cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.6%

Time: 13.2s

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: 50.6% 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, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.102:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x\_m}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x\_m}^{6} + 0.001388888888888889 \cdot {x\_m}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.102)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x_m 2.0))
     (+
      (* -2.48015873015873e-5 (pow x_m 6.0))
      (* 0.001388888888888889 (pow x_m 4.0)))))
   (* (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 <= 0.102) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x_m, 2.0)) + ((-2.48015873015873e-5 * pow(x_m, 6.0)) + (0.001388888888888889 * pow(x_m, 4.0))));
	} 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 <= 0.102d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x_m ** 2.0d0)) + (((-2.48015873015873d-5) * (x_m ** 6.0d0)) + (0.001388888888888889d0 * (x_m ** 4.0d0))))
    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 <= 0.102) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x_m, 2.0)) + ((-2.48015873015873e-5 * Math.pow(x_m, 6.0)) + (0.001388888888888889 * Math.pow(x_m, 4.0))));
	} 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 <= 0.102:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x_m, 2.0)) + ((-2.48015873015873e-5 * math.pow(x_m, 6.0)) + (0.001388888888888889 * math.pow(x_m, 4.0))))
	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 <= 0.102)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x_m ^ 2.0)) + Float64(Float64(-2.48015873015873e-5 * (x_m ^ 6.0)) + Float64(0.001388888888888889 * (x_m ^ 4.0)))));
	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 <= 0.102)
		tmp = 0.5 + ((-0.041666666666666664 * (x_m ^ 2.0)) + ((-2.48015873015873e-5 * (x_m ^ 6.0)) + (0.001388888888888889 * (x_m ^ 4.0))));
	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, 0.102], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.48015873015873e-5 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 0.101999999999999993

   1. Initial program 32.0%

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

   3. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)} \]

   if 0.101999999999999993 < x

   1. Initial program 98.7%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.7%

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

      3. pow2 98.7%

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

      4. pow-flip 99.2%

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

      5. metadata-eval 99.2%

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (+ 0.5 (* -0.041666666666666664 (* x_m x_m)))
   (* (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 <= 0.0056) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0056d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m * x_m))
    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 <= 0.0056) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0056:
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m))
	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 <= 0.0056)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x_m * x_m)));
	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 <= 0.0056)
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	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, 0.0056], N[(0.5 + N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $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 < 0.00559999999999999994

   1. Initial program 32.0%

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

   3. Taylor expanded in x around 0 69.6%

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

      1. pow2 69.6%

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

   5. Applied egg-rr 69.6%

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

   if 0.00559999999999999994 < x

   1. Initial program 98.7%

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

      1. clear-num 98.7%

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

      2. associate-/r/ 98.7%

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

      3. pow2 98.7%

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

      4. pow-flip 99.2%

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

      5. metadata-eval 99.2%

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (+ 0.5 (* -0.041666666666666664 (* x_m x_m)))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0056) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0056d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m * x_m))
    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 <= 0.0056) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0056:
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m))
	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 <= 0.0056)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x_m * x_m)));
	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 <= 0.0056)
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	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, 0.0056], N[(0.5 + N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $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 < 0.00559999999999999994

   1. Initial program 32.0%

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

   3. Taylor expanded in x around 0 69.6%

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

      1. pow2 69.6%

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

   5. Applied egg-rr 69.6%

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

   if 0.00559999999999999994 < x

   1. Initial program 98.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (+ 0.5 (* -0.041666666666666664 (* x_m x_m)))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0056) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0056d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m * x_m))
    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 <= 0.0056) {
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	} 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 <= 0.0056:
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m))
	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 <= 0.0056)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * Float64(x_m * x_m)));
	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 <= 0.0056)
		tmp = 0.5 + (-0.041666666666666664 * (x_m * x_m));
	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, 0.0056], N[(0.5 + N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00559999999999999994

   1. Initial program 32.0%

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

   3. Taylor expanded in x around 0 69.6%

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

      1. pow2 69.6%

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

   5. Applied egg-rr 69.6%

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

   if 0.00559999999999999994 < x

   1. Initial program 98.7%

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

      1. associate-/r* 99.3%

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

      2. div-inv 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 9.4 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x\_m} \cdot \frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 9.4e+76) 0.5 (* (/ 0.0 x_m) (/ 1.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 9.4e+76) {
		tmp = 0.5;
	} else {
		tmp = (0.0 / 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 <= 9.4d+76) then
        tmp = 0.5d0
    else
        tmp = (0.0d0 / 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 <= 9.4e+76) {
		tmp = 0.5;
	} else {
		tmp = (0.0 / x_m) * (1.0 / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 9.4e+76:
		tmp = 0.5
	else:
		tmp = (0.0 / x_m) * (1.0 / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 9.4e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(0.0 / 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 <= 9.4e+76)
		tmp = 0.5;
	else
		tmp = (0.0 / 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, 9.4e+76], 0.5, N[(N[(0.0 / x$95$m), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.4000000000000006e76

   1. Initial program 36.4%

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

   3. Taylor expanded in x around 0 66.2%

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

   if 9.4000000000000006e76 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. expm1-log1p-u 99.3%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x} \cdot \frac{1}{x} \]

   6. Applied egg-rr 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \cos x\right)\right)}}{x} \cdot \frac{1}{x} \]
   7. Step-by-step derivation

      1. expm1-undefine 99.2%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(1 - \cos x\right)} - 1}}{x} \cdot \frac{1}{x} \]

      2. sub-neg 99.2%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(1 - \cos x\right)} + \left(-1\right)}}{x} \cdot \frac{1}{x} \]

      3. log1p-undefine 99.2%

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

      4. rem-exp-log 99.2%

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

      5. associate-+r- 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in x around 0 64.5%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.4 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.9% 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 46.8%

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

3. Taylor expanded in x around 0 55.6%

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

4. Final simplification 55.6%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))