bug500, discussion (missed optimization)

Percentage Accurate: 53.3% → 97.2%

Time: 16.6s

Alternatives: 8

Speedup: 19.3×

• Could not converge on a ground truth

  1. x = 1.0884787882528566e+202

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: 53.3% 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: 97.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right) \cdot 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right) \cdot \left(x \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma x x 0.0)
   (* (fma x x 0.0) 3.08641975308642e-5)
   -0.027777777777777776)
  (*
   x
   (/
    x
    (fma
     x
     (* x (fma (fma x x 0.0) 0.0003527336860670194 -0.005555555555555556))
     -0.16666666666666666)))))

C

double code(double x) {
	return fma(fma(x, x, 0.0), (fma(x, x, 0.0) * 3.08641975308642e-5), -0.027777777777777776) * (x * (x / fma(x, (x * fma(fma(x, x, 0.0), 0.0003527336860670194, -0.005555555555555556)), -0.16666666666666666)));
}

Julia

function code(x)
	return Float64(fma(fma(x, x, 0.0), Float64(fma(x, x, 0.0) * 3.08641975308642e-5), -0.027777777777777776) * Float64(x * Float64(x / fma(x, Float64(x * fma(fma(x, x, 0.0), 0.0003527336860670194, -0.005555555555555556)), -0.16666666666666666))))
end

Wolfram

code[x_] := N[(N[(N[(x * x + 0.0), $MachinePrecision] * N[(N[(x * x + 0.0), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] * N[(x * N[(x / N[(x * N[(x * N[(N[(x * x + 0.0), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

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

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

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0003527336860670194, 0\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \cdot x\right)} \]

   2. flip-+ N/A

      \[\leadsto x \cdot \left(\color{blue}{\frac{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) - \frac{1}{6}}} \cdot x\right) \]

   3. associate-*l/ N/A

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) - \frac{1}{6}}} \]

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

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) - \frac{1}{6}}} \]

7. Applied egg-rr 97.1%

   \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right)\right)\right), -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{32400} \cdot {x}^{4} - \frac{1}{36}\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{32400} \cdot {x}^{4} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   2. *-commutative N/A

      \[\leadsto x \cdot \frac{\left(\color{blue}{{x}^{4} \cdot \frac{1}{32400}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   3. metadata-eval N/A

      \[\leadsto x \cdot \frac{\left({x}^{4} \cdot \frac{1}{32400} + \color{blue}{\frac{-1}{36}}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

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

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{32400}, \frac{-1}{36}\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   5. metadata-eval N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left({x}^{\color{blue}{\left(3 + 1\right)}}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   6. pow-plus N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{3} \cdot x}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   7. *-commutative N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot {x}^{3}}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

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

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot {x}^{3}}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   9. cube-mult N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   10. unpow2 N/A

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right), \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

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

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   12. unpow2 N/A

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   13. *-lowering-*.f64 97.2

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \]

10. Simplified 97.2%

    \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{32400} + \frac{-1}{36}\right) \cdot x}{\left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{2835} + \frac{-1}{180}\right) + \frac{-1}{6}} \cdot x} \]

    2. associate-/l* N/A

       \[\leadsto \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{32400} + \frac{-1}{36}\right) \cdot \frac{x}{\left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{2835} + \frac{-1}{180}\right) + \frac{-1}{6}}\right)} \cdot x \]

    3. associate-*l* N/A

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{32400} + \frac{-1}{36}\right) \cdot \left(\frac{x}{\left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{2835} + \frac{-1}{180}\right) + \frac{-1}{6}} \cdot x\right)} \]

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

       \[\leadsto \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{32400} + \frac{-1}{36}\right) \cdot \left(\frac{x}{\left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \frac{1}{2835} + \frac{-1}{180}\right) + \frac{-1}{6}} \cdot x\right)} \]

12. Applied egg-rr 97.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right) \cdot 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right) \cdot \left(\frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \cdot x\right)} \]

13. Final simplification 97.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x, 0\right) \cdot 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right) \cdot \left(x \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)}\right) \]
14. Add Preprocessing

Alternative 2: 97.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   (* x (fma (* x (* x (* x x))) 3.08641975308642e-5 -0.027777777777777776))
   (fma
    (fma x x 0.0)
    (fma (fma x x 0.0) 0.0003527336860670194 -0.005555555555555556)
    -0.16666666666666666))))

C

double code(double x) {
	return x * ((x * fma((x * (x * (x * x))), 3.08641975308642e-5, -0.027777777777777776)) / fma(fma(x, x, 0.0), fma(fma(x, x, 0.0), 0.0003527336860670194, -0.005555555555555556), -0.16666666666666666));
}

