expq2 (section 3.11)

Percentage Accurate: 37.4% → 100.0%

Time: 8.9s

Alternatives: 17

Speedup: 17.9×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{-1 + e^{x}} \leq 1.5:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ (exp x) (+ -1.0 (exp x))) 1.5)
   (/ -24.0 (* x (* x (* x x))))
   (fma
    x
    (fma x (* x -0.001388888888888889) 0.08333333333333333)
    (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if ((exp(x) / (-1.0 + exp(x))) <= 1.5) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = fma(x, fma(x, (x * -0.001388888888888889), 0.08333333333333333), (0.5 + (1.0 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) / Float64(-1.0 + exp(x))) <= 1.5)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = fma(x, fma(x, Float64(x * -0.001388888888888889), 0.08333333333333333), Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] / N[(-1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 x) (-.f64 (exp.f64 x) #s(literal 1 binary64))) < 1.5

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 77.2

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

   7. Simplified 77.2%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 77.2

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

   10. Simplified 77.2%

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

   if 1.5 < (/.f64 (exp.f64 x) (-.f64 (exp.f64 x) #s(literal 1 binary64)))

   1. Initial program 5.1%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

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

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

   5. Simplified 99.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{-1 + e^{x}} \leq 1.5:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

3. Taylor expanded in x around 0

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

5. Simplified 98.6%

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

Alternative 4: 95.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.18518518518518517, -0.3888888888888889\right), 0.6666666666666666\right), -2\right), -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   x
   (fma
    (*
     x
     (fma
      (fma x 0.041666666666666664 -0.16666666666666666)
      (* (fma x 0.041666666666666664 -0.16666666666666666) (* x x))
      -0.25))
    (fma
     x
     (fma x (fma x 0.18518518518518517 -0.3888888888888889) 0.6666666666666666)
     -2.0)
    -1.0))))

C

double code(double x) {
	return -1.0 / (x * fma((x * fma(fma(x, 0.041666666666666664, -0.16666666666666666), (fma(x, 0.041666666666666664, -0.16666666666666666) * (x * x)), -0.25)), fma(x, fma(x, fma(x, 0.18518518518518517, -0.3888888888888889), 0.6666666666666666), -2.0), -1.0));
}

Julia

function code(x)
	return Float64(-1.0 / Float64(x * fma(Float64(x * fma(fma(x, 0.041666666666666664, -0.16666666666666666), Float64(fma(x, 0.041666666666666664, -0.16666666666666666) * Float64(x * x)), -0.25)), fma(x, fma(x, fma(x, 0.18518518518518517, -0.3888888888888889), 0.6666666666666666), -2.0), -1.0)))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(N[(x * N[(N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] * N[(N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * 0.18518518518518517 + -0.3888888888888889), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] + -2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 91.0

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

7. Simplified 91.0%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

9. Applied egg-rr 74.0%

   \[\leadsto \frac{-1}{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), -0.5\right)}, -1\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{x \cdot \left(\frac{2}{3} + x \cdot \left(\frac{5}{27} \cdot x - \frac{7}{18}\right)\right) - 2}, -1\right)} \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{x \cdot \left(\frac{2}{3} + x \cdot \left(\frac{5}{27} \cdot x - \frac{7}{18}\right)\right) + \left(\mathsf{neg}\left(2\right)\right)}, -1\right)} \]

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

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{3} + x \cdot \left(\frac{5}{27} \cdot x - \frac{7}{18}\right), \mathsf{neg}\left(2\right)\right)}, -1\right)} \]

    3. +-commutative N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{5}{27} \cdot x - \frac{7}{18}\right) + \frac{2}{3}}, \mathsf{neg}\left(2\right)\right), -1\right)} \]

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

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{5}{27} \cdot x - \frac{7}{18}, \frac{2}{3}\right)}, \mathsf{neg}\left(2\right)\right), -1\right)} \]

    5. sub-neg N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{5}{27} \cdot x + \left(\mathsf{neg}\left(\frac{7}{18}\right)\right)}, \frac{2}{3}\right), \mathsf{neg}\left(2\right)\right), -1\right)} \]

