cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.6%

Time: 12.7s

Alternatives: 7

Speedup: 34.9×

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: 50.8% 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: 99.6% accurate, 0.3× speedup

Math

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

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)) 1.0) -1.0))))
   (* (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)) + 1.0) + -1.0)));
	} 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)) + 1.0d0) + (-1.0d0))))
    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)) + 1.0) + -1.0)));
	} 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)) + 1.0) + -1.0)))
	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(Float64(Float64(0.001388888888888889 * (x_m ^ 4.0)) + 1.0) + -1.0))));
	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)) + 1.0) + -1.0)));
	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[(N[(N[(0.001388888888888889 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $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 36.2%

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

   2. Taylor expanded in x around 0 65.1%

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

      1. expm1-log1p-u 65.1%

         \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.001388888888888889 \cdot {x}^{4}\right)\right)}\right)\right) \]

      2. expm1-udef 65.1%

         \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \color{blue}{\left(e^{\mathsf{log1p}\left(0.001388888888888889 \cdot {x}^{4}\right)} - 1\right)}\right)\right) \]

   4. Applied egg-rr 65.1%

      \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \color{blue}{\left(e^{\mathsf{log1p}\left(0.001388888888888889 \cdot {x}^{4}\right)} - 1\right)}\right)\right) \]

   5. Taylor expanded in x around 0 65.1%

      \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \left(\color{blue}{\left(1 + 0.001388888888888889 \cdot {x}^{4}\right)} - 1\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative 65.1%

         \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \left(\color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + 1\right)} - 1\right)\right)\right) \]

   7. Simplified 65.1%

      \[\leadsto 0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + \left(\color{blue}{\left(0.001388888888888889 \cdot {x}^{4} + 1\right)} - 1\right)\right)\right) \]

   if 0.101999999999999993 < x

   1. Initial program 98.4%

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

      1. clear-num 98.3%

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

      2. associate-/r/ 98.4%

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

      3. pow2 98.4%

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

      4. pow-flip 99.1%

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

      5. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

4. Final simplification 72.6%

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

Alternative 2: 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 36.2%

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

   2. Taylor expanded in x around 0 65.1%

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

   if 0.101999999999999993 < x

   1. Initial program 98.4%

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

      1. clear-num 98.3%

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

      2. associate-/r/ 98.4%

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

      3. pow2 98.4%

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

      4. pow-flip 99.1%

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

      5. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

4. Final simplification 72.6%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0046:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x_m}^{-2} \cdot \left(1 - \cos x_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0046)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (* (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 <= 0.0046) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} 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 <= 0.0046d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    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 <= 0.0046) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} 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 <= 0.0046:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	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 <= 0.0046)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	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 <= 0.0046)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	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, 0.0046], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $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 < 0.0045999999999999999

   1. Initial program 35.9%

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

   2. Taylor expanded in x around 0 65.4%

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

   if 0.0045999999999999999 < x

   1. Initial program 98.2%

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

      1. clear-num 98.1%

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

      2. associate-/r/ 98.2%

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

      3. pow2 98.2%

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

      4. pow-flip 98.9%

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

      5. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0046:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x_m}{x_m \cdot x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0046)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0046) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} 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 <= 0.0046d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    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 <= 0.0046) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} 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 <= 0.0046:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	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 <= 0.0046)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	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 <= 0.0046)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	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, 0.0046], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $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 < 0.0045999999999999999

   1. Initial program 35.9%

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

   2. Taylor expanded in x around 0 65.4%

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

   if 0.0045999999999999999 < x

   1. Initial program 98.2%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0046:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x_m}{x_m}}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0046)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0046) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} 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 <= 0.0046d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    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 <= 0.0046) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} 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 <= 0.0046:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	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 <= 0.0046)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	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 <= 0.0046)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	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, 0.0046], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $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 < 0.0045999999999999999

   1. Initial program 35.9%

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

   2. Taylor expanded in x around 0 65.4%

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

   if 0.0045999999999999999 < x

   1. Initial program 98.2%

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

      1. associate-/r* 98.9%

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

      2. div-inv 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. un-div-inv 98.9%

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

   5. Applied egg-rr 98.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0046:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 6: 75.5% accurate, 34.9× 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 40.3%

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

   2. Taylor expanded in x around 0 62.4%

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

   if 1.25000000000000001e77 < x

   1. Initial program 98.1%

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

      1. add-log-exp 98.1%

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

   3. Applied egg-rr 98.1%

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

   4. Taylor expanded in x around 0 65.9%

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

   5. Taylor expanded in x around 0 65.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 27.0% 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 49.8%

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

   1. add-log-exp 49.7%

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

3. Applied egg-rr 49.7%

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

4. Taylor expanded in x around 0 24.5%

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

5. Taylor expanded in x around 0 25.3%

   \[\leadsto \color{blue}{0} \]

6. Final simplification 25.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))