expq2 (section 3.11)

Percentage Accurate: 37.6% → 100.0%

Time: 9.0s

Alternatives: 15

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: 37.6% 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 38.5%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

   7. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

   11. div-sub N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

   12. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

   13. *-inverses N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. neg-lowering-neg.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

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: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75:\\ \;\;\;\;\frac{e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.75)
   (/ (exp x) x)
   (fma
    x
    (fma x (* x -0.001388888888888889) 0.08333333333333333)
    (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -3.75) {
		tmp = exp(x) / x;
	} else {
		tmp = fma(x, fma(x, (x * -0.001388888888888889), 0.08333333333333333), ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.75)
		tmp = Float64(exp(x) / x);
	else
		tmp = fma(x, fma(x, Float64(x * -0.001388888888888889), 0.08333333333333333), Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.75], N[(N[Exp[x], $MachinePrecision] / x), $MachinePrecision], N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   if -3.75 < x

   1. Initial program 6.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

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

          \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)}, \left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)\right) \]

   5. Simplified 99.7%

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

Alternative 3: 94.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, -0.004629629629629629\right), \frac{1}{0.027777777777777776}, 0.5\right), -1\right), 0\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (fma
   x
   (fma
    x
    (fma
     (* x (fma (* x (* x x)) 7.233796296296296e-5 -0.004629629629629629))
     (/ 1.0 0.027777777777777776)
     0.5)
    -1.0)
   0.0)))

C

double code(double x) {
	return -1.0 / fma(x, fma(x, fma((x * fma((x * (x * x)), 7.233796296296296e-5, -0.004629629629629629)), (1.0 / 0.027777777777777776), 0.5), -1.0), 0.0);
}

Julia

function code(x)
	return Float64(-1.0 / fma(x, fma(x, fma(Float64(x * fma(Float64(x * Float64(x * x)), 7.233796296296296e-5, -0.004629629629629629)), Float64(1.0 / 0.027777777777777776), 0.5), -1.0), 0.0))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(x * N[(N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 7.233796296296296e-5 + -0.004629629629629629), $MachinePrecision]), $MachinePrecision] * N[(1.0 / 0.027777777777777776), $MachinePrecision] + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

   7. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

   11. div-sub N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

   12. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

   13. *-inverses N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. neg-lowering-neg.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + \color{blue}{0}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}, 0\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), 0\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}, -1\right), 0\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \color{blue}{\frac{1}{2}}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   11. accelerator-lowering-fma.f64 87.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

7. Simplified 87.6%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) \cdot x + \frac{1}{2}\right), -1\right), 0\right)\right) \]

   2. flip3-+ N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{{\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{-1}{6}}^{3}}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{-1}{6}\right)} \cdot x + \frac{1}{2}\right), -1\right), 0\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{\left({\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{-1}{6}}^{3}\right) \cdot x}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{-1}{6}\right)} + \frac{1}{2}\right), -1\right), 0\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\left(\left({\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{-1}{6}}^{3}\right) \cdot x\right) \cdot \frac{1}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{-1}{6}\right)} + \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\left(\left({\left(x \cdot \frac{1}{24}\right)}^{3} + {\frac{-1}{6}}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{1}{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(x \cdot \frac{1}{24}\right) \cdot \frac{-1}{6}\right)}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

9. Applied egg-rr 75.9%

   \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, -0.004629629629629629\right) \cdot x, \frac{1}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, 0.027777777777777776 + x \cdot 0.006944444444444444\right)}, 0.5\right)}, -1\right), 0\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{13824}, \frac{-1}{216}\right), x\right), \mathsf{/.f64}\left(1, \color{blue}{\frac{1}{36}}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]
11. Step-by-step derivation

12. Simplified 91.5%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, -0.004629629629629629\right) \cdot x, \frac{1}{\color{blue}{0.027777777777777776}}, 0.5\right), -1\right), 0\right)} \]

13. Final simplification 91.5%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot x\right), 7.233796296296296 \cdot 10^{-5}, -0.004629629629629629\right), \frac{1}{0.027777777777777776}, 0.5\right), -1\right), 0\right)} \]
14. Add Preprocessing