    6. *-commutative N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{5}{27}} + \left(\mathsf{neg}\left(\frac{7}{18}\right)\right), \frac{2}{3}\right), \mathsf{neg}\left(2\right)\right), -1\right)} \]

    7. metadata-eval N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{5}{27} + \color{blue}{\frac{-7}{18}}, \frac{2}{3}\right), \mathsf{neg}\left(2\right)\right), -1\right)} \]

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

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{5}{27}, \frac{-7}{18}\right)}, \frac{2}{3}\right), \mathsf{neg}\left(2\right)\right), -1\right)} \]

    9. metadata-eval 93.9

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.18518518518518517, -0.3888888888888889\right), 0.6666666666666666\right), \color{blue}{-2}\right), -1\right)} \]

12. Simplified 93.9%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.18518518518518517, -0.3888888888888889\right), 0.6666666666666666\right), -2\right)}, -1\right)} \]

13. Final simplification 93.9%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.18518518518518517, -0.3888888888888889\right), 0.6666666666666666\right), -2\right), -1\right)} \]
14. Add Preprocessing

Alternative 5: 95.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.3888888888888889, 0.6666666666666666\right), -2\right), -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   x
   (fma
    (*
     x
     (fma
      (fma x 0.041666666666666664 -0.16666666666666666)
      (* (fma x 0.041666666666666664 -0.16666666666666666) (* x x))
      -0.25))
    (fma x (fma x -0.3888888888888889 0.6666666666666666) -2.0)
    -1.0))))

C

double code(double x) {
	return -1.0 / (x * fma((x * fma(fma(x, 0.041666666666666664, -0.16666666666666666), (fma(x, 0.041666666666666664, -0.16666666666666666) * (x * x)), -0.25)), fma(x, fma(x, -0.3888888888888889, 0.6666666666666666), -2.0), -1.0));
}

Julia

function code(x)
	return Float64(-1.0 / Float64(x * fma(Float64(x * fma(fma(x, 0.041666666666666664, -0.16666666666666666), Float64(fma(x, 0.041666666666666664, -0.16666666666666666) * Float64(x * x)), -0.25)), fma(x, fma(x, -0.3888888888888889, 0.6666666666666666), -2.0), -1.0)))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(N[(x * N[(N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] * N[(N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * -0.3888888888888889 + 0.6666666666666666), $MachinePrecision] + -2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 91.0

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

7. Simplified 91.0%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

9. Applied egg-rr 74.0%

   \[\leadsto \frac{-1}{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), -0.5\right)}, -1\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{x \cdot \left(\frac{2}{3} + \frac{-7}{18} \cdot x\right) - 2}, -1\right)} \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{x \cdot \left(\frac{2}{3} + \frac{-7}{18} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right)}, -1\right)} \]

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

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{3} + \frac{-7}{18} \cdot x, \mathsf{neg}\left(2\right)\right)}, -1\right)} \]

    3. +-commutative N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \color{blue}{\frac{-7}{18} \cdot x + \frac{2}{3}}, \mathsf{neg}\left(2\right)\right), -1\right)} \]

    4. *-commutative N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-7}{18}} + \frac{2}{3}, \mathsf{neg}\left(2\right)\right), -1\right)} \]

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

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{-7}{18}, \frac{2}{3}\right)}, \mathsf{neg}\left(2\right)\right), -1\right)} \]

    6. metadata-eval 93.2

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.3888888888888889, 0.6666666666666666\right), \color{blue}{-2}\right), -1\right)} \]

12. Simplified 93.2%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.3888888888888889, 0.6666666666666666\right), -2\right)}, -1\right)} \]

13. Final simplification 93.2%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.3888888888888889, 0.6666666666666666\right), -2\right), -1\right)} \]
14. Add Preprocessing

Alternative 6: 94.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right), \mathsf{fma}\left(x, 0.6666666666666666, -2\right), -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   x
   (fma
    (*
     x
     (fma
      (fma x 0.041666666666666664 -0.16666666666666666)
      (* (fma x 0.041666666666666664 -0.16666666666666666) (* x x))
      -0.25))
    (fma x 0.6666666666666666 -2.0)
    -1.0))))

