expq2 (section 3.11)

Percentage Accurate: 38.2% → 100.0%

Time: 8.4s

Alternatives: 16

Speedup: 17.9×

Specification

\[710 > x\]

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(exp(x) / Float64(exp(x) - 1.0))
end

MATLAB

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

Wolfram

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

Initial Program: 38.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{e^{x} - 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(exp(x) / Float64(exp(x) - 1.0))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(0 - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.0 (expm1 (- 0.0 x))))

C

double code(double x) {
	return -1.0 / expm1((0.0 - x));
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1((0.0 - x));
}

Python

def code(x):
	return -1.0 / math.expm1((0.0 - x))

Julia

function code(x)
	return Float64(-1.0 / expm1(Float64(0.0 - x)))
end

Wolfram

code[x_] := N[(-1.0 / N[(Exp[N[(0.0 - x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

   5. distribute-neg-frac N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

   6. neg-sub0 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

   7. associate-+l- N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

   8. neg-sub0 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

   9. +-commutative N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

   10. sub-neg N/A

       \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

   11. div-sub N/A

       \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

   12. rec-exp N/A

       \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

   13. *-inverses N/A

       \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

   14. accelerator-lowering-expm1.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

   15. neg-lowering-neg.f64 100.0

       \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

5. Final simplification 100.0%

   \[\leadsto \frac{-1}{\mathsf{expm1}\left(0 - x\right)} \]
6. Add Preprocessing

Alternative 2: 91.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)\right), 0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5\right), 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/
    -1.0
    (fma x (* x (* x (fma x 0.041666666666666664 -0.16666666666666666))) 0.0))
   (/
    (fma
     x
     (fma x (fma x (* x -0.001388888888888889) 0.08333333333333333) 0.5)
     1.0)
    x)))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = -1.0 / fma(x, (x * (x * fma(x, 0.041666666666666664, -0.16666666666666666))), 0.0);
	} else {
		tmp = fma(x, fma(x, fma(x, (x * -0.001388888888888889), 0.08333333333333333), 0.5), 1.0) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(-1.0 / fma(x, Float64(x * Float64(x * fma(x, 0.041666666666666664, -0.16666666666666666))), 0.0));
	else
		tmp = Float64(fma(x, fma(x, fma(x, Float64(x * -0.001388888888888889), 0.08333333333333333), 0.5), 1.0) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(-1.0 / N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + 0}} \]

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

         \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1, 0\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, 0\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{-1}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right), -1\right)}, 0\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \frac{1}{2}}, -1\right), 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{24} \cdot x - \frac{1}{6}, \frac{1}{2}\right)}, -1\right), 0\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, \frac{1}{2}\right), -1\right), 0\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24} + \color{blue}{\frac{-1}{6}}, \frac{1}{2}\right), -1\right), 0\right)} \]

      11. accelerator-lowering-fma.f64 70.5

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}, 0.5\right), -1\right), 0\right)} \]

   7. Simplified 70.5%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)}, 0\right)} \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right), 0\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right), 0\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, 0\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right), 0\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}, 0\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}\right)\right), 0\right)} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24} + x \cdot \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}\right), 0\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x} \cdot \frac{1}{6}}\right)\right)\right)\right), 0\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + x \cdot \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right)\right), 0\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + x \cdot \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right), 0\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \color{blue}{\left(x \cdot \frac{1}{x}\right) \cdot \frac{-1}{6}}\right)\right), 0\right)} \]

      14. rgt-mult-inverse N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \color{blue}{1} \cdot \frac{-1}{6}\right)\right), 0\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \color{blue}{\frac{-1}{6}}\right)\right), 0\right)} \]

      16. accelerator-lowering-fma.f64 70.5

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}\right), 0\right)} \]

   10. Simplified 70.5%

       \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)\right)}, 0\right)} \]

   if 0.0 < (exp.f64 x)

