expq2 (section 3.11)

Percentage Accurate: 38.1% → 100.0%

Time: 6.4s

Alternatives: 14

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: 38.1% 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 39.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. lift-exp.f64 N/A

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

   15. rec-exp N/A

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

   16. *-inverses N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 10^{-101}:\\ \;\;\;\;\frac{-1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right), x, 0.5\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 1e-101)
   (/
    -1.0
    (* (* (fma (fma 0.041666666666666664 x -0.16666666666666666) x 0.5) x) x))
   (fma
    (fma (* x x) -0.001388888888888889 0.08333333333333333)
    x
    (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 1e-101) {
		tmp = -1.0 / ((fma(fma(0.041666666666666664, x, -0.16666666666666666), x, 0.5) * x) * x);
	} else {
		tmp = fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 1e-101)
		tmp = Float64(-1.0 / Float64(Float64(fma(fma(0.041666666666666664, x, -0.16666666666666666), x, 0.5) * x) * x));
	else
		tmp = fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 1e-101], N[(-1.0 / N[(N[(N[(N[(0.041666666666666664 * x + -0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 1.00000000000000005e-101

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. lift-exp.f64 N/A

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

      15. rec-exp N/A

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

      16. *-inverses N/A

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

      17. lower-expm1.f64 N/A

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

      18. lower-neg.f64 100.0

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

   4. Applied rewrites 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 76.1

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

   7. Applied rewrites 76.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 76.1%

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

   if 1.00000000000000005e-101 < (exp.f64 x)

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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. *-commutative N/A

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

   5. Applied rewrites 98.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 10^{-101}:\\ \;\;\;\;\frac{-1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right), x, 0.5\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, \frac{1}{x} + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 10^{-101}:\\ \;\;\;\;\frac{-1}{\left(\left(\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 1e-101)
   (/ -1.0 (* (* (* (fma 0.041666666666666664 x -0.16666666666666666) x) x) x))
   (fma
    (fma (* x x) -0.001388888888888889 0.08333333333333333)
    x
    (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 1e-101) {
		tmp = -1.0 / (((fma(0.041666666666666664, x, -0.16666666666666666) * x) * x) * x);
	} else {
		tmp = fma(fma((x * x), -0.001388888888888889, 0.08333333333333333), x, ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 1e-101)
		tmp = Float64(-1.0 / Float64(Float64(Float64(fma(0.041666666666666664, x, -0.16666666666666666) * x) * x) * x));
	else
		tmp = fma(fma(Float64(x * x), -0.001388888888888889, 0.08333333333333333), x, Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 1e-101], N[(-1.0 / N[(N[(N[(N[(0.041666666666666664 * x + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.08333333333333333), $MachinePrecision] * x + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 1.00000000000000005e-101

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. lift-exp.f64 N/A

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

      15. rec-exp N/A

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

      16. *-inverses N/A

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

      17. lower-expm1.f64 N/A

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

      18. lower-neg.f64 100.0

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

   4. Applied rewrites 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 76.1

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

   7. Applied rewrites 76.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 76.1%

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

   if 1.00000000000000005e-101 < (exp.f64 x)

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{12} + \frac{-1}{720} \cdot {x}^{2}\right)\right)}{x}} \]
   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. *-commutative N/A

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

   5. Applied rewrites 98.3%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 10^{-101}:\\ \;\;\;\;\frac{-1}{\left(\left(\mathsf{fma}\left(0.041666666666666664, x, -0.16666666666666666\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.08333333333333333\right), x, \frac{1}{x} + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} \leq 10^{-101}:\\ \;\;\;\;\frac{-1}{\left(0.5 \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (exp x) 1e-101)
   (/ -1.0 (* (* 0.5 x) x))
   (fma 0.08333333333333333 x (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (exp(x) <= 1e-101) {
		tmp = -1.0 / ((0.5 * x) * x);
	} else {
		tmp = fma(0.08333333333333333, x, ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (exp(x) <= 1e-101)
		tmp = Float64(-1.0 / Float64(Float64(0.5 * x) * x));
	else
		tmp = fma(0.08333333333333333, x, Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Exp[x], $MachinePrecision], 1e-101], N[(-1.0 / N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 * x + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 1.00000000000000005e-101

