cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.6%

Time: 8.9s

Alternatives: 9

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: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.095:\\ \;\;\;\;0.5 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.001388888888888889 + {x\_m}^{2} \cdot -2.48015873015873 \cdot 10^{-5}\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.095)
   (+
    0.5
    (*
     (pow x_m 2.0)
     (-
      (*
       (pow x_m 2.0)
       (+ 0.001388888888888889 (* (pow x_m 2.0) -2.48015873015873e-5)))
      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 <= 0.095) {
		tmp = 0.5 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * (0.001388888888888889 + (pow(x_m, 2.0) * -2.48015873015873e-5))) - 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 <= 0.095d0) then
        tmp = 0.5d0 + ((x_m ** 2.0d0) * (((x_m ** 2.0d0) * (0.001388888888888889d0 + ((x_m ** 2.0d0) * (-2.48015873015873d-5)))) - 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 <= 0.095) {
		tmp = 0.5 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * (0.001388888888888889 + (Math.pow(x_m, 2.0) * -2.48015873015873e-5))) - 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 <= 0.095:
		tmp = 0.5 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * (0.001388888888888889 + (math.pow(x_m, 2.0) * -2.48015873015873e-5))) - 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 <= 0.095)
		tmp = Float64(0.5 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.001388888888888889 + Float64((x_m ^ 2.0) * -2.48015873015873e-5))) - 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 <= 0.095)
		tmp = 0.5 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * (0.001388888888888889 + ((x_m ^ 2.0) * -2.48015873015873e-5))) - 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, 0.095], N[(0.5 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.001388888888888889 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -2.48015873015873e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $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 < 0.095000000000000001

   1. Initial program 37.3%

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

   3. Taylor expanded in x around 0 65.8%

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

   if 0.095000000000000001 < x

   1. Initial program 99.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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/ 99.3%

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

      3. pow2 99.3%

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

      4. pow-flip 99.3%

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

      5. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.001388888888888889 + {x}^{2} \cdot -2.48015873015873 \cdot 10^{-5}\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.029)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* 0.001388888888888889 (* x_m x_m)) 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 <= 0.029) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 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 <= 0.029d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((0.001388888888888889d0 * (x_m * x_m)) - 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 <= 0.029) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 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 <= 0.029:
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 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 <= 0.029)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(0.001388888888888889 * Float64(x_m * x_m)) - 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 <= 0.029)
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 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, 0.029], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $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 < 0.0290000000000000015

   1. Initial program 37.3%

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

   3. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 66.0%

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

   5. Applied egg-rr 66.0%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. pow2 66.0%

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

   7. Applied egg-rr 66.0%

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

   if 0.0290000000000000015 < x

   1. Initial program 99.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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/ 99.3%

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

      3. pow2 99.3%

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

      4. pow-flip 99.3%

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

      5. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m} \cdot \frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.029)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* 0.001388888888888889 (* x_m x_m)) 0.041666666666666664)))
   (* (/ (- 1.0 (cos x_m)) x_m) (/ 1.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.029) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} else {
		tmp = ((1.0 - cos(x_m)) / 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 <= 0.029d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((0.001388888888888889d0 * (x_m * x_m)) - 0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x_m)) / 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 <= 0.029) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) * (1.0 / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.029:
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664))
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) * (1.0 / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.029)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(0.001388888888888889 * Float64(x_m * x_m)) - 0.041666666666666664)));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / 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 <= 0.029)
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	else
		tmp = ((1.0 - cos(x_m)) / 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, 0.029], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 37.3%

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

   3. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 66.0%

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

   5. Applied egg-rr 66.0%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. pow2 66.0%

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

   7. Applied egg-rr 66.0%

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

   if 0.0290000000000000015 < x

   1. Initial program 99.3%

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

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.029)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* 0.001388888888888889 (* x_m x_m)) 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 <= 0.029) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 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 <= 0.029d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((0.001388888888888889d0 * (x_m * x_m)) - 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 <= 0.029) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 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 <= 0.029:
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 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 <= 0.029)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(0.001388888888888889 * Float64(x_m * x_m)) - 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 <= 0.029)
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 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, 0.029], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $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 < 0.0290000000000000015