Julia

function code(x)
	return Float64(x * Float64(Float64(x * fma(Float64(x * Float64(x * Float64(x * x))), 3.08641975308642e-5, -0.027777777777777776)) / fma(fma(x, x, 0.0), fma(fma(x, x, 0.0), 0.0003527336860670194, -0.005555555555555556), -0.16666666666666666)))
end

Wolfram

code[x_] := N[(x * N[(N[(x * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5 + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x + 0.0), $MachinePrecision] * N[(N[(x * x + 0.0), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

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

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

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0003527336860670194, 0\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \cdot x\right)} \]

   2. flip-+ N/A

      \[\leadsto x \cdot \left(\color{blue}{\frac{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}}{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) - \frac{1}{6}}} \cdot x\right) \]

   3. associate-*l/ N/A

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) - \frac{1}{6}}} \]

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

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) - \frac{1}{6}}} \]

7. Applied egg-rr 97.1%

   \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right)\right)\right), -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{32400} \cdot {x}^{4} - \frac{1}{36}\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto x \cdot \frac{\color{blue}{\left(\frac{1}{32400} \cdot {x}^{4} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   2. *-commutative N/A

      \[\leadsto x \cdot \frac{\left(\color{blue}{{x}^{4} \cdot \frac{1}{32400}} + \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   3. metadata-eval N/A

      \[\leadsto x \cdot \frac{\left({x}^{4} \cdot \frac{1}{32400} + \color{blue}{\frac{-1}{36}}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

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

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{4}, \frac{1}{32400}, \frac{-1}{36}\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   5. metadata-eval N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left({x}^{\color{blue}{\left(3 + 1\right)}}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   6. pow-plus N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(\color{blue}{{x}^{3} \cdot x}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   7. *-commutative N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot {x}^{3}}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

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

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot {x}^{3}}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   9. cube-mult N/A

      \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   10. unpow2 N/A

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right), \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

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

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}, \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   12. unpow2 N/A

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), \frac{1}{32400}, \frac{-1}{36}\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{2835}, \frac{-1}{180}\right), \frac{-1}{6}\right)} \]

   13. *-lowering-*.f64 97.2

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \]

10. Simplified 97.2%

    \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right)} \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \]

11. Final simplification 97.2%

    \[\leadsto x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 3.08641975308642 \cdot 10^{-5}, -0.027777777777777776\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right), -0.16666666666666666\right)} \]
12. Add Preprocessing

Alternative 3: 97.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.16666666666666666, 0\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (fma x x 0.0)
  (* x (* x (fma (fma x x 0.0) 0.0003527336860670194 -0.005555555555555556)))
  (fma (fma x x 0.0) 0.16666666666666666 0.0)))

C

double code(double x) {
	return fma(fma(x, x, 0.0), (x * (x * fma(fma(x, x, 0.0), 0.0003527336860670194, -0.005555555555555556))), fma(fma(x, x, 0.0), 0.16666666666666666, 0.0));
}

Julia

function code(x)
	return fma(fma(x, x, 0.0), Float64(x * Float64(x * fma(fma(x, x, 0.0), 0.0003527336860670194, -0.005555555555555556))), fma(fma(x, x, 0.0), 0.16666666666666666, 0.0))
end

Wolfram

code[x_] := N[(N[(x * x + 0.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x + 0.0), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x + 0.0), $MachinePrecision] * 0.16666666666666666 + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

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

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

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0003527336860670194, 0\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) + \frac{1}{6}\right)} \]

   2. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(x \cdot x + 0\right)} \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) + \left(x \cdot x + 0\right) \cdot \frac{1}{6}} \]

   4. *-commutative N/A

      \[\leadsto \left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) + \color{blue}{\frac{1}{6} \cdot \left(x \cdot x + 0\right)} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x + 0, \left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right), \frac{1}{6} \cdot \left(x \cdot x + 0\right)\right)} \]

7. Applied egg-rr 97.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right)\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.16666666666666666, 0\right)\right)} \]
8. Add Preprocessing

Alternative 4: 97.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 0.16666666666666666, x, \mathsf{fma}\left(x, x, 0\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* x 0.16666666666666666)
  x
  (*
   (fma x x 0.0)
   (*
    x
    (* x (fma (fma x x 0.0) 0.0003527336860670194 -0.005555555555555556))))))

C

double code(double x) {
	return fma((x * 0.16666666666666666), x, (fma(x, x, 0.0) * (x * (x * fma(fma(x, x, 0.0), 0.0003527336860670194, -0.005555555555555556)))));
}

Julia

function code(x)
	return fma(Float64(x * 0.16666666666666666), x, Float64(fma(x, x, 0.0) * Float64(x * Float64(x * fma(fma(x, x, 0.0), 0.0003527336860670194, -0.005555555555555556)))))
end

Wolfram

code[x_] := N[(N[(x * 0.16666666666666666), $MachinePrecision] * x + N[(N[(x * x + 0.0), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x + 0.0), $MachinePrecision] * 0.0003527336860670194 + -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