Alternative 4: 93.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right)}{0.16666666666666666}, 0.5\right), -1\right), 0\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (fma
   x
   (fma
    x
    (fma
     x
     (/
      (fma (* x x) 0.001736111111111111 -0.027777777777777776)
      0.16666666666666666)
     0.5)
    -1.0)
   0.0)))

C

double code(double x) {
	return -1.0 / fma(x, fma(x, fma(x, (fma((x * x), 0.001736111111111111, -0.027777777777777776) / 0.16666666666666666), 0.5), -1.0), 0.0);
}

Julia

function code(x)
	return Float64(-1.0 / fma(x, fma(x, fma(x, Float64(fma(Float64(x * x), 0.001736111111111111, -0.027777777777777776) / 0.16666666666666666), 0.5), -1.0), 0.0))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(x * N[(x * N[(N[(N[(x * x), $MachinePrecision] * 0.001736111111111111 + -0.027777777777777776), $MachinePrecision] / 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

   7. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

   11. div-sub N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

   12. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

   13. *-inverses N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. neg-lowering-neg.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + \color{blue}{0}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}, 0\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), 0\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}, -1\right), 0\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \color{blue}{\frac{1}{2}}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   11. accelerator-lowering-fma.f64 87.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

7. Simplified 87.6%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \frac{-1}{6}}{\color{blue}{x \cdot \frac{1}{24} - \frac{-1}{6}}}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \frac{-1}{6}\right), \color{blue}{\left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(\color{blue}{x \cdot \frac{1}{24}} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   4. swap-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{24}\right) + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(\color{blue}{x} \cdot \frac{1}{24} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} \cdot \frac{1}{24}\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(\color{blue}{x \cdot \frac{1}{24}} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot \frac{1}{24}\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(\color{blue}{x} \cdot \frac{1}{24} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(x \cdot \color{blue}{\frac{1}{24}} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)\right), \left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \left(x \cdot \frac{1}{24} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   12. metadata-eval 87.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \mathsf{fma.f64}\left(x, \frac{1}{24}, \frac{1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

9. Applied egg-rr 87.6%

   \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right)}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}}, 0.5\right), -1\right), 0\right)} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \color{blue}{\frac{1}{6}}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]
11. Step-by-step derivation

12. Simplified 89.3%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right)}{\color{blue}{0.16666666666666666}}, 0.5\right), -1\right), 0\right)} \]
13. Add Preprocessing

Alternative 5: 92.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;\frac{-1}{x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6)
   (/ -1.0 (* x (* x (* x (fma x 0.041666666666666664 -0.16666666666666666)))))
   (fma
    x
    (fma x (* x -0.001388888888888889) 0.08333333333333333)
    (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -1.0 / (x * (x * (x * fma(x, 0.041666666666666664, -0.16666666666666666))));
	} else {
		tmp = fma(x, fma(x, (x * -0.001388888888888889), 0.08333333333333333), ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = Float64(-1.0 / Float64(x * Float64(x * Float64(x * fma(x, 0.041666666666666664, -0.16666666666666666)))));
	else
		tmp = fma(x, fma(x, Float64(x * -0.001388888888888889), 0.08333333333333333), Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], N[(-1.0 / N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

      11. div-sub N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

      12. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

      13. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. neg-lowering-neg.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + \color{blue}{0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}, 0\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), 0\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right), 0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}, -1\right), 0\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \color{blue}{\frac{1}{2}}\right), -1\right), 0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      11. accelerator-lowering-fma.f64 65.2%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   7. Simplified 65.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, 0\right)\right) \]

   10. Simplified 65.2%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot x} - \frac{1}{6}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(x \cdot {x}^{2}\right) \cdot \left(\frac{1}{24} \cdot \color{blue}{x} - \frac{1}{6}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{1}{24} \cdot x} - \frac{1}{6}\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(\left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) \cdot x\right)}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x + \frac{-1}{6}\right)\right)\right)\right)\right) \]

       12. distribute-lft-in N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right) + \color{blue}{x \cdot \frac{-1}{6}}\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right) + x \cdot \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right) + x \cdot \left(\mathsf{neg}\left(\frac{1}{6} \cdot 1\right)\right)\right)\right)\right)\right) \]