   1. Initial program 5.8%

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

      1. accelerator-lowering-expm1.f64 100.0

         \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   6. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5\right), 1\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)\right), 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/
    -1.0
    (fma x (* x (* x (fma x 0.041666666666666664 -0.16666666666666666))) 0.0))
   (+
    0.5
    (fma
     x
     (fma -0.001388888888888889 (* x x) 0.08333333333333333)
     (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = -1.0 / fma(x, (x * (x * fma(x, 0.041666666666666664, -0.16666666666666666))), 0.0);
	} else {
		tmp = 0.5 + fma(x, fma(-0.001388888888888889, (x * x), 0.08333333333333333), (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(-1.0 / fma(x, Float64(x * Float64(x * fma(x, 0.041666666666666664, -0.16666666666666666))), 0.0));
	else
		tmp = Float64(0.5 + fma(x, fma(-0.001388888888888889, Float64(x * x), 0.08333333333333333), Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(-1.0 / N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(x * N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + 0}} \]

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

         \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1, 0\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, 0\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{-1}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right), -1\right)}, 0\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \frac{1}{2}}, -1\right), 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{24} \cdot x - \frac{1}{6}, \frac{1}{2}\right)}, -1\right), 0\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, \frac{1}{2}\right), -1\right), 0\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24} + \color{blue}{\frac{-1}{6}}, \frac{1}{2}\right), -1\right), 0\right)} \]

      11. accelerator-lowering-fma.f64 70.5

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}, 0.5\right), -1\right), 0\right)} \]

   7. Simplified 70.5%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{{x}^{3} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)}, 0\right)} \]
   9. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right), 0\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right), 0\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, 0\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right), 0\right)} \]

      6. associate-*l* N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}, 0\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}\right)\right), 0\right)} \]

      9. distribute-lft-in N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{24} + x \cdot \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}\right), 0\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + x \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{x} \cdot \frac{1}{6}}\right)\right)\right)\right), 0\right)} \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + x \cdot \color{blue}{\left(\frac{1}{x} \cdot \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right)\right), 0\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + x \cdot \left(\frac{1}{x} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right), 0\right)} \]

      13. associate-*r* N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \color{blue}{\left(x \cdot \frac{1}{x}\right) \cdot \frac{-1}{6}}\right)\right), 0\right)} \]

      14. rgt-mult-inverse N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \color{blue}{1} \cdot \frac{-1}{6}\right)\right), 0\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \color{blue}{\frac{-1}{6}}\right)\right), 0\right)} \]

      16. accelerator-lowering-fma.f64 70.5

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}\right), 0\right)} \]

   10. Simplified 70.5%

       \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)\right)}, 0\right)} \]

   if 0.0 < (exp.f64 x)

   1. Initial program 5.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}} + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{1}{2} \cdot x}}{x} + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot x\right) \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      12. lft-mult-inverse N/A

          \[\leadsto \color{blue}{1} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), \frac{1}{x} + 0.5\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}\right) + \frac{1}{x}\right) + \frac{1}{2}} \]

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

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}\right) + \frac{1}{x}\right) + \frac{1}{2}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}, \frac{1}{x}\right)} + \frac{1}{2} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{720}} + \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{720} \cdot \left(x \cdot x\right)} + \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2} \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{12}\right)}, \frac{1}{x}\right) + \frac{1}{2} \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{720}, \color{blue}{x \cdot x}, \frac{1}{12}\right), \frac{1}{x}\right) + \frac{1}{2} \]

      8. /-lowering-/.f64 99.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \color{blue}{\frac{1}{x}}\right) + 0.5 \]

   7. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \frac{1}{x}\right) + 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)\right), 0\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ -24.0 (* x (* x (* x x))))
   (+
    0.5
    (fma
     x
     (fma -0.001388888888888889 (* x x) 0.08333333333333333)
     (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = 0.5 + fma(x, fma(-0.001388888888888889, (x * x), 0.08333333333333333), (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(0.5 + fma(x, fma(-0.001388888888888889, Float64(x * x), 0.08333333333333333), Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(x * N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + 0}} \]

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

         \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1, 0\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, 0\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{-1}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right), -1\right)}, 0\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \frac{1}{2}}, -1\right), 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{24} \cdot x - \frac{1}{6}, \frac{1}{2}\right)}, -1\right), 0\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, \frac{1}{2}\right), -1\right), 0\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24} + \color{blue}{\frac{-1}{6}}, \frac{1}{2}\right), -1\right), 0\right)} \]

