cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.5%

Time: 10.5s

Alternatives: 8

Speedup: 17.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.002:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(-0.041666666666666664 + \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m}{x\_m \cdot x\_m} \cdot \tan \left(\frac{x\_m}{2}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.002)
   (+
    0.5
    (*
     (* x_m x_m)
     (+ -0.041666666666666664 (* (* x_m x_m) 0.001388888888888889))))
   (* (/ (sin x_m) (* x_m x_m)) (tan (/ x_m 2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.002) {
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	} else {
		tmp = (sin(x_m) / (x_m * x_m)) * tan((x_m / 2.0));
	}
	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 <= 0.002d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((-0.041666666666666664d0) + ((x_m * x_m) * 0.001388888888888889d0)))
    else
        tmp = (sin(x_m) / (x_m * x_m)) * tan((x_m / 2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.002) {
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	} else {
		tmp = (Math.sin(x_m) / (x_m * x_m)) * Math.tan((x_m / 2.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.002:
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)))
	else:
		tmp = (math.sin(x_m) / (x_m * x_m)) * math.tan((x_m / 2.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.002)
		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(sin(x_m) / Float64(x_m * x_m)) * tan(Float64(x_m / 2.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.002)
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	else
		tmp = (sin(x_m) / (x_m * x_m)) * tan((x_m / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.002], 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[Sin[x$95$m], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-3

   1. Initial program 30.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 72.1%

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

   5. Simplified 72.1%

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

   if 2e-3 < x

   1. Initial program 99.0%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. metadata-eval N/A

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

      4. 1-sub-cos N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. hang-0p-tan N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(x, x\right)\right), \tan \left(\frac{x}{2}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right) \]

      12. /-lowering-/.f64 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{*.f64}\left(x, x\right)\right), \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.032:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(-0.041666666666666664 + \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{x\_m \cdot \frac{-1}{-1 + \cos x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.032)
   (+
    0.5
    (*
     (* x_m x_m)
     (+ -0.041666666666666664 (* (* x_m x_m) 0.001388888888888889))))
   (/ (/ 1.0 x_m) (* x_m (/ -1.0 (+ -1.0 (cos x_m)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.032) {
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	} else {
		tmp = (1.0 / x_m) / (x_m * (-1.0 / (-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 <= 0.032d0) then
        tmp = 0.5d0 + ((x_m * x_m) * ((-0.041666666666666664d0) + ((x_m * x_m) * 0.001388888888888889d0)))
    else
        tmp = (1.0d0 / x_m) / (x_m * ((-1.0d0) / ((-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 <= 0.032) {
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	} else {
		tmp = (1.0 / x_m) / (x_m * (-1.0 / (-1.0 + Math.cos(x_m))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.032:
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)))
	else:
		tmp = (1.0 / x_m) / (x_m * (-1.0 / (-1.0 + math.cos(x_m))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.032)
		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 / x_m) / Float64(x_m * Float64(-1.0 / 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 <= 0.032)
		tmp = 0.5 + ((x_m * x_m) * (-0.041666666666666664 + ((x_m * x_m) * 0.001388888888888889)));
	else
		tmp = (1.0 / x_m) / (x_m * (-1.0 / (-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, 0.032], 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 / x$95$m), $MachinePrecision] / N[(x$95$m * N[(-1.0 / N[(-1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 30.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 72.1%

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

   5. Simplified 72.1%

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

   if 0.032000000000000001 < x

   1. Initial program 99.0%

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

   4. Applied egg-rr 99.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{x \cdot \frac{-1}{-1 + \cos x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.032:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot \left(-0.041666666666666664 + \left(x\_m \cdot x\_m\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.032)
   (+
    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 <= 0.032) {
		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 <= 0.032d0) 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 <= 0.032) {
		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 <= 0.032:
		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 <= 0.032)
		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 <= 0.032)
		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, 0.032], 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 < 0.032000000000000001

   1. Initial program 30.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 72.1%

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

   5. Simplified 72.1%

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

   if 0.032000000000000001 < x

   1. Initial program 99.0%

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

Alternative 4: 78.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.3:\\ \;\;\;\;0.5 + \left(x\_m \cdot x\_m\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.3)
   (+ 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 <= 3.3) {
		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 <= 3.3d0) 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 <= 3.3) {
		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 <= 3.3:
		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 <= 3.3)
		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 <= 3.3)
		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, 3.3], 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 < 3.2999999999999998

   1. Initial program 30.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 71.8%

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

   5. Simplified 71.8%

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

   if 3.2999999999999998 < x

   1. Initial program 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. neg-mul-1 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

      8. cos-lowering-cos.f64 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. metadata-eval 59.9%

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

   9. Simplified 59.9%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   11. 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 59.9%

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

   12. Simplified 59.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% 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 30.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 72.1%

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

   if 3.5 < x

   1. Initial program 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. neg-mul-1 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

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

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

      8. cos-lowering-cos.f64 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. metadata-eval 59.9%

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

   9. Simplified 59.9%

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

   10. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   11. 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 59.9%

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

   12. Simplified 59.9%

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

Alternative 6: 78.5% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.4%

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

4. Applied egg-rr 49.2%

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

   1. neg-mul-1 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

   6. /-lowering-/.f64 N/A

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

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

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

   8. cos-lowering-cos.f64 49.2%

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

6. Applied egg-rr 49.2%

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

7. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. metadata-eval 81.6%

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

9. Simplified 81.6%

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

Alternative 7: 75.9% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.6e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.6e+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 <= 1.6d+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 <= 1.6e+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 <= 1.6e+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 <= 1.6e+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 <= 1.6e+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, 1.6e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001e77

   1. Initial program 36.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 66.7%

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

   if 1.6000000000000001e77 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{1}\right), \mathsf{*.f64}\left(x, x\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 72.1%

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

      1. metadata-eval N/A

         \[\leadsto \frac{0}{\color{blue}{x} \cdot x} \]

      2. div0 72.1%

         \[\leadsto 0 \]

   7. Applied egg-rr 72.1%

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

Alternative 8: 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 48.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \color{blue}{1}\right), \mathsf{*.f64}\left(x, x\right)\right) \]
4. Step-by-step derivation

5. Simplified 27.5%

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

   1. metadata-eval N/A

      \[\leadsto \frac{0}{\color{blue}{x} \cdot x} \]

   2. div0 28.3%

      \[\leadsto 0 \]

7. Applied egg-rr 28.3%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))