bug500, discussion (missed optimization)

Percentage Accurate: 52.3% → 97.1%

Time: 17.7s

Alternatives: 8

Speedup: 40.6×

• could not determine a ground truth

  1. x = 1.6418759226625191e+258

Specification

Math

\[\begin{array}{l} \\ \log \left(\frac{\sinh x}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ (sinh x) x)))

C

double code(double x) {
	return log((sinh(x) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((sinh(x) / x))
end function

Java

public static double code(double x) {
	return Math.log((Math.sinh(x) / x));
}

Python

def code(x):
	return math.log((math.sinh(x) / x))

Julia

function code(x)
	return log(Float64(sinh(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = log((sinh(x) / x));
end

Wolfram

code[x_] := N[Log[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Initial Program: 52.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\sinh x}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (/ (sinh x) x)))

C

double code(double x) {
	return log((sinh(x) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((sinh(x) / x))
end function

Java

public static double code(double x) {
	return Math.log((Math.sinh(x) / x));
}

Python

def code(x):
	return math.log((math.sinh(x) / x))

Julia

function code(x)
	return log(Float64(sinh(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = log((sinh(x) / x));
end

Wolfram

code[x_] := N[Log[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 97.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\\ \frac{x}{\frac{0.027777777777777776 + \left(x \cdot x\right) \cdot \left(t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0 + -0.16666666666666666\right)\right)}{x \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot t\_0\right) \cdot \left(\left(x \cdot x\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ -0.005555555555555556 (* x (* x 0.0003527336860670194)))))
   (/
    x
    (/
     (+
      0.027777777777777776
      (* (* x x) (* t_0 (+ (* (* x x) t_0) -0.16666666666666666))))
     (*
      x
      (+
       0.004629629629629629
       (* (* x (* x x)) (* (* x t_0) (* (* x x) 3.08641975308642e-5)))))))))

C

double code(double x) {
	double t_0 = -0.005555555555555556 + (x * (x * 0.0003527336860670194));
	return x / ((0.027777777777777776 + ((x * x) * (t_0 * (((x * x) * t_0) + -0.16666666666666666)))) / (x * (0.004629629629629629 + ((x * (x * x)) * ((x * t_0) * ((x * x) * 3.08641975308642e-5))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (-0.005555555555555556d0) + (x * (x * 0.0003527336860670194d0))
    code = x / ((0.027777777777777776d0 + ((x * x) * (t_0 * (((x * x) * t_0) + (-0.16666666666666666d0))))) / (x * (0.004629629629629629d0 + ((x * (x * x)) * ((x * t_0) * ((x * x) * 3.08641975308642d-5))))))
end function

Java

public static double code(double x) {
	double t_0 = -0.005555555555555556 + (x * (x * 0.0003527336860670194));
	return x / ((0.027777777777777776 + ((x * x) * (t_0 * (((x * x) * t_0) + -0.16666666666666666)))) / (x * (0.004629629629629629 + ((x * (x * x)) * ((x * t_0) * ((x * x) * 3.08641975308642e-5))))));
}

Python

def code(x):
	t_0 = -0.005555555555555556 + (x * (x * 0.0003527336860670194))
	return x / ((0.027777777777777776 + ((x * x) * (t_0 * (((x * x) * t_0) + -0.16666666666666666)))) / (x * (0.004629629629629629 + ((x * (x * x)) * ((x * t_0) * ((x * x) * 3.08641975308642e-5))))))

Julia

function code(x)
	t_0 = Float64(-0.005555555555555556 + Float64(x * Float64(x * 0.0003527336860670194)))
	return Float64(x / Float64(Float64(0.027777777777777776 + Float64(Float64(x * x) * Float64(t_0 * Float64(Float64(Float64(x * x) * t_0) + -0.16666666666666666)))) / Float64(x * Float64(0.004629629629629629 + Float64(Float64(x * Float64(x * x)) * Float64(Float64(x * t_0) * Float64(Float64(x * x) * 3.08641975308642e-5)))))))
end

MATLAB

function tmp = code(x)
	t_0 = -0.005555555555555556 + (x * (x * 0.0003527336860670194));
	tmp = x / ((0.027777777777777776 + ((x * x) * (t_0 * (((x * x) * t_0) + -0.16666666666666666)))) / (x * (0.004629629629629629 + ((x * (x * x)) * ((x * t_0) * ((x * x) * 3.08641975308642e-5))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.005555555555555556 + N[(x * N[(x * 0.0003527336860670194), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x / N[(N[(0.027777777777777776 + N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * N[(N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(0.004629629629629629 + N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * t$95$0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.2%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \frac{-1}{180}\right)\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{180} + \color{blue}{\frac{1}{2835} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \left({x}^{2} \cdot \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 97.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}\right) \cdot x}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 97.0%