C

double code(double x) {
	return -1.0 / (x * fma((x * fma(fma(x, 0.041666666666666664, -0.16666666666666666), (fma(x, 0.041666666666666664, -0.16666666666666666) * (x * x)), -0.25)), fma(x, 0.6666666666666666, -2.0), -1.0));
}

Julia

function code(x)
	return Float64(-1.0 / Float64(x * fma(Float64(x * fma(fma(x, 0.041666666666666664, -0.16666666666666666), Float64(fma(x, 0.041666666666666664, -0.16666666666666666) * Float64(x * x)), -0.25)), fma(x, 0.6666666666666666, -2.0), -1.0)))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(N[(x * N[(N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] * N[(N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] * N[(x * 0.6666666666666666 + -2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 91.0

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

7. Simplified 91.0%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

9. Applied egg-rr 74.0%

   \[\leadsto \frac{-1}{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), -0.5\right)}, -1\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{\frac{2}{3} \cdot x - 2}, -1\right)} \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{\frac{2}{3} \cdot x + \left(\mathsf{neg}\left(2\right)\right)}, -1\right)} \]

    2. *-commutative N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{x \cdot \frac{2}{3}} + \left(\mathsf{neg}\left(2\right)\right), -1\right)} \]

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

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{2}{3}, \mathsf{neg}\left(2\right)\right)}, -1\right)} \]

    4. metadata-eval 93.1

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \mathsf{fma}\left(x, 0.6666666666666666, \color{blue}{-2}\right), -1\right)} \]

12. Simplified 93.1%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \color{blue}{\mathsf{fma}\left(x, 0.6666666666666666, -2\right)}, -1\right)} \]

13. Final simplification 93.1%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right), \mathsf{fma}\left(x, 0.6666666666666666, -2\right), -1\right)} \]
14. Add Preprocessing

Alternative 7: 93.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right), -2, -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   x
   (fma
    (*
     x
     (fma
      (fma x 0.041666666666666664 -0.16666666666666666)
      (* (fma x 0.041666666666666664 -0.16666666666666666) (* x x))
      -0.25))
    -2.0
    -1.0))))

C

double code(double x) {
	return -1.0 / (x * fma((x * fma(fma(x, 0.041666666666666664, -0.16666666666666666), (fma(x, 0.041666666666666664, -0.16666666666666666) * (x * x)), -0.25)), -2.0, -1.0));
}

Julia

function code(x)
	return Float64(-1.0 / Float64(x * fma(Float64(x * fma(fma(x, 0.041666666666666664, -0.16666666666666666), Float64(fma(x, 0.041666666666666664, -0.16666666666666666) * Float64(x * x)), -0.25)), -2.0, -1.0)))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(N[(x * N[(N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] * N[(N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.25), $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 91.0

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

7. Simplified 91.0%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

9. Applied egg-rr 74.0%

   \[\leadsto \frac{-1}{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), -0.5\right)}, -1\right)}} \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right), \mathsf{fma}\left(x, \frac{1}{24}, \frac{-1}{6}\right) \cdot \left(x \cdot x\right), \frac{-1}{4}\right) \cdot x, \color{blue}{-2}, -1\right)} \]
11. Step-by-step derivation

12. Simplified 93.1%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right) \cdot x, \color{blue}{-2}, -1\right)} \]

13. Final simplification 93.1%

    \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right) \cdot \left(x \cdot x\right), -0.25\right), -2, -1\right)} \]
14. Add Preprocessing

