bug500, discussion (missed optimization)

Percentage Accurate: 52.2% → 97.3%

Time: 17.2s

Alternatives: 10

Speedup: 19.3×

• Could not converge on a ground truth

  1. x = 9.972661012474258e+254

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: 52.2% 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: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\\ t_1 := x \cdot \left(x \cdot t\_0\right)\\ \mathsf{log1p}\left(t\_1 \cdot \left(t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_1\right)\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(t\_0 \cdot t\_0\right), t\_0 \cdot \left(x \cdot \left(-x\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          (* x x)
          (fma x (* x 0.0001984126984126984) 0.008333333333333333)
          0.16666666666666666))
        (t_1 (* x (* x t_0))))
   (-
    (log1p (* t_1 (* t_0 (* (* x x) t_1))))
    (log1p (fma (* x x) (* (* x x) (* t_0 t_0)) (* t_0 (* x (- x))))))))

C

double code(double x) {
	double t_0 = fma((x * x), fma(x, (x * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666);
	double t_1 = x * (x * t_0);
	return log1p((t_1 * (t_0 * ((x * x) * t_1)))) - log1p(fma((x * x), ((x * x) * (t_0 * t_0)), (t_0 * (x * -x))));
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666)
	t_1 = Float64(x * Float64(x * t_0))
	return Float64(log1p(Float64(t_1 * Float64(t_0 * Float64(Float64(x * x) * t_1)))) - log1p(fma(Float64(x * x), Float64(Float64(x * x) * Float64(t_0 * t_0)), Float64(t_0 * Float64(x * Float64(-x))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(N[Log[1 + N[(t$95$1 * N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \log \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

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

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   14. *-lowering-*.f64 54.0

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

5. Simplified 54.0%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]

7. Applied egg-rr 95.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right) \cdot \left(\left(\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right)} \]

8. Final simplification 95.5%

   \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right)\right)\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right) \cdot \left(x \cdot \left(-x\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 2: 97.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 3.936759889140842 \cdot 10^{-8}, -6.944444444444444 \cdot 10^{-5}\right)}{\mathsf{fma}\left(x, x \cdot 0.0001984126984126984, -0.008333333333333333\right)}, 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log1p
  (*
   x
   (*
    x
    (fma
     (* x x)
     (/
      (fma (* x (* x (* x x))) 3.936759889140842e-8 -6.944444444444444e-5)
      (fma x (* x 0.0001984126984126984) -0.008333333333333333))
     0.16666666666666666)))))

C

double code(double x) {
	return log1p((x * (x * fma((x * x), (fma((x * (x * (x * x))), 3.936759889140842e-8, -6.944444444444444e-5) / fma(x, (x * 0.0001984126984126984), -0.008333333333333333)), 0.16666666666666666))));
}

Julia

function code(x)
	return log1p(Float64(x * Float64(x * fma(Float64(x * x), Float64(fma(Float64(x * Float64(x * Float64(x * x))), 3.936759889140842e-8, -6.944444444444444e-5) / fma(x, Float64(x * 0.0001984126984126984), -0.008333333333333333)), 0.16666666666666666))))
end

Wolfram

code[x_] := N[Log[1 + N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.936759889140842e-8 + -6.944444444444444e-5), $MachinePrecision] / N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \log \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

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

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   14. *-lowering-*.f64 54.0

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

5. Simplified 54.0%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)} \]

   2. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)}\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)} + \frac{1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}, \frac{1}{6}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}, \frac{1}{6}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right)\right)\right) \]

   11. *-lowering-*.f64 95.4

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right) \]

7. Applied egg-rr 95.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right)} \]
8. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}}, \frac{1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) - \frac{1}{120} \cdot \frac{1}{120}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}}, \frac{1}{6}\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{\left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{5040}\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{5040}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\left(\left(x \cdot x\right) \cdot \frac{1}{5040}\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{5040}\right)} + \left(\mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   6. swap-sqr N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{5040} \cdot \frac{1}{5040}\right)} + \left(\mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{5040} \cdot \frac{1}{5040}, \mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \frac{1}{5040} \cdot \frac{1}{5040}, \mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   9. cube-unmult N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{{x}^{3}}, \frac{1}{5040} \cdot \frac{1}{5040}, \mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(\color{blue}{x \cdot {x}^{3}}, \frac{1}{5040} \cdot \frac{1}{5040}, \mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   11. cube-unmult N/A