   \[\leadsto x \cdot \color{blue}{\frac{\left(0.004629629629629629 + \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)\right)\right)\right) \cdot x}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right) - 0.16666666666666666\right)}} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}\right)}\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{0.027777777777777776 + \left(x \cdot x\right) \cdot \left(\left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) + -0.16666666666666666\right)\right)}{x \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(\left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right)\right)\right)\right)\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \color{blue}{\left(\frac{1}{32400} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \left({x}^{2} \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right)\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{32400}\right)\right)\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f64 97.1%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{216}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2835}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{32400}\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 97.1%

    \[\leadsto \frac{x}{\frac{0.027777777777777776 + \left(x \cdot x\right) \cdot \left(\left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) + -0.16666666666666666\right)\right)}{x \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right)\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right)\right)}} \]
13. Add Preprocessing

Alternative 2: 97.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\\ t_1 := x \cdot t\_0\\ x \cdot \frac{x \cdot \left(0.027777777777777776 - x \cdot \left(t\_0 \cdot t\_1\right)\right)}{0.16666666666666666 - t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ -0.005555555555555556 (* (* x x) 0.0003527336860670194))))
        (t_1 (* x t_0)))
   (*
    x
    (/
     (* x (- 0.027777777777777776 (* x (* t_0 t_1))))
     (- 0.16666666666666666 t_1)))))

C

double code(double x) {
	double t_0 = x * (-0.005555555555555556 + ((x * x) * 0.0003527336860670194));
	double t_1 = x * t_0;
	return x * ((x * (0.027777777777777776 - (x * (t_0 * t_1)))) / (0.16666666666666666 - t_1));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = x * ((-0.005555555555555556d0) + ((x * x) * 0.0003527336860670194d0))
    t_1 = x * t_0
    code = x * ((x * (0.027777777777777776d0 - (x * (t_0 * t_1)))) / (0.16666666666666666d0 - t_1))
end function

Java

public static double code(double x) {
	double t_0 = x * (-0.005555555555555556 + ((x * x) * 0.0003527336860670194));
	double t_1 = x * t_0;
	return x * ((x * (0.027777777777777776 - (x * (t_0 * t_1)))) / (0.16666666666666666 - t_1));
}

Python

def code(x):
	t_0 = x * (-0.005555555555555556 + ((x * x) * 0.0003527336860670194))
	t_1 = x * t_0
	return x * ((x * (0.027777777777777776 - (x * (t_0 * t_1)))) / (0.16666666666666666 - t_1))

Julia

function code(x)
	t_0 = Float64(x * Float64(-0.005555555555555556 + Float64(Float64(x * x) * 0.0003527336860670194)))
	t_1 = Float64(x * t_0)
	return Float64(x * Float64(Float64(x * Float64(0.027777777777777776 - Float64(x * Float64(t_0 * t_1)))) / Float64(0.16666666666666666 - t_1)))
end

MATLAB

function tmp = code(x)
	t_0 = x * (-0.005555555555555556 + ((x * x) * 0.0003527336860670194));
	t_1 = x * t_0;
	tmp = x * ((x * (0.027777777777777776 - (x * (t_0 * t_1)))) / (0.16666666666666666 - t_1));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(-0.005555555555555556 + N[(N[(x * x), $MachinePrecision] * 0.0003527336860670194), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, N[(x * N[(N[(x * N[(0.027777777777777776 - N[(x * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 53.2%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \frac{-1}{180}\right)\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{180} + \color{blue}{\frac{1}{2835} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \left({x}^{2} \cdot \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 97.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}{\frac{1}{6} - \left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot x}{\color{blue}{\frac{1}{6} - \left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} - \left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}\right)\right) \]

7. Applied egg-rr 97.0%

   \[\leadsto x \cdot \color{blue}{\frac{\left(0.027777777777777776 - x \cdot \left(\left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)\right)\right) \cdot x}{0.16666666666666666 - x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)}} \]

8. Final simplification 97.0%

   \[\leadsto x \cdot \frac{x \cdot \left(0.027777777777777776 - x \cdot \left(\left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)\right)\right)}{0.16666666666666666 - x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)} \]
9. Add Preprocessing

Alternative 3: 97.0% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (+
    0.16666666666666666
    (* (* x x) (+ -0.005555555555555556 (* (* x x) 0.0003527336860670194)))))))

C

double code(double x) {
	return x * (x * (0.16666666666666666 + ((x * x) * (-0.005555555555555556 + ((x * x) * 0.0003527336860670194)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (0.16666666666666666d0 + ((x * x) * ((-0.005555555555555556d0) + ((x * x) * 0.0003527336860670194d0)))))
end function