   1. Initial program 37.3%

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

   3. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 66.0%

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

   5. Applied egg-rr 66.0%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. pow2 66.0%

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

   7. Applied egg-rr 66.0%

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

   if 0.0290000000000000015 < x

   1. Initial program 99.3%

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

Alternative 5: 76.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 8 \cdot 10^{+38}:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(0.001388888888888889 \cdot \left(x\_m \cdot x\_m\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 8e+38)
   (+
    0.5
    (*
     (* x_m x_m)
     (- (* 0.001388888888888889 (* x_m x_m)) 0.041666666666666664)))
   (/ 0.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 8e+38) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} else {
		tmp = 0.0 / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 8d+38) then
        tmp = 0.5d0 + ((x_m * x_m) * ((0.001388888888888889d0 * (x_m * x_m)) - 0.041666666666666664d0))
    else
        tmp = 0.0d0 / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 8e+38) {
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	} else {
		tmp = 0.0 / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 8e+38:
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664))
	else:
		tmp = 0.0 / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 8e+38)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(Float64(0.001388888888888889 * Float64(x_m * x_m)) - 0.041666666666666664)));
	else
		tmp = Float64(0.0 / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 8e+38)
		tmp = 0.5 + ((x_m * x_m) * ((0.001388888888888889 * (x_m * x_m)) - 0.041666666666666664));
	else
		tmp = 0.0 / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 8e+38], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999982e38

   1. Initial program 39.4%

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

   3. Taylor expanded in x around 0 64.1%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 64.1%

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

   5. Applied egg-rr 64.1%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]
   6. Step-by-step derivation

      1. pow2 64.1%

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

   7. Applied egg-rr 64.1%

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

   if 7.99999999999999982e38 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 62.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{+38}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5)
   (+ 0.5 (* (* x_m x_m) -0.041666666666666664))
   (/ 0.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	} else {
		tmp = 0.0 / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 3.5d0) then
        tmp = 0.5d0 + ((x_m * x_m) * (-0.041666666666666664d0))
    else
        tmp = 0.0d0 / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	} else {
		tmp = 0.0 / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.5:
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664)
	else:
		tmp = 0.0 / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * -0.041666666666666664));
	else
		tmp = Float64(0.0 / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 3.5)
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	else
		tmp = 0.0 / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.5], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(0.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 37.6%

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

      1. pow2 37.6%

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

      2. add-cube-cbrt 37.4%

         \[\leadsto \frac{1 - \cos x}{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{2}} \]

      3. pow3 37.3%

         \[\leadsto \frac{1 - \cos x}{{\color{blue}{\left({\left(\sqrt[3]{x}\right)}^{3}\right)}}^{2}} \]

      4. pow-pow 37.3%

         \[\leadsto \frac{1 - \cos x}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(3 \cdot 2\right)}}} \]

      5. metadata-eval 37.3%

         \[\leadsto \frac{1 - \cos x}{{\left(\sqrt[3]{x}\right)}^{\color{blue}{6}}} \]

   4. Applied egg-rr 37.3%

      \[\leadsto \frac{1 - \cos x}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{6}}} \]

   5. Taylor expanded in x around 0 65.2%

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

      1. +-commutative 65.2%

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

   7. Simplified 65.2%

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

      1. pow2 65.9%

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

   9. Applied egg-rr 65.2%

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

   if 3.5 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 55.9%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5) (+ 0.5 (* (* x_m x_m) -0.041666666666666664)) 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	} 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 <= 3.5d0) then
        tmp = 0.5d0 + ((x_m * x_m) * (-0.041666666666666664d0))
    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 <= 3.5) {
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.5:
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664)
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * -0.041666666666666664));
	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 <= 3.5)
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	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, 3.5], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 37.6%

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

      1. pow2 37.6%

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

      2. add-cube-cbrt 37.4%

         \[\leadsto \frac{1 - \cos x}{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{2}} \]

      3. pow3 37.3%

         \[\leadsto \frac{1 - \cos x}{{\color{blue}{\left({\left(\sqrt[3]{x}\right)}^{3}\right)}}^{2}} \]

      4. pow-pow 37.3%

         \[\leadsto \frac{1 - \cos x}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(3 \cdot 2\right)}}} \]

      5. metadata-eval 37.3%

         \[\leadsto \frac{1 - \cos x}{{\left(\sqrt[3]{x}\right)}^{\color{blue}{6}}} \]

   4. Applied egg-rr 37.3%

      \[\leadsto \frac{1 - \cos x}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{6}}} \]

   5. Taylor expanded in x around 0 65.2%

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

      1. +-commutative 65.2%

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

   7. Simplified 65.2%

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

      1. pow2 65.9%

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

   9. Applied egg-rr 65.2%

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

   if 3.5 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 55.9%

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

   4. Taylor expanded in x around 0 55.9%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 3e+76) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3e+76) {
		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 <= 3d+76) 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 <= 3e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3e+76)
		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 <= 3e+76)
		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, 3e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.9999999999999998e76

   1. Initial program 41.9%

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

   3. Taylor expanded in x around 0 61.3%

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

   if 2.9999999999999998e76 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 74.8%

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

   4. Taylor expanded in x around 0 74.8%

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

Alternative 9: 27.9% 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 51.1%

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

3. Taylor expanded in x around 0 28.3%

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

4. Taylor expanded in x around 0 28.9%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))