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, \frac{1}{5040} \cdot \frac{1}{5040}, \mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, \frac{1}{5040} \cdot \frac{1}{5040}, \mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right), \frac{1}{5040} \cdot \frac{1}{5040}, \mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \color{blue}{\frac{1}{25401600}}, \mathsf{neg}\left(\frac{1}{120} \cdot \frac{1}{120}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \frac{1}{25401600}, \mathsf{neg}\left(\color{blue}{\frac{1}{14400}}\right)\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \frac{1}{25401600}, \color{blue}{\frac{-1}{14400}}\right)}{x \cdot \left(x \cdot \frac{1}{5040}\right) - \frac{1}{120}}, \frac{1}{6}\right)\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \frac{1}{25401600}, \frac{-1}{14400}\right)}{\color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right) + \left(\mathsf{neg}\left(\frac{1}{120}\right)\right)}}, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \frac{1}{25401600}, \frac{-1}{14400}\right)}{\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \mathsf{neg}\left(\frac{1}{120}\right)\right)}}, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), \frac{1}{25401600}, \frac{-1}{14400}\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{5040}}, \mathsf{neg}\left(\frac{1}{120}\right)\right)}, \frac{1}{6}\right)\right)\right) \]

   20. metadata-eval 95.4

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 3.936759889140842 \cdot 10^{-8}, -6.944444444444444 \cdot 10^{-5}\right)}{\mathsf{fma}\left(x, x \cdot 0.0001984126984126984, \color{blue}{-0.008333333333333333}\right)}, 0.16666666666666666\right)\right)\right) \]

9. Applied egg-rr 95.4%

   \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right), 3.936759889140842 \cdot 10^{-8}, -6.944444444444444 \cdot 10^{-5}\right)}{\mathsf{fma}\left(x, x \cdot 0.0001984126984126984, -0.008333333333333333\right)}}, 0.16666666666666666\right)\right)\right) \]
10. Add Preprocessing

Alternative 3: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log1p
  (*
   x
   (*
    x
    (fma
     (* x x)
     (fma x (* x 0.0001984126984126984) 0.008333333333333333)
     0.16666666666666666)))))

C

double code(double x) {
	return log1p((x * (x * fma((x * x), fma(x, (x * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666))));
}

Julia

function code(x)
	return log1p(Float64(x * Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666))))
end

Wolfram

code[x_] := N[Log[1 + N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \log \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

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

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), 1\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}}, 1\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right) + \frac{1}{6}, 1\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)} + \frac{1}{6}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), \frac{1}{6}\right)}, 1\right)\right) \]

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

      \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{5040} \cdot {x}^{2} + \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{5040}} + \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

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

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right), 1\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5040}, \frac{1}{120}\right), \frac{1}{6}\right), 1\right)\right) \]

   14. *-lowering-*.f64 54.0

       \[\leadsto \log \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right) \]

5. Simplified 54.0%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), 1\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \log \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)} \]

   2. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)} \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{log1p}\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)\right) + \frac{1}{6}\right)\right)}\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}\right)} + \frac{1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}, \frac{1}{6}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(x \cdot x\right) \cdot \frac{1}{5040} + \frac{1}{120}, \frac{1}{6}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{5040}\right)} + \frac{1}{120}, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5040}, \frac{1}{120}\right)}, \frac{1}{6}\right)\right)\right) \]

   11. *-lowering-*.f64 95.4

       \[\leadsto \mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0001984126984126984}, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right) \]

7. Applied egg-rr 95.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right)\right)\right)} \]
8. Add Preprocessing

Alternative 4: 97.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   x
   (/
    1.0
    (fma
     x
     (* x (fma x (* x 0.0003527336860670194) -0.005555555555555556))
     0.16666666666666666)))))

