cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.6%

Time: 19.4s

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, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.028:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(-0.041666666666666664 + \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.028)
   (+
    0.5
    (*
     (* x_m x_m)
     (+ -0.041666666666666664 (* (* x_m x_m) 0.001388888888888889))))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.028) {
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	} 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 <= 0.028d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((-0.041666666666666664d0) + ((x_m * x_m) * 0.001388888888888889d0)))
    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 <= 0.028) {
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	} 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 <= 0.028:
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)))
	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 <= 0.028)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(-0.041666666666666664 + Float64(Float64(x_m * x_m) * 0.001388888888888889))));
	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 <= 0.028)
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	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, 0.028], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(-0.041666666666666664 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $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 < 0.0280000000000000006

   1. Initial program 33.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \frac{-1}{24}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{24} + \color{blue}{\frac{1}{720} \cdot {x}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right) \]

   5. Simplified 69.4%

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

   if 0.0280000000000000006 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - \cos x}{x}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 - \cos x\right), x\right), x\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \cos x\right), x\right), x\right) \]

      6. cos-lowering-cos.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right), x\right), x\right) \]

   5. Simplified 99.2%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.028:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(-0.041666666666666664 + \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.028)
   (+
    0.5
    (*
     (* x_m x_m)
     (+ -0.041666666666666664 (* (* x_m x_m) 0.001388888888888889))))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.028) {
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	} 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 <= 0.028d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((-0.041666666666666664d0) + ((x_m * x_m) * 0.001388888888888889d0)))
    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 <= 0.028) {
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	} 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 <= 0.028:
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)))
	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 <= 0.028)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * Float64(-0.041666666666666664 + Float64(Float64(x_m * x_m) * 0.001388888888888889))));
	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 <= 0.028)
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	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, 0.028], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(-0.041666666666666664 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $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 < 0.0280000000000000006

   1. Initial program 33.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \frac{-1}{24}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{24} + \color{blue}{\frac{1}{720} \cdot {x}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 69.4%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right) \]

   5. Simplified 69.4%

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

   if 0.0280000000000000006 < x

   1. Initial program 98.7%

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

Alternative 3: 78.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.2:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.2)
   (+ 0.5 (* (* x_m x_m) -0.041666666666666664))
   (/ 6.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.2) {
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	} else {
		tmp = 6.0 / (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 <= 3.2d0) then
        tmp = 0.5d0 + ((x_m * x_m) * (-0.041666666666666664d0))
    else
        tmp = 6.0d0 / (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 <= 3.2) {
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	} else {
		tmp = 6.0 / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.2:
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664)
	else:
		tmp = 6.0 / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.2)
		tmp = Float64(0.5 + Float64(Float64(x_m * x_m) * -0.041666666666666664));
	else
		tmp = Float64(6.0 / 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 <= 3.2)
		tmp = 0.5 + ((x_m * x_m) * -0.041666666666666664);
	else
		tmp = 6.0 / (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, 3.2], N[(0.5 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(6.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 33.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 68.9%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 68.9%

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

   if 3.2000000000000002 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - \cos x}{x}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 - \cos x\right), x\right), x\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \cos x\right), x\right), x\right) \]

      6. cos-lowering-cos.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right), x\right), x\right) \]

   5. Simplified 99.2%

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

      1. associate-/l/ N/A

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

      2. sub-div N/A

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

      3. associate-/r* N/A

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

      4. frac-sub N/A

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

      5. cube-unmult N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{{x}^{3}}{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}\right)}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({x}^{3}\right), \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x\right)}\right)\right) \]

      9. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{\frac{1}{x} \cdot \left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\color{blue}{\frac{1}{x} \cdot \left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{x} \cdot \color{blue}{\left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

      12. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{-1} \cdot \left(x \cdot x\right) - x \cdot \cos x\right)\right)\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{-1} \cdot {x}^{2} - x \cdot \cos x\right)\right)\right) \]

      14. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{\left(-1 + 2\right)} - \color{blue}{x} \cdot \cos x\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{1} - x \cdot \cos x\right)\right)\right) \]

      16. unpow1 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(x - \color{blue}{x} \cdot \cos x\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(x \cdot \cos x\right)}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\cos x}\right)\right)\right)\right) \]

      19. cos-lowering-cos.f64 86.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(x\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 86.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + \frac{1}{6} \cdot {x}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right) \]

      5. *-lowering-*.f64 56.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right) \]

   10. Simplified 56.1%

       \[\leadsto \frac{1}{\color{blue}{2 + \left(x \cdot x\right) \cdot 0.16666666666666666}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 56.1%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Simplified 56.1%

       \[\leadsto \color{blue}{\frac{6}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 3.5) 0.5 (/ 6.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (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
    else
        tmp = 6.0d0 / (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;
	} else {
		tmp = 6.0 / (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
	else:
		tmp = 6.0 / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = 0.5;
	else
		tmp = Float64(6.0 / 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;
	else
		tmp = 6.0 / (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], 0.5, N[(6.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 33.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 69.4%