      11. accelerator-lowering-fma.f64 70.5

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}, 0.5\right), -1\right), 0\right)} \]

   7. Simplified 70.5%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-24}{{x}^{4}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-24}{{x}^{4}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-24}{{x}^{\color{blue}{\left(3 + 1\right)}}} \]

      3. pow-plus N/A

         \[\leadsto \frac{-24}{\color{blue}{{x}^{3} \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{-24}{\color{blue}{x \cdot {x}^{3}}} \]

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

         \[\leadsto \frac{-24}{\color{blue}{x \cdot {x}^{3}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{-24}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}} \]

      7. unpow2 N/A

         \[\leadsto \frac{-24}{x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)} \]

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

         \[\leadsto \frac{-24}{x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{-24}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

      10. *-lowering-*.f64 70.5

          \[\leadsto \frac{-24}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   10. Simplified 70.5%

       \[\leadsto \color{blue}{\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]

   if 0.0 < (exp.f64 x)

   1. Initial program 5.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}} + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{1}{2} \cdot x}}{x} + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot x\right) \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      12. lft-mult-inverse N/A

          \[\leadsto \color{blue}{1} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), \frac{1}{x} + 0.5\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}\right) + \frac{1}{x}\right) + \frac{1}{2}} \]

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

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}\right) + \frac{1}{x}\right) + \frac{1}{2}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}, \frac{1}{x}\right)} + \frac{1}{2} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{720}} + \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{720} \cdot \left(x \cdot x\right)} + \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2} \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{12}\right)}, \frac{1}{x}\right) + \frac{1}{2} \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{720}, \color{blue}{x \cdot x}, \frac{1}{12}\right), \frac{1}{x}\right) + \frac{1}{2} \]

      8. /-lowering-/.f64 99.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \color{blue}{\frac{1}{x}}\right) + 0.5 \]

   7. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \frac{1}{x}\right) + 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ -24.0 (* x (* x (* x x))))
   (+ 0.5 (/ (fma x (* x 0.08333333333333333) 1.0) x))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = 0.5 + (fma(x, (x * 0.08333333333333333), 1.0) / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(0.5 + Float64(fma(x, Float64(x * 0.08333333333333333), 1.0) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(x * N[(x * 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + 0}} \]

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

         \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1, 0\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, 0\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{-1}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right), -1\right)}, 0\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \frac{1}{2}}, -1\right), 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{24} \cdot x - \frac{1}{6}, \frac{1}{2}\right)}, -1\right), 0\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, \frac{1}{2}\right), -1\right), 0\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24} + \color{blue}{\frac{-1}{6}}, \frac{1}{2}\right), -1\right), 0\right)} \]

      11. accelerator-lowering-fma.f64 70.5

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}, 0.5\right), -1\right), 0\right)} \]

   7. Simplified 70.5%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-24}{{x}^{4}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-24}{{x}^{4}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{-24}{{x}^{\color{blue}{\left(3 + 1\right)}}} \]

      3. pow-plus N/A

         \[\leadsto \frac{-24}{\color{blue}{{x}^{3} \cdot x}} \]

      4. *-commutative N/A

         \[\leadsto \frac{-24}{\color{blue}{x \cdot {x}^{3}}} \]

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

         \[\leadsto \frac{-24}{\color{blue}{x \cdot {x}^{3}}} \]

      6. cube-mult N/A

         \[\leadsto \frac{-24}{x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}} \]

      7. unpow2 N/A

         \[\leadsto \frac{-24}{x \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)} \]

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

         \[\leadsto \frac{-24}{x \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}} \]

      9. unpow2 N/A

         \[\leadsto \frac{-24}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

      10. *-lowering-*.f64 70.5

          \[\leadsto \frac{-24}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   10. Simplified 70.5%

       \[\leadsto \color{blue}{\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}} \]

   if 0.0 < (exp.f64 x)

