cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.6%

Time: 8.9s

Alternatives: 6

Speedup: 17.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0054:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0054)
   (+ 0.5 (* (pow x_m 2.0) -0.041666666666666664))
   (* (pow x_m -2.0) (- 1.0 (cos x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0054) {
		tmp = 0.5 + (pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = pow(x_m, -2.0) * (1.0 - cos(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 <= 0.0054d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (x_m ** (-2.0d0)) * (1.0d0 - cos(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 <= 0.0054) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * -0.041666666666666664);
	} else {
		tmp = Math.pow(x_m, -2.0) * (1.0 - Math.cos(x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0054:
		tmp = 0.5 + (math.pow(x_m, 2.0) * -0.041666666666666664)
	else:
		tmp = math.pow(x_m, -2.0) * (1.0 - math.cos(x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0054)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64((x_m ^ -2.0) * Float64(1.0 - cos(x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0054)
		tmp = 0.5 + ((x_m ^ 2.0) * -0.041666666666666664);
	else
		tmp = (x_m ^ -2.0) * (1.0 - cos(x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0054], 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[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054000000000000003

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 65.7%

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

      1. *-commutative 65.7%

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

   5. Simplified 65.7%

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

   if 0.0054000000000000003 < x

   1. Initial program 97.3%

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

      1. clear-num 97.3%

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

      2. associate-/r/ 97.2%

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

      3. pow2 97.2%

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

      4. pow-flip 99.6%

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

      5. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0054:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0054)
   (+ 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 <= 0.0054) {
		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 <= 0.0054d0) 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 <= 0.0054) {
		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 <= 0.0054:
		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 <= 0.0054)
		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 <= 0.0054)
		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, 0.0054], 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 < 0.0054000000000000003

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 65.7%

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

      1. *-commutative 65.7%

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

   5. Simplified 65.7%

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

   if 0.0054000000000000003 < x

   1. Initial program 97.3%

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

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

      2. div-sub 99.4%

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

      3. div-sub 99.4%

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

      4. associate-/r* 97.2%

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

      5. pow2 97.2%

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

      6. pow-flip 99.4%

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

      7. metadata-eval 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. metadata-eval 99.4%

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

      2. pow-flip 97.2%

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

      3. pow2 97.2%

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

      4. associate-/l/ 97.2%

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

      5. div-sub 97.3%

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

      6. associate-/r* 99.5%

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

   8. Applied egg-rr 99.5%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0054:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0054)
   (+ 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 <= 0.0054) {
		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 <= 0.0054d0) 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 <= 0.0054) {
		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 <= 0.0054:
		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 <= 0.0054)
		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 <= 0.0054)
		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, 0.0054], 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 < 0.0054000000000000003

   1. Initial program 35.8%

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

   3. Taylor expanded in x around 0 65.7%

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

      1. *-commutative 65.7%

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

   5. Simplified 65.7%

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

   if 0.0054000000000000003 < x

   1. Initial program 97.3%

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

Alternative 4: 75.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.22 \cdot 10^{+84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x\_m} \cdot \frac{-1}{x\_m} + \frac{\frac{-1}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.22e+84)
   0.5
   (+ (* (/ -1.0 x_m) (/ -1.0 x_m)) (/ (/ -1.0 x_m) x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.22e+84) {
		tmp = 0.5;
	} else {
		tmp = ((-1.0 / x_m) * (-1.0 / 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 <= 1.22d+84) then
        tmp = 0.5d0
    else
        tmp = (((-1.0d0) / x_m) * ((-1.0d0) / 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 <= 1.22e+84) {
		tmp = 0.5;
	} else {
		tmp = ((-1.0 / x_m) * (-1.0 / 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 <= 1.22e+84:
		tmp = 0.5
	else:
		tmp = ((-1.0 / x_m) * (-1.0 / 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 <= 1.22e+84)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(-1.0 / x_m) * Float64(-1.0 / x_m)) + Float64(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 <= 1.22e+84)
		tmp = 0.5;
	else
		tmp = ((-1.0 / x_m) * (-1.0 / 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, 1.22e+84], 0.5, N[(N[(N[(-1.0 / x$95$m), $MachinePrecision] * N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2200000000000001e84

   1. Initial program 41.5%

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

   3. Taylor expanded in x around 0 61.0%

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

   if 1.2200000000000001e84 < x

   1. Initial program 96.7%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. un-div-inv 99.7%

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

      2. div-sub 99.6%

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

      3. div-sub 99.5%

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

      4. associate-/r* 96.5%

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

      5. pow2 96.5%

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

      6. pow-flip 99.6%

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

      7. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

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

      2. pow-prod-up 99.4%

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

      3. inv-pow 99.4%

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

      4. inv-pow 99.4%

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

      5. clear-num 99.4%

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

      6. clear-num 99.4%

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

      7. frac-2neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. /-rgt-identity 99.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 68.7%

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

      12. sqr-neg 68.7%

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

      13. sqrt-prod 68.6%

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

      14. add-sqr-sqrt 68.6%

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

      15. frac-2neg 68.6%

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

      16. metadata-eval 68.6%

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

      17. /-rgt-identity 68.6%

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

      18. add-sqr-sqrt 0.0%

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

      19. sqrt-unprod 96.4%

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

      20. sqr-neg 96.4%

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

      21. sqrt-prod 99.4%

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

      22. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in x around 0 69.4%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{+84}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} \cdot \frac{-1}{x} + \frac{\frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.25e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.25e+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 <= 1.25d+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 <= 1.25e+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 <= 1.25e+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 <= 1.25e+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 <= 1.25e+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, 1.25e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.25000000000000001e77

   1. Initial program 41.3%

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

   3. Taylor expanded in x around 0 61.3%

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

   if 1.25000000000000001e77 < x

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0 68.1%

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

   4. Taylor expanded in x around 0 68.1%

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

Alternative 6: 28.0% 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 53.8%

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

3. Taylor expanded in x around 0 27.7%

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

4. Taylor expanded in x around 0 28.4%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))