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

   if 3.5 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - \cos x}{x}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 - \cos x\right), x\right), x\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \cos x\right), x\right), x\right) \]

      6. cos-lowering-cos.f64 99.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right), x\right), x\right) \]

   5. Simplified 99.2%

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

      1. associate-/l/ N/A

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

      2. sub-div N/A

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

      3. associate-/r* N/A

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

      4. frac-sub N/A

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

      5. cube-unmult N/A

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

      6. clear-num N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{{x}^{3}}{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}\right)}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({x}^{3}\right), \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x\right)}\right)\right) \]

      9. cube-unmult N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{\frac{1}{x} \cdot \left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\color{blue}{\frac{1}{x} \cdot \left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{x} \cdot \color{blue}{\left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

      12. inv-pow N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{-1} \cdot \left(x \cdot x\right) - x \cdot \cos x\right)\right)\right) \]

      13. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{-1} \cdot {x}^{2} - x \cdot \cos x\right)\right)\right) \]

      14. pow-prod-up N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{\left(-1 + 2\right)} - \color{blue}{x} \cdot \cos x\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{1} - x \cdot \cos x\right)\right)\right) \]

      16. unpow1 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(x - \color{blue}{x} \cdot \cos x\right)\right)\right) \]

      17. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(x \cdot \cos x\right)}\right)\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\cos x}\right)\right)\right)\right) \]

      19. cos-lowering-cos.f64 86.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(x\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 86.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + \frac{1}{6} \cdot {x}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right) \]

      5. *-lowering-*.f64 56.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right) \]

   10. Simplified 56.1%

       \[\leadsto \frac{1}{\color{blue}{2 + \left(x \cdot x\right) \cdot 0.16666666666666666}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(6, \left(x \cdot \color{blue}{x}\right)\right) \]

       3. *-lowering-*.f64 56.1%

          \[\leadsto \mathsf{/.f64}\left(6, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Simplified 56.1%

       \[\leadsto \color{blue}{\frac{6}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{2 + \left(x\_m \cdot x\_m\right) \cdot 0.16666666666666666} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (+ 2.0 (* (* x_m x_m) 0.16666666666666666))))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 / (2.0 + ((x_m * x_m) * 0.16666666666666666));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0 / (2.0d0 + ((x_m * x_m) * 0.16666666666666666d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 / (2.0 + ((x_m * x_m) * 0.16666666666666666));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 / (2.0 + ((x_m * x_m) * 0.16666666666666666))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / Float64(2.0 + Float64(Float64(x_m * x_m) * 0.16666666666666666)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 / (2.0 + ((x_m * x_m) * 0.16666666666666666));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(2.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1 - \cos x}{{x}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - \cos x}{x}\right), \color{blue}{x}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 - \cos x\right), x\right), x\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \cos x\right), x\right), x\right) \]

   6. cos-lowering-cos.f64 53.5%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right), x\right), x\right) \]

5. Simplified 53.5%

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

   1. associate-/l/ N/A

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

   2. sub-div N/A

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

   3. associate-/r* N/A

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

   4. frac-sub N/A

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

   5. cube-unmult N/A

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

   6. clear-num N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{{x}^{3}}{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}\right)}\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left({x}^{3}\right), \color{blue}{\left(\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x\right)}\right)\right) \]

   9. cube-unmult N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \left(\color{blue}{\frac{1}{x} \cdot \left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \left(\color{blue}{\frac{1}{x} \cdot \left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{x} \cdot \color{blue}{\left(x \cdot x\right)} - x \cdot \cos x\right)\right)\right) \]

   12. inv-pow N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{-1} \cdot \left(x \cdot x\right) - x \cdot \cos x\right)\right)\right) \]

   13. pow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{-1} \cdot {x}^{2} - x \cdot \cos x\right)\right)\right) \]

   14. pow-prod-up N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{\left(-1 + 2\right)} - \color{blue}{x} \cdot \cos x\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left({x}^{1} - x \cdot \cos x\right)\right)\right) \]

   16. unpow1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \left(x - \color{blue}{x} \cdot \cos x\right)\right)\right) \]

   17. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \color{blue}{\left(x \cdot \cos x\right)}\right)\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\cos x}\right)\right)\right)\right) \]

   19. cos-lowering-cos.f64 45.5%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{cos.f64}\left(x\right)\right)\right)\right)\right) \]

7. Applied egg-rr 45.5%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(2 + \frac{1}{6} \cdot {x}^{2}\right)}\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{6} \cdot {x}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{6}\right)\right)\right) \]

   5. *-lowering-*.f64 75.6%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{6}\right)\right)\right) \]

10. Simplified 75.6%

    \[\leadsto \frac{1}{\color{blue}{2 + \left(x \cdot x\right) \cdot 0.16666666666666666}} \]
11. Add Preprocessing

Alternative 6: 51.8% 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 52.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Simplified 50.3%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))