Alternative 8: 91.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 91.0

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

7. Simplified 91.0%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   7. accelerator-lowering-fma.f64 91.0

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

9. Applied egg-rr 91.0%

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

Alternative 9: 91.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 91.0

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

7. Simplified 91.0%

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

   1. clear-num N/A

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

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

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. /-rgt-identity N/A

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

   6. distribute-lft-in N/A

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

   7. distribute-neg-in N/A

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

   8. *-commutative N/A

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

   9. neg-mul-1 N/A

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

   10. remove-double-neg N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

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

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

9. Applied egg-rr 91.0%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. associate-*r* N/A

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

    4. unpow2 N/A

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

    5. *-rgt-identity N/A

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

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

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

12. Simplified 91.0%

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

Alternative 10: 91.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 91.0

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

7. Simplified 91.0%

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

Alternative 11: 90.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 91.0

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

7. Simplified 91.0%

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

8. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

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

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

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

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

   11. mul-1-neg N/A

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

   12. associate-*l* N/A

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

   13. mul-1-neg N/A

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

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

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

   15. lft-mult-inverse N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

10. Simplified 90.4%

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

Alternative 12: 91.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.08333333333333333, 0.5\right) - \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (/ -24.0 (* x (* x (* x x))))
   (- (fma x 0.08333333333333333 0.5) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = fma(x, 0.08333333333333333, 0.5) - (-1.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = Float64(fma(x, 0.08333333333333333, 0.5) - Float64(-1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.08333333333333333 + 0.5), $MachinePrecision] - N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 78.0

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

   7. Simplified 78.0%

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

   8. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. cube-mult N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 78.0

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

   10. Simplified 78.0%

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

   if -4.20000000000000018 < x

   1. Initial program 5.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

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

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

      18. *-lft-identity N/A

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

      19. associate-*l/ N/A

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

      20. distribute-rgt-in N/A

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

   5. Simplified 98.8%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. /-lowering-/.f64 98.8

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

   7. Applied egg-rr 98.8%

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

Alternative 13: 82.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4.5

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 52.3

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

   7. Simplified 52.3%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 52.3

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

   10. Simplified 52.3%

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

   if -4.5 < x

   1. Initial program 5.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. *-commutative N/A

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

      15. associate-*l/ N/A

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

      16. *-lft-identity N/A

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

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

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

      18. *-lft-identity N/A

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

      19. associate-*l/ N/A

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

      20. distribute-rgt-in N/A

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

   5. Simplified 98.8%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. unsub-neg N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. /-lowering-/.f64 98.8

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

   7. Applied egg-rr 98.8%

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

Alternative 14: 82.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.76:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.76) (/ -2.0 (* x x)) (+ 0.5 (/ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.76) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.76d0)) then
        tmp = (-2.0d0) / (x * x)
    else
        tmp = 0.5d0 + (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.76) {
		tmp = -2.0 / (x * x);
	} else {
		tmp = 0.5 + (1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.76:
		tmp = -2.0 / (x * x)
	else:
		tmp = 0.5 + (1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.76)
		tmp = Float64(-2.0 / Float64(x * x));
	else
		tmp = Float64(0.5 + Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.76)
		tmp = -2.0 / (x * x);
	else
		tmp = 0.5 + (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.76], N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.76000000000000001

   1. Initial program 100.0%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

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

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-sub0 N/A

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

      7. associate-+l- N/A

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

      8. neg-sub0 N/A

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

      9. +-commutative N/A

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

      10. sub-neg N/A

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

      11. div-sub N/A

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

      12. rec-exp N/A

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

      13. *-inverses N/A

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

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

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

      15. neg-lowering-neg.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. accelerator-lowering-fma.f64 52.3

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

   7. Simplified 52.3%

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

   8. Taylor expanded in x around inf

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 52.3

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

   10. Simplified 52.3%

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

   if -1.76000000000000001 < x

   1. Initial program 5.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

      7. associate-*l* N/A

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

      8. rgt-mult-inverse N/A

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

      9. metadata-eval 98.6

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

   5. Simplified 98.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.76:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.4% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

   1. clear-num N/A

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

   2. frac-2neg N/A

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

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

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-sub0 N/A

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

   7. associate-+l- N/A

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

   8. neg-sub0 N/A

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

   9. +-commutative N/A

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

   10. sub-neg N/A

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

   11. div-sub N/A

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

   12. rec-exp N/A

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

   13. *-inverses N/A

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

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

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

   15. neg-lowering-neg.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

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

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

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. accelerator-lowering-fma.f64 81.3

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

7. Simplified 81.3%

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

Alternative 16: 67.1% accurate, 17.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.7%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 63.5

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

5. Simplified 63.5%

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

Alternative 17: 3.4% accurate, 215.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 40.7%

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

3. Taylor expanded in x around 0

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0

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

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

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

   2. +-lowering-+.f64 62.7

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

8. Simplified 62.7%

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

9. Taylor expanded in x around inf

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

11. Simplified 3.7%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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)))