Java

public static double code(double x) {
	return x * (x * (0.16666666666666666 + ((x * x) * (-0.005555555555555556 + ((x * x) * 0.0003527336860670194)))));
}

Python

def code(x):
	return x * (x * (0.16666666666666666 + ((x * x) * (-0.005555555555555556 + ((x * x) * 0.0003527336860670194)))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(Float64(x * x) * Float64(-0.005555555555555556 + Float64(Float64(x * x) * 0.0003527336860670194))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (0.16666666666666666 + ((x * x) * (-0.005555555555555556 + ((x * x) * 0.0003527336860670194)))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(-0.005555555555555556 + N[(N[(x * x), $MachinePrecision] * 0.0003527336860670194), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \frac{-1}{180}\right)\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{180} + \color{blue}{\frac{1}{2835} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \left({x}^{2} \cdot \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 97.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)} \]
6. Add Preprocessing

Alternative 4: 97.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{6 + \left(x \cdot x\right) \cdot 0.2}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (/ (+ 6.0 (* (* x x) 0.2)) x)))

C

double code(double x) {
	return x / ((6.0 + ((x * x) * 0.2)) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((6.0d0 + ((x * x) * 0.2d0)) / x)
end function

Java

public static double code(double x) {
	return x / ((6.0 + ((x * x) * 0.2)) / x);
}

Python

def code(x):
	return x / ((6.0 + ((x * x) * 0.2)) / x)

Julia

function code(x)
	return Float64(x / Float64(Float64(6.0 + Float64(Float64(x * x) * 0.2)) / x))
end

MATLAB

function tmp = code(x)
	tmp = x / ((6.0 + ((x * x) * 0.2)) / x);
end

Wolfram

code[x_] := N[(x / N[(N[(6.0 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \frac{-1}{180}\right)\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{180} + \color{blue}{\frac{1}{2835} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \left({x}^{2} \cdot \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 97.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}\right) \cdot x}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 97.0%

   \[\leadsto x \cdot \color{blue}{\frac{\left(0.004629629629629629 + \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)\right)\right)\right) \cdot x}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right) - 0.16666666666666666\right)}} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}\right)}\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{0.027777777777777776 + \left(x \cdot x\right) \cdot \left(\left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) + -0.16666666666666666\right)\right)}{x \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(\left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right)\right)\right)\right)\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{6 + \frac{1}{5} \cdot {x}^{2}}{x}\right)}\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(6 + \frac{1}{5} \cdot {x}^{2}\right), \color{blue}{x}\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \left(\frac{1}{5} \cdot {x}^{2}\right)\right), x\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \left({x}^{2} \cdot \frac{1}{5}\right)\right), x\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{5}\right)\right), x\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right), x\right)\right) \]

    6. *-lowering-*.f64 97.0%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(6, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right), x\right)\right) \]

12. Simplified 97.0%

    \[\leadsto \frac{x}{\color{blue}{\frac{6 + \left(x \cdot x\right) \cdot 0.2}{x}}} \]
13. Add Preprocessing

Alternative 5: 96.6% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* x (+ 0.16666666666666666 (* (* x x) -0.005555555555555556)))))

C

double code(double x) {
	return x * (x * (0.16666666666666666 + ((x * x) * -0.005555555555555556)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (0.16666666666666666d0 + ((x * x) * (-0.005555555555555556d0))))
end function

Java

public static double code(double x) {
	return x * (x * (0.16666666666666666 + ((x * x) * -0.005555555555555556)));
}

Python

def code(x):
	return x * (x * (0.16666666666666666 + ((x * x) * -0.005555555555555556)))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(0.16666666666666666 + Float64(Float64(x * x) * -0.005555555555555556))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (0.16666666666666666 + ((x * x) * -0.005555555555555556)));
end

Wolfram

code[x_] := N[(x * N[(x * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{-1}{180} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-1}{180} \cdot {x}^{2}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{180}}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{180}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 96.6%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{180}\right)\right)\right)\right) \]

5. Simplified 96.6%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot -0.005555555555555556\right)\right)} \]
6. Add Preprocessing

Alternative 6: 96.4% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{6}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (/ 6.0 x)))

C

double code(double x) {
	return x / (6.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (6.0d0 / x)
end function

Java

public static double code(double x) {
	return x / (6.0 / x);
}

Python

def code(x):
	return x / (6.0 / x)

Julia

function code(x)
	return Float64(x / Float64(6.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = x / (6.0 / x);
end

Wolfram

code[x_] := N[(x / N[(6.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)\right)}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2} - \frac{1}{180}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{2835} \cdot {x}^{2}} - \frac{1}{180}\right)\right)\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{180}\right)\right)}\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{2835} \cdot {x}^{2} + \frac{-1}{180}\right)\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{180} + \color{blue}{\frac{1}{2835} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \color{blue}{\left(\frac{1}{2835} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \left({x}^{2} \cdot \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{2835}}\right)\right)\right)\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 97.0%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{180}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{2835}\right)\right)\right)\right)\right)\right) \]