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

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

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0003527336860670194, 0\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) + \frac{1}{6}\right)} \]

   2. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(x \cdot x + 0\right)} \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right) + \frac{1}{6}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(x \cdot x + 0\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right)} \]

   4. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(x \cdot x + 0\right) \cdot \frac{1}{6} + \left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot x + 0\right)} + \left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \]

   6. +-rgt-identity N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(x \cdot x\right)} + \left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} + \left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{6}\right)} \cdot x + \left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right) \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{1}{6}, x, \left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right)\right)} \]

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

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{1}{6}}, x, \left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{6}, x, \color{blue}{\left(x \cdot x + 0\right) \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{2835} + 0\right) + \frac{-1}{180}\right)\right)}\right) \]

7. Applied egg-rr 97.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.16666666666666666, x, \mathsf{fma}\left(x, x, 0\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), 0.0003527336860670194, -0.005555555555555556\right)\right)\right)\right)} \]
8. Add Preprocessing

Alternative 5: 97.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

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

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

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0003527336860670194, 0\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. *-lowering-*.f64 97.0

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0003527336860670194}, -0.005555555555555556\right), 0.16666666666666666\right)\right) \]

8. Simplified 97.0%

   \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}\right) \]
9. Add Preprocessing

Alternative 6: 96.7% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(-0.005555555555555556, x \cdot x, 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma x x 0.0) (fma -0.005555555555555556 (* x x) 0.16666666666666666)))

C

double code(double x) {
	return fma(x, x, 0.0) * fma(-0.005555555555555556, (x * x), 0.16666666666666666);
}

Julia

function code(x)
	return Float64(fma(x, x, 0.0) * fma(-0.005555555555555556, Float64(x * x), 0.16666666666666666))
end

Wolfram

code[x_] := N[(N[(x * x + 0.0), $MachinePrecision] * N[(-0.005555555555555556 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right)} \]

   2. remove-double-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

   3. +-rgt-identity N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right)\right) \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

   5. remove-double-neg N/A

      \[\leadsto \left(\color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right)\right) \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right)\right) \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(x \cdot x + \color{blue}{0}\right) \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}\right) + \frac{1}{6}\right)} \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{2835} + \frac{-1}{37800} \cdot {x}^{2}\right) - \frac{1}{180}, \frac{1}{6}\right)} \]

5. Simplified 96.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), -2.6455026455026456 \cdot 10^{-5}, 0.0003527336860670194\right), -0.005555555555555556\right), 0.16666666666666666\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\left(\frac{1}{6} + \frac{-1}{180} \cdot {x}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\left(\frac{-1}{180} \cdot {x}^{2} + \frac{1}{6}\right)} \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{180}, {x}^{2}, \frac{1}{6}\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(\frac{-1}{180}, \color{blue}{x \cdot x}, \frac{1}{6}\right) \]

   4. *-lowering-*.f64 96.7

      \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \mathsf{fma}\left(-0.005555555555555556, \color{blue}{x \cdot x}, 0.16666666666666666\right) \]

8. Simplified 96.7%

   \[\leadsto \mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\mathsf{fma}\left(-0.005555555555555556, x \cdot x, 0.16666666666666666\right)} \]
9. Add Preprocessing

Alternative 7: 96.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

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

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

5. Simplified 97.0%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.0003527336860670194, 0\right), -0.005555555555555556\right), 0.16666666666666666\right)\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 96.7

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(-0.005555555555555556, \color{blue}{x \cdot x}, 0.16666666666666666\right)\right) \]

8. Simplified 96.7%

   \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(-0.005555555555555556, x \cdot x, 0.16666666666666666\right)}\right) \]
9. Add Preprocessing

Alternative 8: 96.4% accurate, 19.3× 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 53.0%

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

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]

   2. remove-double-neg N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right)} \]

   3. +-rgt-identity N/A

      \[\leadsto \frac{1}{6} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) + 0\right)}\right)\right) \]

   4. distribute-neg-in N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) + \left(\mathsf{neg}\left(0\right)\right)\right)} \]

   5. remove-double-neg N/A

      \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{{x}^{2}} + \left(\mathsf{neg}\left(0\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{1}{6} \cdot \left(\color{blue}{x \cdot x} + \left(\mathsf{neg}\left(0\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{6} \cdot \left(x \cdot x + \color{blue}{0}\right) \]

   8. accelerator-lowering-fma.f64 96.3

      \[\leadsto 0.16666666666666666 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)} \]

5. Simplified 96.3%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \mathsf{fma}\left(x, x, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 96.3

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

7. Applied egg-rr 96.3%

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

8. Final simplification 96.3%

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

Developer Target 1: 97.8% accurate, 1.0× 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 2024195 
(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)))