   1. Initial program 5.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}} + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{1}{2} \cdot x}}{x} + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot x\right) \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      12. lft-mult-inverse N/A

          \[\leadsto \color{blue}{1} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), \frac{1}{x} + 0.5\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}\right) + \frac{1}{x}\right) + \frac{1}{2}} \]

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

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}\right) + \frac{1}{x}\right) + \frac{1}{2}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}, \frac{1}{x}\right)} + \frac{1}{2} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{720}} + \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{720} \cdot \left(x \cdot x\right)} + \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2} \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{12}\right)}, \frac{1}{x}\right) + \frac{1}{2} \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{720}, \color{blue}{x \cdot x}, \frac{1}{12}\right), \frac{1}{x}\right) + \frac{1}{2} \]

      8. /-lowering-/.f64 99.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \color{blue}{\frac{1}{x}}\right) + 0.5 \]

   7. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \frac{1}{x}\right) + 0.5} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{12} \cdot {x}^{2}}{x}} + \frac{1}{2} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{12} \cdot {x}^{2}}{x}} + \frac{1}{2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{12} \cdot {x}^{2} + 1}}{x} + \frac{1}{2} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{1}{12} \cdot \color{blue}{\left(x \cdot x\right)} + 1}{x} + \frac{1}{2} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot x} + 1}{x} + \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{12} \cdot x\right)} + 1}{x} + \frac{1}{2} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{12} \cdot x, 1\right)}}{x} + \frac{1}{2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{12}}, 1\right)}{x} + \frac{1}{2} \]

      8. *-lowering-*.f64 98.9

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.08333333333333333}, 1\right)}{x} + 0.5 \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)}{x}} + 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ 6.0 (* x (* x x)))
   (+ 0.5 (/ (fma x (* x 0.08333333333333333) 1.0) x))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = 0.5 + (fma(x, (x * 0.08333333333333333), 1.0) / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(6.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(0.5 + Float64(fma(x, Float64(x * 0.08333333333333333), 1.0) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(6.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(N[(x * N[(x * 0.08333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 62.1

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)}, -1\right), 0\right)} \]

   7. Simplified 62.1%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 62.1

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

   10. Simplified 62.1%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 5.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \color{blue}{\left(x \cdot \frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(1 + \left(\color{blue}{\frac{1}{2} \cdot x} + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot x\right) + x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      7. distribute-lft-in N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right)} \]

      8. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}} + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      9. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{1}{2} \cdot x}}{x} + \frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)\right) + \frac{1 + \frac{1}{2} \cdot x}{x}} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot x\right) \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      12. lft-mult-inverse N/A

          \[\leadsto \color{blue}{1} \cdot \left(x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right) + \frac{1 + \frac{1}{2} \cdot x}{x} \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)} + \frac{1 + \frac{1}{2} \cdot x}{x} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), \frac{1}{x} + 0.5\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}\right) + \frac{1}{x}\right) + \frac{1}{2}} \]

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

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}\right) + \frac{1}{x}\right) + \frac{1}{2}} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \frac{-1}{720}\right) + \frac{1}{12}, \frac{1}{x}\right)} + \frac{1}{2} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot \frac{-1}{720}} + \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{720} \cdot \left(x \cdot x\right)} + \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2} \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{-1}{720}, x \cdot x, \frac{1}{12}\right)}, \frac{1}{x}\right) + \frac{1}{2} \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(\frac{-1}{720}, \color{blue}{x \cdot x}, \frac{1}{12}\right), \frac{1}{x}\right) + \frac{1}{2} \]

      8. /-lowering-/.f64 99.1

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \color{blue}{\frac{1}{x}}\right) + 0.5 \]

   7. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(-0.001388888888888889, x \cdot x, 0.08333333333333333\right), \frac{1}{x}\right) + 0.5} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{12} \cdot {x}^{2}}{x}} + \frac{1}{2} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{12} \cdot {x}^{2}}{x}} + \frac{1}{2} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{12} \cdot {x}^{2} + 1}}{x} + \frac{1}{2} \]