       15. lft-mult-inverse N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right) + x \cdot \left(\mathsf{neg}\left(\frac{1}{6} \cdot \left(\frac{1}{x} \cdot x\right)\right)\right)\right)\right)\right)\right) \]

       16. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right) + x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{6} \cdot \frac{1}{x}\right) \cdot x\right)\right)\right)\right)\right)\right) \]

       17. distribute-lft-neg-out N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x\right) + x \cdot \left(\left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

       18. distribute-lft-in N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot x + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) \cdot x\right)}\right)\right)\right)\right) \]

       19. distribute-rgt-in N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}\right)\right)\right)\right)\right) \]

       20. sub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \left(\frac{1}{24} - \color{blue}{\frac{1}{6} \cdot \frac{1}{x}}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 65.2%

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

   if -3.60000000000000009 < x

   1. Initial program 6.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

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

          \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)}, \left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)\right) \]

   5. Simplified 99.7%

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

Alternative 6: 92.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.8)
   (/ -24.0 (* x (* x (* x x))))
   (fma
    x
    (fma x (* x -0.001388888888888889) 0.08333333333333333)
    (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -3.8) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = fma(x, fma(x, (x * -0.001388888888888889), 0.08333333333333333), ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.8)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = fma(x, fma(x, Float64(x * -0.001388888888888889), 0.08333333333333333), Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.8], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

      11. div-sub N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

      12. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

      13. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. neg-lowering-neg.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + \color{blue}{0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}, 0\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), 0\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right), 0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}, -1\right), 0\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \color{blue}{\frac{1}{2}}\right), -1\right), 0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      11. accelerator-lowering-fma.f64 65.2%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   7. Simplified 65.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(-24, \color{blue}{\left({x}^{4}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

   10. Simplified 65.2%

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

   if -3.7999999999999998 < x

   1. Initial program 6.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

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

          \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)}, \left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)\right) \]

   5. Simplified 99.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), \frac{1}{x} + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.5%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

   7. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

   11. div-sub N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

   12. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

   13. *-inverses N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. neg-lowering-neg.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + \color{blue}{0}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}, 0\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), 0\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}, -1\right), 0\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \color{blue}{\frac{1}{2}}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   11. accelerator-lowering-fma.f64 87.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

7. Simplified 87.6%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \frac{-1}{6}}{\color{blue}{x \cdot \frac{1}{24} - \frac{-1}{6}}}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) - \frac{-1}{6} \cdot \frac{-1}{6}\right), \color{blue}{\left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot \frac{1}{24}\right) \cdot \left(x \cdot \frac{1}{24}\right) + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(\color{blue}{x \cdot \frac{1}{24}} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   4. swap-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{24} \cdot \frac{1}{24}\right) + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(\color{blue}{x} \cdot \frac{1}{24} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} \cdot \frac{1}{24}\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(\color{blue}{x \cdot \frac{1}{24}} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot \frac{1}{24}\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(\color{blue}{x} \cdot \frac{1}{24} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)\right), \left(x \cdot \color{blue}{\frac{1}{24}} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \left(\mathsf{neg}\left(\frac{1}{36}\right)\right)\right), \left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \left(x \cdot \frac{1}{24} - \frac{-1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \left(x \cdot \frac{1}{24} + \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   12. metadata-eval 87.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{/.f64}\left(\mathsf{fma.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{576}, \frac{-1}{36}\right), \mathsf{fma.f64}\left(x, \frac{1}{24}, \frac{1}{6}\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

9. Applied egg-rr 87.6%

   \[\leadsto \frac{-1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right)}{\mathsf{fma}\left(x, 0.041666666666666664, 0.16666666666666666\right)}}, 0.5\right), -1\right), 0\right)} \]
10. Step-by-step derivation

    1. +-rgt-identity N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{576} + \frac{-1}{36}}{x \cdot \frac{1}{24} + \frac{1}{6}} + \frac{1}{2}\right) + -1\right)}\right)\right) \]

    2. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(x \cdot \left(x \cdot \frac{\left(x \cdot x\right) \cdot \frac{1}{576} + \frac{-1}{36}}{x \cdot \frac{1}{24} + \frac{1}{6}} + \frac{1}{2}\right)\right) \cdot x + \color{blue}{-1 \cdot x}\right)\right) \]