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{{\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}\right) \cdot x}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 97.0%

   \[\leadsto x \cdot \color{blue}{\frac{\left(0.004629629629629629 + \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right)\right)\right)\right) \cdot x}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(-0.005555555555555556 + \left(x \cdot x\right) \cdot 0.0003527336860670194\right)\right) - 0.16666666666666666\right)}} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{36} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) - \frac{1}{6}\right)}{\left(\frac{1}{216} + \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right) \cdot \left(x \cdot \left(\left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-1}{180} + \left(x \cdot x\right) \cdot \frac{1}{2835}\right)\right)\right)\right)\right)\right) \cdot x}\right)}\right) \]

9. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{0.027777777777777776 + \left(x \cdot x\right) \cdot \left(\left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) + -0.16666666666666666\right)\right)}{x \cdot \left(0.004629629629629629 + \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right)\right) \cdot \left(\left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(-0.005555555555555556 + x \cdot \left(x \cdot 0.0003527336860670194\right)\right)\right)\right)\right)\right)}}} \]

10. Taylor expanded in x around 0

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

    1. /-lowering-/.f64 96.5%

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

12. Simplified 96.5%

    \[\leadsto \frac{x}{\color{blue}{\frac{6}{x}}} \]
13. Add Preprocessing

Alternative 7: 96.4% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 0.16666666666666666)))

C

double code(double x) {
	return x * (x * 0.16666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * 0.16666666666666666d0)
end function

Java

public static double code(double x) {
	return x * (x * 0.16666666666666666);
}

Python

def code(x):
	return x * (x * 0.16666666666666666)

Julia

function code(x)
	return Float64(x * Float64(x * 0.16666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 0.16666666666666666);
end

Wolfram

code[x_] := N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
4. Step-by-step derivation

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 96.5%

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

5. Simplified 96.5%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot x\right) \cdot \color{blue}{x} \]

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 96.5%

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

7. Applied egg-rr 96.5%

   \[\leadsto \color{blue}{\left(x \cdot 0.16666666666666666\right) \cdot x} \]

8. Final simplification 96.5%

   \[\leadsto x \cdot \left(x \cdot 0.16666666666666666\right) \]
9. Add Preprocessing

Alternative 8: 96.3% accurate, 40.6× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) 0.16666666666666666))

C

double code(double x) {
	return (x * x) * 0.16666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * 0.16666666666666666d0
end function

Java

public static double code(double x) {
	return (x * x) * 0.16666666666666666;
}

Python

def code(x):
	return (x * x) * 0.16666666666666666

Julia

function code(x)
	return Float64(Float64(x * x) * 0.16666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * 0.16666666666666666;
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 53.2%

   \[\log \left(\frac{\sinh x}{x}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
4. Step-by-step derivation

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 96.5%

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

5. Simplified 96.5%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]

6. Final simplification 96.5%

   \[\leadsto \left(x \cdot x\right) \cdot 0.16666666666666666 \]
7. Add Preprocessing

Developer Target 1: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.085:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.6455026455026456 \cdot 10^{-5}, x \cdot x, 0.0003527336860670194\right), x \cdot x, -0.005555555555555556\right), x \cdot x, 0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sinh x}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.085)
   (*
    (* x x)
    (fma
     (fma
      (fma -2.6455026455026456e-5 (* x x) 0.0003527336860670194)
      (* x x)
      -0.005555555555555556)
     (* x x)
     0.16666666666666666))
   (log (/ (sinh x) x))))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.085) {
		tmp = (x * x) * fma(fma(fma(-2.6455026455026456e-5, (x * x), 0.0003527336860670194), (x * x), -0.005555555555555556), (x * x), 0.16666666666666666);
	} else {
		tmp = log((sinh(x) / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.085)
		tmp = Float64(Float64(x * x) * fma(fma(fma(-2.6455026455026456e-5, Float64(x * x), 0.0003527336860670194), Float64(x * x), -0.005555555555555556), Float64(x * x), 0.16666666666666666));
	else
		tmp = log(Float64(sinh(x) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.085], N[(N[(x * x), $MachinePrecision] * N[(N[(N[(-2.6455026455026456e-5 * N[(x * x), $MachinePrecision] + 0.0003527336860670194), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.005555555555555556), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Sinh[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "bug500, discussion (missed optimization)"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs x) 17/200) (let ((x2 (* x x))) (* x2 (fma (fma (fma -1/37800 x2 1/2835) x2 -1/180) x2 1/6))) (log (/ (sinh x) x))))

  (log (/ (sinh x) x)))