expq2 (section 3.11)

Percentage Accurate: 38.4% → 100.0%

Time: 7.6s

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: 38.4% 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

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

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(-1.0 / expm1(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 35.6%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ (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 (exp(x) <= 0.0) {
		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 (exp(x) <= 0.0)
		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[N[Exp[x], $MachinePrecision], 0.0], 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 (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

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

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

   5. Simplified 98.7%

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

Alternative 3: 91.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ -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 (exp(x) <= 0.0) {
		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 (exp(x) <= 0.0)
		tmp = Float64(-1.0 / Float64(x * Float64(Float64(x * 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[N[Exp[x], $MachinePrecision], 0.0], N[(-1.0 / N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * 0.041666666666666664 + -0.16666666666666666), $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 (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 80.6

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

   7. Simplified 80.6%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

         \[\leadsto \frac{-1}{x \cdot \left({x}^{3} \cdot \color{blue}{\left(\frac{1}{24} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right)\right)}\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{-1}{x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) \cdot {x}^{3}\right)}} \]

      3. cube-mult N/A

         \[\leadsto \frac{-1}{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) \cdot {x}^{3}\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{-1}{x \cdot \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) \cdot {x}^{3}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{x \cdot \left(\color{blue}{\left(\frac{1}{24} \cdot x\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{x}\right)\right) \cdot {x}^{3}\right)} \]

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

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

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

      10. distribute-rgt-neg-out N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*l* N/A

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

      13. neg-mul-1 N/A

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

      14. distribute-rgt-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

   10. Simplified 80.6%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

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

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

   5. Simplified 98.7%

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

Alternative 4: 91.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ -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 (exp(x) <= 0.0) {
		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 (exp(x) <= 0.0)
		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[N[Exp[x], $MachinePrecision], 0.0], 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 (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 80.6

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

   7. Simplified 80.6%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

         \[\leadsto \frac{-24}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 80.6

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

   10. Simplified 80.6%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

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

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

   5. Simplified 98.7%

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

Alternative 5: 91.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		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 (exp(x) <= 0.0)
		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[N[Exp[x], $MachinePrecision], 0.0], 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 (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 80.6

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

   7. Simplified 80.6%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

         \[\leadsto \frac{-24}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 80.6

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

   10. Simplified 80.6%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

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

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

      18. *-lft-identity N/A

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

      19. associate-*l/ N/A

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

      20. distribute-rgt-in N/A

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

   5. Simplified 98.5%

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

Alternative 6: 82.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0)
   (/ -2.0 (* x x))
   (fma x 0.08333333333333333 (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = fma(x, 0.08333333333333333, ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 0.0)
		tmp = Float64(-2.0 / 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[N[Exp[x], $MachinePrecision], 0.0], N[(-2.0 / N[(x * x), $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 (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 61.9

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

   7. Simplified 61.9%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 61.9

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

   10. Simplified 61.9%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

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

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

      18. *-lft-identity N/A

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

      19. associate-*l/ N/A

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

      20. distribute-rgt-in N/A

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

   5. Simplified 98.5%

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

Alternative 7: 82.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.0) (/ -2.0 (* x x)) (+ (/ 1.0 x) 0.5)))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 0.0) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (exp(x) <= 0.0d0) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = (1.0d0 / x) + 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.exp(x) <= 0.0) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.exp(x) <= 0.0:
		tmp = -2.0 / (x * x)
	else:
		tmp = (1.0 / x) + 0.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (exp(x) <= 0.0)
		tmp = -2.0 / (x * x);
	else
		tmp = (1.0 / x) + 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 0.0], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 61.9

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

   7. Simplified 61.9%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 61.9

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

   10. Simplified 61.9%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval 97.6

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

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \mathsf{fma}\left(t\_1, t\_1, -1\right)}{\mathsf{fma}\left(x, t\_0, 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma x (fma x 0.041666666666666664 -0.16666666666666666) 0.5))
        (t_1 (* x t_0)))
   (if (<= x -1e+103)
     (/ -24.0 (* x (* x (* x x))))
     (/ -1.0 (/ (* x (fma t_1 t_1 -1.0)) (fma x t_0 1.0))))))

C

double code(double x) {
	double t_0 = fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5);
	double t_1 = x * t_0;
	double tmp;
	if (x <= -1e+103) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = -1.0 / ((x * fma(t_1, t_1, -1.0)) / fma(x, t_0, 1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5)
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (x <= -1e+103)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(-1.0 / Float64(Float64(x * fma(t_1, t_1, -1.0)) / fma(x, t_0, 1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[x, -1e+103], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(x * N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1e103

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

         \[\leadsto \frac{-24}{\color{blue}{{x}^{2} \cdot {x}^{2}}} \]

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 100.0

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

   10. Simplified 100.0%

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

   if -1e103 < x

   1. Initial program 18.4%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 91.0

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

   7. Simplified 91.0%

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

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   9. Applied egg-rr 94.6%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right), 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.2% 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 35.6%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 92.9

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

7. Simplified 92.9%

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

Alternative 10: 90.7% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 92.9

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

7. Simplified 92.9%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 92.3

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

10. Simplified 92.3%

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

Alternative 11: 82.4% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. accelerator-lowering-fma.f64 86.7

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

7. Simplified 86.7%

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

Alternative 12: 66.2% 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 35.6%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. distribute-rgt-in N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

   7. associate-*l* N/A

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

   8. rgt-mult-inverse N/A

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

   9. metadata-eval 69.2

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

5. Simplified 69.2%

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

Alternative 13: 66.1% accurate, 17.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 69.1

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

5. Simplified 69.1%

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

Alternative 14: 3.3% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 35.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 97.5%

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

6. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 68.5

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

8. Simplified 68.5%

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

9. Taylor expanded in x around inf

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

11. Simplified 4.1%

    \[\leadsto \color{blue}{1} \]
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 35.6%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. lft-mult-inverse N/A

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

   13. *-lft-identity N/A

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

   14. *-commutative N/A

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

   15. associate-*l/ N/A

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

   16. *-lft-identity N/A

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

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

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

   18. *-lft-identity N/A

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

   19. associate-*l/ N/A

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

   20. distribute-rgt-in N/A

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

5. Simplified 69.5%

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

6. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. accelerator-lowering-fma.f64 3.1

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

8. Simplified 3.1%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
10. Step-by-step derivation

11. Simplified 3.4%

    \[\leadsto \color{blue}{0.5} \]
12. 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 2024199 
(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)))