate-*r/ 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x + -1\right) \cdot \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0052:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (+ 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 <= 0.0052) {
		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 <= 0.0052d0) 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 <= 0.0052) {
		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 <= 0.0052:
		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 <= 0.0052)
		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 <= 0.0052)
		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, 0.0052], 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 < 0.0051999999999999998

   1. Initial program 30.7%

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

   3. Taylor expanded in x around 0 71.1%

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

      1. *-commutative 71.1%

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

   5. Simplified 71.1%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0052:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (+ 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 <= 0.0052) {
		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 <= 0.0052d0) 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 <= 0.0052) {
		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 <= 0.0052:
		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 <= 0.0052)
		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 <= 0.0052)
		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, 0.0052], 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 < 0.0051999999999999998

   1. Initial program 30.7%

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

   3. Taylor expanded in x around 0 71.1%

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

      1. *-commutative 71.1%

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

   5. Simplified 71.1%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.2%

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

      1. associate-/r* 99.5%

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

      2. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. un-div-inv 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 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.45:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.45)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (/ (/ 2.0 x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.45) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = (2.0 / 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 <= 2.45d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (2.0d0 / 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 <= 2.45) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = (2.0 / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.45:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = (2.0 / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.45)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(2.0 / 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 <= 2.45)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = (2.0 / 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, 2.45], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000002

   1. Initial program 31.1%

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

   3. Taylor expanded in x around 0 70.9%

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

      1. *-commutative 70.9%

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

   5. Simplified 70.9%

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

   if 2.4500000000000002 < x

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

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

      3. sqrt-div 99.1%

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

      4. sqrt-prod 99.0%

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

      5. add-sqr-sqrt 99.3%

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

   4. Applied egg-rr 99.3%

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

   6. Applied egg-rr 64.9%

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

   7. Taylor expanded in x around 0 65.7%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.0) 0.5 (/ (/ 2.0 x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / 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 <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = (2.0d0 / 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 <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.0:
		tmp = 0.5
	else:
		tmp = (2.0 / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(2.0 / 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 <= 2.0)
		tmp = 0.5;
	else
		tmp = (2.0 / 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, 2.0], 0.5, N[(N[(2.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 31.1%

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

   3. Taylor expanded in x around 0 71.3%

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

   if 2 < x

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

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

      3. sqrt-div 99.1%

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

      4. sqrt-prod 99.0%

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

      5. add-sqr-sqrt 99.3%

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

   4. Applied egg-rr 99.3%

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

   6. Applied egg-rr 64.9%

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

   7. Taylor expanded in x around 0 65.7%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.6% 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 49.4%

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

3. Taylor expanded in x around 0 53.2%

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

4. Final simplification 53.2%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))