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. lift-exp.f64 N/A

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

      15. rec-exp N/A

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

      16. *-inverses N/A

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

      17. lower-expm1.f64 N/A

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

      18. lower-neg.f64 100.0

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

   4. Applied rewrites 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 51.3

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

   7. Applied rewrites 51.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 51.3%

       \[\leadsto \frac{-1}{\left(0.5 \cdot x\right) \cdot x} \]

   if 1.00000000000000005e-101 < (exp.f64 x)

   1. Initial program 6.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-/l* N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

      20. lower-+.f64 N/A

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

   5. Applied rewrites 98.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 10^{-101}:\\ \;\;\;\;\frac{-1}{\left(0.5 \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.09375, x, -0.375\right), x, 1.5\right), x, -6\right) \cdot \left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right), x, 0.5\right), x, -1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   (fma
    (fma
     (*
      (fma (fma (fma 0.09375 x -0.375) x 1.5) x -6.0)
      (- (fma 0.001736111111111111 (* x x) -0.027777777777777776)))
     x
     0.5)
    x
    -1.0)
   x)))

C

double code(double x) {
	return -1.0 / (fma(fma((fma(fma(fma(0.09375, x, -0.375), x, 1.5), x, -6.0) * -fma(0.001736111111111111, (x * x), -0.027777777777777776)), x, 0.5), x, -1.0) * x);
}

Julia

function code(x)
	return Float64(-1.0 / Float64(fma(fma(Float64(fma(fma(fma(0.09375, x, -0.375), x, 1.5), x, -6.0) * Float64(-fma(0.001736111111111111, Float64(x * x), -0.027777777777777776))), x, 0.5), x, -1.0) * x))
end

Wolfram

code[x_] := N[(-1.0 / N[(N[(N[(N[(N[(N[(N[(0.09375 * x + -0.375), $MachinePrecision] * x + 1.5), $MachinePrecision] * x + -6.0), $MachinePrecision] * (-N[(0.001736111111111111 * N[(x * x), $MachinePrecision] + -0.027777777777777776), $MachinePrecision])), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. lift-exp.f64 N/A

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

   15. rec-exp N/A

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

   16. *-inverses N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 90.2

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

7. Applied rewrites 90.2%

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

9. Applied rewrites 90.2%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-\mathsf{fma}\left(\frac{1}{576}, x \cdot x, \frac{-1}{36}\right)\right) \cdot \left(x \cdot \left(\frac{3}{2} + x \cdot \left(\frac{3}{32} \cdot x - \frac{3}{8}\right)\right) - 6\right), x, \frac{1}{2}\right), x, -1\right) \cdot x} \]
11. Step-by-step derivation

12. Applied rewrites 93.5%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.09375, x, -0.375\right), x, 1.5\right), x, -6\right), x, 0.5\right), x, -1\right) \cdot x} \]

13. Final simplification 93.5%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.09375, x, -0.375\right), x, 1.5\right), x, -6\right) \cdot \left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right), x, 0.5\right), x, -1\right) \cdot x} \]
14. Add Preprocessing

Alternative 6: 94.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.375, x, 1.5\right), x, -6\right) \cdot \left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right), x, 0.5\right), x, -1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   (fma
    (fma
     (*
      (fma (fma -0.375 x 1.5) x -6.0)
      (- (fma 0.001736111111111111 (* x x) -0.027777777777777776)))
     x
     0.5)
    x
    -1.0)
   x)))

C

double code(double x) {
	return -1.0 / (fma(fma((fma(fma(-0.375, x, 1.5), x, -6.0) * -fma(0.001736111111111111, (x * x), -0.027777777777777776)), x, 0.5), x, -1.0) * x);
}