11. Applied egg-rr 87.6%

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

12. Final simplification 87.6%

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

Alternative 8: 91.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(-1.0 / Float64(x * fma(x, fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5), -1.0)))
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]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

   7. associate-+l- N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

   11. div-sub N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

   12. rec-exp N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

   13. *-inverses N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   15. neg-lowering-neg.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + \color{blue}{0}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}, 0\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), 0\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}, -1\right), 0\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \color{blue}{\frac{1}{2}}\right), -1\right), 0\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   11. accelerator-lowering-fma.f64 87.6%

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

7. Simplified 87.6%

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

   1. +-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right) + -1\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right) + -1\right) \cdot \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right) + -1\right), \color{blue}{x}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fma.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right) + \frac{1}{2}\right), -1\right), x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), x\right)\right) \]

   6. accelerator-lowering-fma.f64 87.6%

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(\mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), x\right)\right) \]

9. Applied egg-rr 87.6%

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

10. Final simplification 87.6%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right)} \]
11. Add Preprocessing

Alternative 9: 92.0% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.0)
   (/ -24.0 (* x (* x (* x x))))
   (fma x 0.08333333333333333 (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = fma(x, 0.08333333333333333, ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = fma(x, 0.08333333333333333, Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.0], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 0.08333333333333333 + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

      11. div-sub N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

      12. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

      13. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. neg-lowering-neg.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + \color{blue}{0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}, 0\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), 0\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right), 0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}, -1\right), 0\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \color{blue}{\frac{1}{2}}\right), -1\right), 0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      11. accelerator-lowering-fma.f64 65.2%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   7. Simplified 65.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \mathsf{/.f64}\left(-24, \color{blue}{\left({x}^{4}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \left({x}^{\left(3 + \color{blue}{1}\right)}\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \left({x}^{3} \cdot \color{blue}{x}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{x}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), x\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), x\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), x\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), x\right)\right) \]

      9. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{/.f64}\left(-24, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), x\right)\right) \]

   10. Simplified 65.2%

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

   if -4 < x

   1. Initial program 6.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

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

          \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{12}}, \left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right) \]

      20. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(1 \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{x}\right)\right) \]

   5. Simplified 99.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, \frac{1}{x} + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (/ 6.0 (* x (* x x)))
   (fma x 0.08333333333333333 (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = 6.0 / (x * (x * x));
	} else {
		tmp = fma(x, 0.08333333333333333, ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(6.0 / Float64(x * Float64(x * x)));
	else
		tmp = fma(x, 0.08333333333333333, Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(6.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 0.08333333333333333 + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(\mathsf{neg}\left(\frac{e^{x} - 1}{e^{x}}\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\color{blue}{\frac{e^{x} - 1}{e^{x}}}\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\mathsf{neg}\left(\left(e^{x} - 1\right)\right)}{\color{blue}{e^{x}}}\right)\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{0 - \left(e^{x} - 1\right)}{e^{\color{blue}{x}}}\right)\right) \]

      7. associate-+l- N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(0 - e^{x}\right) + 1}{e^{\color{blue}{x}}}\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{\left(\mathsf{neg}\left(e^{x}\right)\right) + 1}{e^{x}}\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 + \left(\mathsf{neg}\left(e^{x}\right)\right)}{e^{\color{blue}{x}}}\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1 - e^{x}}{e^{\color{blue}{x}}}\right)\right) \]

      11. div-sub N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(\frac{1}{e^{x}} - \color{blue}{\frac{e^{x}}{e^{x}}}\right)\right) \]

      12. rec-exp N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{e^{x}}}{e^{x}}\right)\right) \]

      13. *-inverses N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(e^{\mathsf{neg}\left(x\right)} - 1\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      15. neg-lowering-neg.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{expm1.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right) + \color{blue}{0}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) - 1\right)}, 0\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right), 0\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right) + -1\right), 0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right)\right)}, -1\right), 0\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \left(\frac{1}{24} \cdot x - \frac{1}{6}\right) + \color{blue}{\frac{1}{2}}\right), -1\right), 0\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\left(\frac{1}{24} \cdot x - \frac{1}{6}\right)}, \frac{1}{2}\right), -1\right), 0\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(\frac{1}{24} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \left(x \cdot \frac{1}{24} + \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