      3. unpow2 N/A

         \[\leadsto \frac{\frac{1}{12} \cdot \color{blue}{\left(x \cdot x\right)} + 1}{x} + \frac{1}{2} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{12} \cdot x\right) \cdot x} + 1}{x} + \frac{1}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{12} \cdot x\right)} + 1}{x} + \frac{1}{2} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{12} \cdot x, 1\right)}}{x} + \frac{1}{2} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{12}}, 1\right)}{x} + \frac{1}{2} \]

      8. *-lowering-*.f64 98.9

         \[\leadsto \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.08333333333333333}, 1\right)}{x} + 0.5 \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)}{x}} + 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{\mathsf{fma}\left(x, x \cdot 0.08333333333333333, 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, 0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ 6.0 (* x (* x x)))
   (fma x 0.08333333333333333 (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = fma(x, 0.08333333333333333, (0.5 + (1.0 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(6.0 / Float64(x * Float64(x * x)));
	else
		tmp = fma(x, 0.08333333333333333, Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(6.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 0.08333333333333333 + N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 62.1

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)}, -1\right), 0\right)} \]

   7. Simplified 62.1%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 62.1

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

   10. Simplified 62.1%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 5.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right) + 1\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(x \cdot \color{blue}{\left(\frac{1}{12} \cdot x + \frac{1}{2}\right)} + 1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot x\right) + x \cdot \frac{1}{2}\right)} + 1\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(\left(x \cdot \left(\frac{1}{12} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot x}\right) + 1\right) \]

      8. associate-+l+ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot x\right) + \left(\frac{1}{2} \cdot x + 1\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{12} \cdot x\right) + \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{12} \cdot x\right)\right) + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot x\right) \cdot \left(\frac{1}{12} \cdot x\right)} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

      12. lft-mult-inverse N/A

          \[\leadsto \color{blue}{1} \cdot \left(\frac{1}{12} \cdot x\right) + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{1}{12} \cdot x} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{12}} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

      15. associate-*l/ N/A

          \[\leadsto x \cdot \frac{1}{12} + \color{blue}{\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}} \]

      16. *-lft-identity N/A

          \[\leadsto x \cdot \frac{1}{12} + \frac{\color{blue}{1 + \frac{1}{2} \cdot x}}{x} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{12}, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \frac{\color{blue}{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}}{x}\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \color{blue}{\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)}\right) \]

      20. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \color{blue}{1 \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{x}}\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.08333333333333333, \frac{1}{x} + 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, 0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return math.exp(x) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{e^{x}}{\color{blue}{x}} \]
4. Step-by-step derivation

5. Simplified 98.7%

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

Alternative 9: 94.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \mathsf{fma}\left(t\_0, x \cdot \left(x \cdot t\_0\right), -1\right)}{\mathsf{fma}\left(x, t\_0, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 -0.16666666666666666) 0.5)))
   (if (<= x -1e+103)
     (/ 6.0 (* x (* x x)))
     (/ -1.0 (/ (* x (fma t_0 (* x (* x t_0)) -1.0)) (fma x t_0 1.0))))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5);
	double tmp;
	if (x <= -1e+103) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = -1.0 / ((x * fma(t_0, (x * (x * t_0)), -1.0)) / fma(x, t_0, 1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5)
	tmp = 0.0
	if (x <= -1e+103)
		tmp = Float64(6.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * fma(t_0, Float64(x * Float64(x * t_0)), -1.0)) / fma(x, t_0, 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[x, -1e+103], N[(6.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(t$95$0 * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e103

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 100.0

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)}, -1\right), 0\right)} \]

   7. Simplified 100.0%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{6}{{x}^{3}}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 100.0

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

   10. Simplified 100.0%

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

   if -1e103 < x

   1. Initial program 20.3%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + 0}} \]

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

         \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1, 0\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, 0\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{-1}, 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right), -1\right)}, 0\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \frac{1}{2}}, -1\right), 0\right)} \]

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

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{24} \cdot x - \frac{1}{6}, \frac{1}{2}\right)}, -1\right), 0\right)} \]

      8. sub-neg N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, \frac{1}{2}\right), -1\right), 0\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24} + \color{blue}{\frac{-1}{6}}, \frac{1}{2}\right), -1\right), 0\right)} \]