Julia

function code(x)
	return Float64(-1.0 / Float64(fma(fma(Float64(fma(fma(-0.375, x, 1.5), x, -6.0) * Float64(-fma(0.001736111111111111, Float64(x * x), -0.027777777777777776))), x, 0.5), x, -1.0) * x))
end

Wolfram

code[x_] := N[(-1.0 / N[(N[(N[(N[(N[(N[(-0.375 * x + 1.5), $MachinePrecision] * x + -6.0), $MachinePrecision] * (-N[(0.001736111111111111 * N[(x * x), $MachinePrecision] + -0.027777777777777776), $MachinePrecision])), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. lift-exp.f64 N/A

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

   15. rec-exp N/A

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

   16. *-inverses N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 90.2

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

7. Applied rewrites 90.2%

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

9. Applied rewrites 90.2%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-\mathsf{fma}\left(\frac{1}{576}, x \cdot x, \frac{-1}{36}\right)\right) \cdot \left(x \cdot \left(\frac{3}{2} + \frac{-3}{8} \cdot x\right) - 6\right), x, \frac{1}{2}\right), x, -1\right) \cdot x} \]
11. Step-by-step derivation

12. Applied rewrites 93.4%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.375, x, 1.5\right), x, -6\right), x, 0.5\right), x, -1\right) \cdot x} \]

13. Final simplification 93.4%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.375, x, 1.5\right), x, -6\right) \cdot \left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right), x, 0.5\right), x, -1\right) \cdot x} \]
14. Add Preprocessing

Alternative 7: 94.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.5, x, -6\right) \cdot \left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right), x, 0.5\right), x, -1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   (fma
    (fma
     (*
      (fma 1.5 x -6.0)
      (- (fma 0.001736111111111111 (* x x) -0.027777777777777776)))
     x
     0.5)
    x
    -1.0)
   x)))

C

double code(double x) {
	return -1.0 / (fma(fma((fma(1.5, x, -6.0) * -fma(0.001736111111111111, (x * x), -0.027777777777777776)), x, 0.5), x, -1.0) * x);
}

Julia

function code(x)
	return Float64(-1.0 / Float64(fma(fma(Float64(fma(1.5, x, -6.0) * Float64(-fma(0.001736111111111111, Float64(x * x), -0.027777777777777776))), x, 0.5), x, -1.0) * x))
end

Wolfram

code[x_] := N[(-1.0 / N[(N[(N[(N[(N[(1.5 * x + -6.0), $MachinePrecision] * (-N[(0.001736111111111111 * N[(x * x), $MachinePrecision] + -0.027777777777777776), $MachinePrecision])), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. lift-exp.f64 N/A

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

   15. rec-exp N/A

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

   16. *-inverses N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 90.2

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

7. Applied rewrites 90.2%

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

9. Applied rewrites 90.2%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-\mathsf{fma}\left(\frac{1}{576}, x \cdot x, \frac{-1}{36}\right)\right) \cdot \left(\frac{3}{2} \cdot x - 6\right), x, \frac{1}{2}\right), x, -1\right) \cdot x} \]
11. Step-by-step derivation

12. Applied rewrites 93.1%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right) \cdot \mathsf{fma}\left(1.5, x, -6\right), x, 0.5\right), x, -1\right) \cdot x} \]

13. Final simplification 93.1%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1.5, x, -6\right) \cdot \left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right), x, 0.5\right), x, -1\right) \cdot x} \]
14. Add Preprocessing

Alternative 8: 92.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(-6 \cdot \left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right), x, 0.5\right), x, -1\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   (fma
    (fma
     (* -6.0 (- (fma 0.001736111111111111 (* x x) -0.027777777777777776)))
     x
     0.5)
    x
    -1.0)
   x)))

C

double code(double x) {
	return -1.0 / (fma(fma((-6.0 * -fma(0.001736111111111111, (x * x), -0.027777777777777776)), x, 0.5), x, -1.0) * x);
}

