bug500, discussion (missed optimization)

Percentage Accurate: 52.7% → 97.0%

Time: 18.0s

Alternatives: 5

Speedup: 40.6×

• could not determine a ground truth

  1. x = 1.9398804743465771e+174

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: 52.7% 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: 97.0% 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 50.7%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 97.5%

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

5. Simplified 97.5%

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

Alternative 2: 97.0% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \frac{-0.027777777777777776}{-0.16666666666666666 - x \cdot \left(x \cdot 0.005555555555555556\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (/
    -0.027777777777777776
    (- -0.16666666666666666 (* x (* x 0.005555555555555556)))))))

C

double code(double x) {
	return x * (x * (-0.027777777777777776 / (-0.16666666666666666 - (x * (x * 0.005555555555555556)))));
}

Fortran

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

Java

public static double code(double x) {
	return x * (x * (-0.027777777777777776 / (-0.16666666666666666 - (x * (x * 0.005555555555555556)))));
}

Python

def code(x):
	return x * (x * (-0.027777777777777776 / (-0.16666666666666666 - (x * (x * 0.005555555555555556)))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(-0.027777777777777776 / Float64(-0.16666666666666666 - Float64(x * Float64(x * 0.005555555555555556))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (-0.027777777777777776 / (-0.16666666666666666 - (x * (x * 0.005555555555555556)))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(-0.027777777777777776 / N[(-0.16666666666666666 - N[(x * N[(x * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.7%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 97.2%

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

5. Simplified 97.2%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

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

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

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

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. swap-sqr N/A

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

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

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

   12. metadata-eval N/A

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

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

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

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

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

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

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

   16. *-commutative N/A

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

7. Applied egg-rr 97.1%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 97.4%

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

10. Simplified 97.4%

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

    1. associate-/l* N/A

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

    2. *-commutative N/A

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

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

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

    4. frac-2neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{36}\right)}{\mathsf{neg}\left(\left(\frac{1}{6} + \frac{1}{180} \cdot \left(x \cdot x\right)\right)\right)}\right), x\right)\right) \]

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

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

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

    9. metadata-eval N/A

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

    10. metadata-eval N/A

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

    11. *-commutative N/A

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

    12. distribute-neg-in N/A

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

    13. unsub-neg N/A

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

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

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

    15. metadata-eval N/A

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

    16. associate-*l* N/A

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

    17. *-commutative N/A

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

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

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

    19. *-commutative N/A

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

    20. *-lowering-*.f64 97.5%

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

12. Applied egg-rr 97.5%

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

13. Final simplification 97.5%

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

Alternative 3: 96.5% 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 50.7%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 97.1%

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

5. Simplified 97.1%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
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 97.2%

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

7. Applied egg-rr 97.2%

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

8. Final simplification 97.2%

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

Alternative 4: 96.5% accurate, 40.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.7%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 97.1%

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

5. Simplified 97.1%

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

Alternative 5: 4.5% accurate, 203.0× speedup

Math

\[\begin{array}{l} \\ 5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 5.0)

C

double code(double x) {
	return 5.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 5.0d0
end function

Java

public static double code(double x) {
	return 5.0;
}

Python

def code(x):
	return 5.0

Julia

function code(x)
	return 5.0
end

MATLAB

function tmp = code(x)
	tmp = 5.0;
end

Wolfram

code[x_] := 5.0

Derivation

1. Initial program 50.7%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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 97.2%

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

5. Simplified 97.2%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

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

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

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

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

   7. metadata-eval N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. swap-sqr N/A

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

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

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

   12. metadata-eval N/A

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

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

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

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

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

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

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

   16. *-commutative N/A

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

7. Applied egg-rr 97.1%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 97.4%

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

10. Simplified 97.4%

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

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{5} \]
12. Step-by-step derivation

13. Simplified 4.3%

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

Developer Target 1: 98.0% 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 2024158 
(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)))