      11. accelerator-lowering-fma.f64 87.7

          \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}, 0.5\right), -1\right), 0\right)} \]

   7. Simplified 87.7%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right), 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right) + -1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right) + -1\right) \cdot x}} \]

      3. flip-+ N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right)\right) - -1 \cdot -1}{x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right) - -1}} \cdot x} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right)\right) - -1 \cdot -1\right) \cdot x}{x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right) - -1}}} \]

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

         \[\leadsto \frac{-1}{\color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right)\right) - -1 \cdot -1\right) \cdot x}{x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right) - -1}}} \]

   9. Applied egg-rr 91.6%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), x \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right)\right), -1\right) \cdot x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), 1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), x \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right)\right), -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), x \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)\right), -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e+154)
   (/ -2.0 (* x x))
   (/
    -1.0
    (/
     (*
      x
      (fma
       (fma x -0.16666666666666666 0.5)
       (* x (* x (fma x -0.16666666666666666 0.5)))
       -1.0))
     (fma x (fma x -0.16666666666666666 0.5) 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -4e+154) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = -1.0 / ((x * fma(fma(x, -0.16666666666666666, 0.5), (x * (x * fma(x, -0.16666666666666666, 0.5))), -1.0)) / fma(x, fma(x, -0.16666666666666666, 0.5), 1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e+154)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * fma(fma(x, -0.16666666666666666, 0.5), Float64(x * Float64(x * fma(x, -0.16666666666666666, 0.5))), -1.0)) / fma(x, fma(x, -0.16666666666666666, 0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4e+154], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] * N[(x * N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * -0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000015e154

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 100.0

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}, 0\right)} \]

   7. Simplified 100.0%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 100.0

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]

   if -4.00000000000000015e154 < x

   1. Initial program 25.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 85.3

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)}, -1\right), 0\right)} \]

   7. Simplified 85.3%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), -1\right), 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. flip-+ N/A

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

      4. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 90.3%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), x \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)\right), -1\right) \cdot x}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), 1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), x \cdot \left(x \cdot \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right)\right), -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.16666666666666666, 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right), 0\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (fma
   x
   (fma x (fma x (fma x 0.041666666666666664 -0.16666666666666666) 0.5) -1.0)
   0.0)))

C

double code(double x) {
	return -1.0 / fma(x, fma(x, fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5), -1.0), 0.0);
}

Julia

function code(x)
	return Float64(-1.0 / fma(x, fma(x, fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5), -1.0), 0.0))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

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

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

   2. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

   5. distribute-neg-frac N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

   6. neg-sub0 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

   7. associate-+l- N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

   8. neg-sub0 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

   9. +-commutative N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

   10. sub-neg N/A

       \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

   11. div-sub N/A

       \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

   12. rec-exp N/A

       \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

   13. *-inverses N/A

       \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

   14. accelerator-lowering-expm1.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

   15. neg-lowering-neg.f64 100.0

       \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + 0}} \]

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

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1, 0\right)}} \]

   3. sub-neg N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}, 0\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{-1}, 0\right)} \]

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

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right), -1\right)}, 0\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \frac{1}{2}}, -1\right), 0\right)} \]

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

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{24} \cdot x - \frac{1}{6}, \frac{1}{2}\right)}, -1\right), 0\right)} \]

   8. sub-neg N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{1}{24} \cdot x + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, \frac{1}{2}\right), -1\right), 0\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{1}{24} + \color{blue}{\frac{-1}{6}}, \frac{1}{2}\right), -1\right), 0\right)} \]

   11. accelerator-lowering-fma.f64 90.0

       \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)}, 0.5\right), -1\right), 0\right)} \]

7. Simplified 90.0%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right), 0\right)}} \]
8. Add Preprocessing

Alternative 12: 83.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, 0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.5)
   (/ -2.0 (* x x))
   (fma x 0.08333333333333333 (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.5) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = fma(x, 0.08333333333333333, (0.5 + (1.0 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.5)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = fma(x, 0.08333333333333333, Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.5], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * 0.08333333333333333 + N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 46.7