Julia

function code(x)
	return Float64(-1.0 / Float64(fma(fma(Float64(-6.0 * Float64(-fma(0.001736111111111111, Float64(x * x), -0.027777777777777776))), x, 0.5), x, -1.0) * x))
end

Wolfram

code[x_] := N[(-1.0 / N[(N[(N[(N[(-6.0 * (-N[(0.001736111111111111 * N[(x * x), $MachinePrecision] + -0.027777777777777776), $MachinePrecision])), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. lift-exp.f64 N/A

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

   15. rec-exp N/A

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

   16. *-inverses N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 90.2

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

7. Applied rewrites 90.2%

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

9. Applied rewrites 90.2%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-\mathsf{fma}\left(\frac{1}{576}, x \cdot x, \frac{-1}{36}\right)\right) \cdot -6, x, \frac{1}{2}\right), x, -1\right) \cdot x} \]
11. Step-by-step derivation

12. Applied rewrites 91.2%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right) \cdot -6, x, 0.5\right), x, -1\right) \cdot x} \]

13. Final simplification 91.2%

    \[\leadsto \frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(-6 \cdot \left(-\mathsf{fma}\left(0.001736111111111111, x \cdot x, -0.027777777777777776\right)\right), x, 0.5\right), x, -1\right) \cdot x} \]
14. Add Preprocessing

Alternative 9: 91.4% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. lift-exp.f64 N/A

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

   15. rec-exp N/A

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

   16. *-inverses N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 90.2

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

7. Applied rewrites 90.2%

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

Alternative 10: 89.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;\frac{-1}{\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.45)
   (/ -1.0 (* (* (fma -0.16666666666666666 x 0.5) x) x))
   (fma 0.08333333333333333 x (+ (/ 1.0 x) 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= -3.45) {
		tmp = -1.0 / ((fma(-0.16666666666666666, x, 0.5) * x) * x);
	} else {
		tmp = fma(0.08333333333333333, x, ((1.0 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.45)
		tmp = Float64(-1.0 / Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) * x));
	else
		tmp = fma(0.08333333333333333, x, Float64(Float64(1.0 / x) + 0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.45], N[(-1.0 / N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(0.08333333333333333 * x + N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4500000000000002

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. lift-exp.f64 N/A

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

      15. rec-exp N/A

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

      16. *-inverses N/A

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

      17. lower-expm1.f64 N/A

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

      18. lower-neg.f64 100.0

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

   4. Applied rewrites 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 76.1

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

   7. Applied rewrites 76.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 76.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{-1}{\left(\mathsf{fma}\left(\frac{-1}{6}, x, \frac{1}{2}\right) \cdot x\right) \cdot x} \]
   12. Step-by-step derivation

   13. Applied rewrites 69.8%

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

   if -3.4500000000000002 < x

   1. Initial program 6.6%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-/l* N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-rgt-identity N/A

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

      17. associate-/l* N/A

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

      18. +-commutative N/A

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

      19. distribute-lft1-in N/A

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

      20. lower-+.f64 N/A

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

   5. Applied rewrites 98.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;\frac{-1}{\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, \frac{1}{x} + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.0% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. lift-exp.f64 N/A

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

   15. rec-exp N/A

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

   16. *-inverses N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 87.9

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

7. Applied rewrites 87.9%

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

Alternative 12: 83.2% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. lift-exp.f64 N/A

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

   15. rec-exp N/A

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

   16. *-inverses N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 81.0

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

7. Applied rewrites 81.0%

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

Alternative 13: 66.4% accurate, 17.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 64.5

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

5. Applied rewrites 64.5%

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

Alternative 14: 3.3% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 39.8%

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

3. Taylor expanded in x around 0

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

   1. *-rgt-identity N/A

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

   2. associate-/l* N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-+.f64 N/A

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

   6. associate-*l* N/A

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

   7. rgt-mult-inverse N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 64.0

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

5. Applied rewrites 64.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{1}{2} \]
7. Step-by-step derivation

8. Applied rewrites 3.3%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024242 
(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)))