expq2 (section 3.11)

Percentage Accurate: 37.3% → 100.0%

Time: 6.5s

Alternatives: 17

Speedup: 14.3×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.3% 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 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. 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. *-inverses N/A

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

      15. lift-exp.f64 N/A

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

      16. rec-exp N/A

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

      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 77.1

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

   7. Applied rewrites 77.1%

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

   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 77.1%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 8.4%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 98.5%

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

4. Final simplification 90.7%

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

Alternative 3: 89.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. 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. *-inverses N/A

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

      15. lift-exp.f64 N/A

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

      16. rec-exp N/A

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

      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 65.3

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 65.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{-1}{{x}^{3} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x} - \frac{1}{6}\right)}} \]
   12. Step-by-step derivation

   13. Applied rewrites 65.3%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 8.4%

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

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

   7. Applied rewrites 98.2%

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

Alternative 4: 83.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 x) < 0.0

   1. Initial program 100.0%

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

      1. 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. *-inverses N/A

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

      15. lift-exp.f64 N/A

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

      16. rec-exp N/A

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

      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 55.5

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

   7. Applied rewrites 55.5%

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

   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 55.5%

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

   if 0.0 < (exp.f64 x)

   1. Initial program 8.4%

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

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

   7. Applied rewrites 98.2%

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

Alternative 5: 67.1% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (pow x -1.0))

C

double code(double x) {
	return pow(x, -1.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, -1.0);
}

Python

def code(x):
	return math.pow(x, -1.0)

Julia

function code(x)
	return x ^ -1.0
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -1.0], $MachinePrecision]

Derivation

1. Initial program 42.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 62.9

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

5. Applied rewrites 62.9%

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

6. Final simplification 62.9%

   \[\leadsto {x}^{-1} \]
7. Add Preprocessing

Alternative 6: 95.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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 90.3

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

7. Applied rewrites 90.3%

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

9. Applied rewrites 90.3%

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

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 95.8%

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

Alternative 7: 95.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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 90.3

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

7. Applied rewrites 90.3%

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

9. Applied rewrites 90.3%

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

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 95.4%

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

Alternative 8: 94.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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 90.3

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

7. Applied rewrites 90.3%

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

9. Applied rewrites 90.3%

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

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 94.7%

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

Alternative 9: 93.2% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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 90.3

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

7. Applied rewrites 90.3%

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

9. Applied rewrites 90.3%

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

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 93.1%

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

Alternative 10: 91.5% 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 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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 90.3

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

7. Applied rewrites 90.3%

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

Alternative 11: 89.1% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-1}{\left(\left(-0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333333, x, 0.5\right) - \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.0)
   (/ -1.0 (* (* (* -0.16666666666666666 x) x) x))
   (- (fma 0.08333333333333333 x 0.5) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -4.0) {
		tmp = -1.0 / (((-0.16666666666666666 * x) * x) * x);
	} else {
		tmp = fma(0.08333333333333333, x, 0.5) - (-1.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.0)
		tmp = Float64(-1.0 / Float64(Float64(Float64(-0.16666666666666666 * x) * x) * x));
	else
		tmp = Float64(fma(0.08333333333333333, x, 0.5) - Float64(-1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.0], N[(-1.0 / N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.08333333333333333 * x + 0.5), $MachinePrecision] - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4

   1. Initial program 100.0%

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

      1. 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. *-inverses N/A

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

      15. lift-exp.f64 N/A

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

      16. rec-exp N/A

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

      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 65.3

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 65.3%

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

   if -4 < x

   1. Initial program 8.4%

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

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

   7. Applied rewrites 98.2%

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

Alternative 12: 90.5% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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 90.3

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

7. Applied rewrites 90.3%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 89.1%

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

Alternative 13: 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 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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 85.9

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

7. Applied rewrites 85.9%

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

Alternative 14: 83.3% 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 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   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 82.0

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

7. Applied rewrites 82.0%

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

Alternative 15: 67.1% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 63.0%

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

7. Applied rewrites 63.0%

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

Alternative 16: 67.0% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 62.9

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

5. Applied rewrites 62.9%

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

6. Final simplification 62.9%

   \[\leadsto 0.5 - \frac{-1}{x} \]
7. Add Preprocessing

Alternative 17: 3.2% accurate, 215.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 42.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 62.9

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

5. Applied rewrites 62.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 3.5%

   \[\leadsto 0.5 \]
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 2024309 
(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)))