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}, 0\right)} \]

   7. Simplified 46.7%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 46.7

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

   10. Simplified 46.7%

       \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]

   if -4.5 < x

   1. Initial program 5.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}{x}} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right) + 1\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(x \cdot \color{blue}{\left(\frac{1}{12} \cdot x + \frac{1}{2}\right)} + 1\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \frac{1}{x} \cdot \left(\color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot x\right) + x \cdot \frac{1}{2}\right)} + 1\right) \]

      7. *-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(\left(x \cdot \left(\frac{1}{12} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot x}\right) + 1\right) \]

      8. associate-+l+ N/A

         \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot x\right) + \left(\frac{1}{2} \cdot x + 1\right)\right)} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{12} \cdot x\right) + \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{12} \cdot x\right)\right) + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]

      11. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot x\right) \cdot \left(\frac{1}{12} \cdot x\right)} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

      12. lft-mult-inverse N/A

          \[\leadsto \color{blue}{1} \cdot \left(\frac{1}{12} \cdot x\right) + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

      13. *-lft-identity N/A

          \[\leadsto \color{blue}{\frac{1}{12} \cdot x} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

      14. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{12}} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

      15. associate-*l/ N/A

          \[\leadsto x \cdot \frac{1}{12} + \color{blue}{\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}} \]

      16. *-lft-identity N/A

          \[\leadsto x \cdot \frac{1}{12} + \frac{\color{blue}{1 + \frac{1}{2} \cdot x}}{x} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{12}, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \frac{\color{blue}{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}}{x}\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \color{blue}{\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)}\right) \]

      20. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \color{blue}{1 \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{x}}\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.08333333333333333, \frac{1}{x} + 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, 0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.8) (/ -2.0 (* x x)) (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.8) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.8d0)) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = 0.5d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.8) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.8:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.8)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = Float64(0.5 + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.8)
		tmp = -2.0 / (x * x);
	else
		tmp = 0.5 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.8], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004

   1. Initial program 100.0%

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

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}} \]

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{e^{x}}}} \]

      6. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \left(e^{x} - 1\right)}}{e^{x}}} \]

      7. associate-+l- N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(0 - e^{x}\right) + 1}}{e^{x}}} \]

      8. neg-sub0 N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\mathsf{neg}\left(e^{x}\right)\right)} + 1}{e^{x}}} \]

      9. +-commutative N/A

         \[\leadsto \frac{-1}{\frac{\color{blue}{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}}{e^{x}}} \]

      10. sub-neg N/A

          \[\leadsto \frac{-1}{\frac{\color{blue}{1 - e^{x}}}{e^{x}}} \]

      11. div-sub N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{1}{e^{x}} - \frac{e^{x}}{e^{x}}}} \]

      12. rec-exp N/A

          \[\leadsto \frac{-1}{\color{blue}{e^{\mathsf{neg}\left(x\right)}} - \frac{e^{x}}{e^{x}}} \]

      13. *-inverses N/A

          \[\leadsto \frac{-1}{e^{\mathsf{neg}\left(x\right)} - \color{blue}{1}} \]

      14. accelerator-lowering-expm1.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\mathsf{expm1}\left(\mathsf{neg}\left(x\right)\right)}} \]

      15. neg-lowering-neg.f64 100.0

          \[\leadsto \frac{-1}{\mathsf{expm1}\left(\color{blue}{-x}\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]

   5. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 46.7

         \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}, 0\right)} \]

   7. Simplified 46.7%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 0\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{-2}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 46.7

         \[\leadsto \frac{-2}{\color{blue}{x \cdot x}} \]

   10. Simplified 46.7%

       \[\leadsto \color{blue}{\frac{-2}{x \cdot x}} \]

   if -1.80000000000000004 < x

   1. Initial program 5.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval 98.6

         \[\leadsto \frac{1}{x} + \color{blue}{0.5} \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.4% accurate, 17.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 x))

C

double code(double x) {
	return 1.0 / x;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / x;
}