C

double code(double x) {
	return x * (x / (1.0 / fma(x, (x * fma(x, (x * 0.0003527336860670194), -0.005555555555555556)), 0.16666666666666666)));
}

Julia

function code(x)
	return Float64(x * Float64(x / Float64(1.0 / fma(x, Float64(x * fma(x, Float64(x * 0.0003527336860670194), -0.005555555555555556)), 0.16666666666666666))))
end

Wolfram

code[x_] := N[(x * N[(x / N[(1.0 / N[(x * N[(x * N[(x * N[(x * 0.0003527336860670194), $MachinePrecision] + -0.005555555555555556), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.8%

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

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

   9. unpow2 N/A

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

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

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. metadata-eval N/A

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

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

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

   17. *-lowering-*.f64 95.1

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

5. Simplified 95.1%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

7. Applied egg-rr 95.2%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}}} \]
8. Add Preprocessing

Alternative 5: 97.1% accurate, 6.4× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

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

   9. unpow2 N/A

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

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

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. metadata-eval N/A

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

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

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

   17. *-lowering-*.f64 95.1

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

5. Simplified 95.1%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   9. *-lowering-*.f64 95.2

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

7. Applied egg-rr 95.2%

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

8. Final simplification 95.2%

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

Alternative 6: 97.2% accurate, 6.4× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

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

   9. unpow2 N/A

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

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

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. metadata-eval N/A

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

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

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

   17. *-lowering-*.f64 95.1

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

5. Simplified 95.1%

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

Alternative 7: 97.2% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

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

   9. unpow2 N/A

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

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

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. metadata-eval N/A

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

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

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

   17. *-lowering-*.f64 95.1

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

5. Simplified 95.1%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

7. Applied egg-rr 95.2%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}}} \]

8. Taylor expanded in x around 0

   \[\leadsto x \cdot \frac{x}{\color{blue}{6 + \frac{1}{5} \cdot {x}^{2}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{1}{5} \cdot {x}^{2} + 6}} \]

   2. unpow2 N/A

      \[\leadsto x \cdot \frac{x}{\frac{1}{5} \cdot \color{blue}{\left(x \cdot x\right)} + 6} \]

   3. associate-*r* N/A

      \[\leadsto x \cdot \frac{x}{\color{blue}{\left(\frac{1}{5} \cdot x\right) \cdot x} + 6} \]

   4. *-commutative N/A

      \[\leadsto x \cdot \frac{x}{\color{blue}{x \cdot \left(\frac{1}{5} \cdot x\right)} + 6} \]

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

      \[\leadsto x \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{5} \cdot x, 6\right)}} \]

   6. *-commutative N/A

      \[\leadsto x \cdot \frac{x}{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{5}}, 6\right)} \]

   7. *-lowering-*.f64 95.1

      \[\leadsto x \cdot \frac{x}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.2}, 6\right)} \]

10. Simplified 95.1%

    \[\leadsto x \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(x, x \cdot 0.2, 6\right)}} \]
11. Add Preprocessing

Alternative 8: 96.8% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 94.5

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

5. Simplified 94.5%

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

Alternative 9: 96.7% accurate, 12.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

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

   9. unpow2 N/A

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

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

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. unpow2 N/A

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

   14. associate-*l* N/A

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

   15. metadata-eval N/A

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

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

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

   17. *-lowering-*.f64 95.1

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

5. Simplified 95.1%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. flip-+ N/A

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

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

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

7. Applied egg-rr 95.2%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0003527336860670194, -0.005555555555555556\right), 0.16666666666666666\right)}}} \]

8. Taylor expanded in x around 0

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

10. Simplified 94.4%

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

Alternative 10: 96.6% 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 54.8%

   \[\log \left(\frac{\sinh x}{x}\right) \]
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. unpow2 N/A

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

   3. *-lowering-*.f64 94.4

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

5. Simplified 94.4%

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

6. Final simplification 94.4%

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

Developer Target 1: 98.0% 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 2024204 
(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)))