      11. accelerator-lowering-fma.f64 65.2%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{24}}, \frac{-1}{6}\right), \frac{1}{2}\right), -1\right), 0\right)\right) \]

   7. Simplified 65.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{fma.f64}\left(x, \color{blue}{\left({x}^{3} \cdot \left(\frac{1}{24} - \frac{1}{6} \cdot \frac{1}{x}\right)\right)}, 0\right)\right) \]

   10. Simplified 65.2%

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

   11. Taylor expanded in x around 0

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

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

          \[\leadsto \mathsf{/.f64}\left(6, \color{blue}{\left({x}^{3}\right)}\right) \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 56.3%

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

   13. Simplified 56.3%

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

   if -4.20000000000000018 < x

   1. Initial program 6.8%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

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

          \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{12}}, \left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)\right) \]

      18. *-lft-identity N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}\right)\right) \]

      19. associate-*l/ N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right) \]

      20. distribute-rgt-in N/A

          \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(1 \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{x}\right)\right) \]

   5. Simplified 99.2%

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

Alternative 11: 67.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.08333333333333333, \frac{1}{x} + 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 0.08333333333333333 (+ (/ 1.0 x) 0.5)))

C

double code(double x) {
	return fma(x, 0.08333333333333333, ((1.0 / x) + 0.5));
}

Julia

function code(x)
	return fma(x, 0.08333333333333333, Float64(Float64(1.0 / x) + 0.5))
end

Wolfram

code[x_] := N[(x * 0.08333333333333333 + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. lft-mult-inverse N/A

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

   13. *-lft-identity N/A

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

   14. *-commutative N/A

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

   15. associate-*l/ N/A

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

   16. *-lft-identity N/A

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

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

       \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{12}}, \left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)\right) \]

   18. *-lft-identity N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}\right)\right) \]

   19. associate-*l/ N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right) \]

   20. distribute-rgt-in N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(1 \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{x}\right)\right) \]

5. Simplified 66.3%

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

Alternative 12: 66.9% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} + 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ 1.0 x) 0.5))

C

double code(double x) {
	return (1.0 / x) + 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) + 0.5d0
end function

Java

public static double code(double x) {
	return (1.0 / x) + 0.5;
}

Python

def code(x):
	return (1.0 / x) + 0.5

Julia

function code(x)
	return Float64(Float64(1.0 / x) + 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / x) + 0.5;
end

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 38.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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

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

   2. associate-*l/ N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

   9. metadata-eval 66.1%

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

5. Simplified 66.1%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
6. Add Preprocessing

Alternative 13: 67.0% 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 38.5%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 65.6%

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

5. Simplified 65.6%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. Add Preprocessing

Alternative 14: 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 38.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. lft-mult-inverse N/A

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

   13. *-lft-identity N/A

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

   14. *-commutative N/A

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

   15. associate-*l/ N/A

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

   16. *-lft-identity N/A

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

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

       \[\leadsto \mathsf{fma.f64}\left(x, \color{blue}{\frac{1}{12}}, \left(\frac{1 + \frac{1}{2} \cdot x}{x}\right)\right) \]

   18. *-lft-identity N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(\frac{1 \cdot \left(1 + \frac{1}{2} \cdot x\right)}{x}\right)\right) \]

   19. associate-*l/ N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(\frac{1}{x} \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)\right) \]

   20. distribute-rgt-in N/A

       \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \left(1 \cdot \frac{1}{x} + \left(\frac{1}{2} \cdot x\right) \cdot \frac{1}{x}\right)\right) \]

5. Simplified 66.3%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma.f64}\left(x, \frac{1}{12}, \color{blue}{\frac{1}{2}}\right) \]
7. Step-by-step derivation

8. Simplified 3.0%

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

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{12} \cdot x} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. *-lowering-*.f64 3.5%

       \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{12}}\right) \]

11. Simplified 3.5%

    \[\leadsto \color{blue}{x \cdot 0.08333333333333333} \]
12. Add Preprocessing

Alternative 15: 3.2% 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 38.5%

   \[\frac{e^{x}}{e^{x} - 1} \]
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

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

   2. associate-*l/ N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

   9. metadata-eval 66.1%

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

5. Simplified 66.1%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]

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 2024193 
(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)))