Python

def code(x):
	return 1.0 / x

Julia

function code(x)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / x;
end

Wolfram

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

Derivation

1. Initial program 35.2%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 69.0

      \[\leadsto \color{blue}{\frac{1}{x}} \]

5. Simplified 69.0%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Add Preprocessing

Alternative 15: 3.3% accurate, 35.8× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.08333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.08333333333333333))

C

double code(double x) {
	return x * 0.08333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.08333333333333333d0
end function

Java

public static double code(double x) {
	return x * 0.08333333333333333;
}

Python

def code(x):
	return x * 0.08333333333333333

Julia

function code(x)
	return Float64(x * 0.08333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.08333333333333333;
end

Wolfram

code[x_] := N[(x * 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]
4. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \color{blue}{1 \cdot \frac{1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)}{x}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)}{x}} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{1}{12} \cdot x\right) + 1\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{1}{x} \cdot \left(x \cdot \color{blue}{\left(\frac{1}{12} \cdot x + \frac{1}{2}\right)} + 1\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \frac{1}{x} \cdot \left(\color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot x\right) + x \cdot \frac{1}{2}\right)} + 1\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{x} \cdot \left(\left(x \cdot \left(\frac{1}{12} \cdot x\right) + \color{blue}{\frac{1}{2} \cdot x}\right) + 1\right) \]

   8. associate-+l+ N/A

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(x \cdot \left(\frac{1}{12} \cdot x\right) + \left(\frac{1}{2} \cdot x + 1\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{12} \cdot x\right) + \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)}\right) \]

   10. distribute-lft-in N/A

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot \left(\frac{1}{12} \cdot x\right)\right) + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)} \]

   11. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot x\right) \cdot \left(\frac{1}{12} \cdot x\right)} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

   12. lft-mult-inverse N/A

       \[\leadsto \color{blue}{1} \cdot \left(\frac{1}{12} \cdot x\right) + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

   13. *-lft-identity N/A

       \[\leadsto \color{blue}{\frac{1}{12} \cdot x} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

   14. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot \frac{1}{12}} + \frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right) \]

   15. associate-*l/ N/A

       \[\leadsto x \cdot \frac{1}{12} + \color{blue}{\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}} \]

   16. *-lft-identity N/A

       \[\leadsto x \cdot \frac{1}{12} + \frac{\color{blue}{1 + \frac{1}{2} \cdot x}}{x} \]

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

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{12}, \frac{1 + \frac{1}{2} \cdot x}{x}\right)} \]

   18. *-lft-identity N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \frac{\color{blue}{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}}{x}\right) \]

   19. associate-*l/ N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \color{blue}{\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)}\right) \]

   20. distribute-rgt-in N/A

       \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \color{blue}{1 \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{x}}\right) \]

5. Simplified 68.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.08333333333333333, \frac{1}{x} + 0.5\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x, \frac{1}{12}, \color{blue}{\frac{1}{2}}\right) \]
7. Step-by-step derivation

8. Simplified 3.1%

   \[\leadsto \mathsf{fma}\left(x, 0.08333333333333333, \color{blue}{0.5}\right) \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{12} \cdot x} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot \frac{1}{12}} \]

    2. *-lowering-*.f64 3.4

       \[\leadsto \color{blue}{x \cdot 0.08333333333333333} \]

11. Simplified 3.4%

    \[\leadsto \color{blue}{x \cdot 0.08333333333333333} \]
12. Add Preprocessing

Alternative 16: 3.3% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 35.2%

   \[\frac{e^{x}}{e^{x} - 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

   9. metadata-eval 68.8

      \[\leadsto \frac{1}{x} + \color{blue}{0.5} \]

5. Simplified 68.8%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

6. Taylor expanded in x around inf

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

8. Simplified 3.3%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0) (expm1 (- x))))

C

double code(double x) {
	return -1.0 / expm1(-x);
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1(-x);
}

Python

def code(x):
	return -1.0 / math.expm1(-x)

Julia

function code(x)
	return Float64(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

code[x_] := N[((-1.0) / N[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024195 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :alt
  (! :herbie-platform default (/ (- 1) (expm1 (- x))))

  (/ (exp x) (- (exp x) 1.0)))