expq2 (section 3.11)

Percentage Accurate: 37.1% → 100.0%

Time: 7.2s

Alternatives: 18

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: 37.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 95.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f64 86.7

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

7. Applied rewrites 86.7%

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

9. Applied rewrites 86.7%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 91.6%

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

Alternative 3: 94.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f64 86.7

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

7. Applied rewrites 86.7%

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

9. Applied rewrites 86.7%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 91.2%

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

Alternative 4: 93.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f64 86.7

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

7. Applied rewrites 86.7%

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

9. Applied rewrites 86.7%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 90.4%

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

Alternative 5: 92.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f64 86.7

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

7. Applied rewrites 86.7%

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

9. Applied rewrites 86.7%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 88.8%

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

Alternative 6: 91.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;\frac{-1}{x \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.4)
   (/
    -1.0
    (* x (* x (fma x (fma x 0.041666666666666664 -0.16666666666666666) 0.5))))
   (fma
    x
    (fma x (* x -0.001388888888888889) 0.08333333333333333)
    (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = -1.0 / (x * (x * fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5)));
	} 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 (x <= -3.4)
		tmp = Float64(-1.0 / Float64(x * Float64(x * fma(x, fma(x, 0.041666666666666664, -0.16666666666666666), 0.5))));
	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[x, -3.4], N[(-1.0 / N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + -0.16666666666666666), $MachinePrecision] + 0.5), $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 x < -3.39999999999999991

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. *-inverses N/A

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

      15. lift-exp.f64 N/A

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

      16. rec-exp N/A

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

      17. lower-expm1.f64 N/A

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

      18. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.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. lower-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. lower-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. lower-fma.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 67.5%

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

   if -3.39999999999999991 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. lower-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. Applied rewrites 99.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;\frac{-1}{x \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right), 0.5\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. *-inverses N/A

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

      15. lift-exp.f64 N/A

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

      16. rec-exp N/A

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

      17. lower-expm1.f64 N/A

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

      18. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.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. lower-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. lower-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. lower-fma.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 67.5%

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

   if -3.60000000000000009 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. lower-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. Applied rewrites 99.7%

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

4. Final simplification 86.8%

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

Alternative 8: 91.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{-1}{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.8)
   (/ -1.0 (* 0.041666666666666664 (* 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 (x <= -3.8) {
		tmp = -1.0 / (0.041666666666666664 * (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 (x <= -3.8)
		tmp = Float64(-1.0 / Float64(0.041666666666666664 * 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[x, -3.8], N[(-1.0 / N[(0.041666666666666664 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $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 x < -3.7999999999999998

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. neg-sub0 N/A

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

      8. lift--.f64 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. div-sub N/A

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

      14. *-inverses N/A

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

      15. lift-exp.f64 N/A

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

      16. rec-exp N/A

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

      17. lower-expm1.f64 N/A

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

      18. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.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. lower-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. lower-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. lower-fma.f64 67.5

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{\frac{1}{24} \cdot \color{blue}{{x}^{4}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.5%

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

   if -3.7999999999999998 < x

   1. Initial program 6.6%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-/l* N/A

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

      3. associate-*l/ N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. associate-+r+ N/A

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

      7. distribute-lft-in N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. +-commutative N/A

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

      11. associate-*r* N/A

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

      12. lft-mult-inverse N/A

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

      13. *-lft-identity N/A

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

      14. lower-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. Applied rewrites 99.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8:\\ \;\;\;\;\frac{-1}{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.4% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f64 86.7

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

7. Applied rewrites 86.7%

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

9. Applied rewrites 86.8%

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

10. Final simplification 86.8%

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

Alternative 10: 91.4% 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 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f64 86.7

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

7. Applied rewrites 86.7%

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

Alternative 11: 88.8% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.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. lower-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. lower-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. lower-fma.f64 86.7

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

7. Applied rewrites 86.7%

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

9. Applied rewrites 86.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 83.6%

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

13. Final simplification 83.6%

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

Alternative 12: 88.8% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 83.6

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

7. Applied rewrites 83.6%

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

Alternative 13: 83.8% 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 43.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. neg-sub0 N/A

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

   8. lift--.f64 N/A

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

   9. associate-+l- N/A

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

   10. neg-sub0 N/A

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

   11. +-commutative N/A

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

   12. sub-neg N/A

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

   13. div-sub N/A

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

   14. *-inverses N/A

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

   15. lift-exp.f64 N/A

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

   16. rec-exp N/A

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

   17. lower-expm1.f64 N/A

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

   18. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.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. lower-fma.f64 76.9

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

7. Applied rewrites 76.9%

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

Alternative 14: 67.5% accurate, 10.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   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. lower-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. Applied rewrites 60.9%

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

6. Final simplification 60.9%

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

Alternative 15: 67.4% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

3. Taylor expanded in x around 0

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

   1. *-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. lower-+.f64 N/A

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

   6. lower-/.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 60.6

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

5. Applied rewrites 60.6%

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

6. Final simplification 60.6%

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

Alternative 16: 67.4% accurate, 17.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 60.5

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

5. Applied rewrites 60.5%

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

Alternative 17: 3.4% accurate, 35.8× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.08333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.08333333333333333))

C

double code(double x) {
	return x * 0.08333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.08333333333333333d0
end function

Java

public static double code(double x) {
	return x * 0.08333333333333333;
}

Python

def code(x):
	return x * 0.08333333333333333

Julia

function code(x)
	return Float64(x * 0.08333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.08333333333333333;
end

Wolfram

code[x_] := N[(x * 0.08333333333333333), $MachinePrecision]

Derivation

1. Initial program 43.8%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   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. lower-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. Applied rewrites 60.9%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 3.3%

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

Alternative 18: 3.3% accurate, 215.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 43.8%

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

3. Taylor expanded in x around 0

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

   1. *-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. lower-+.f64 N/A

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

   6. lower-/.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 60.6

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

5. Applied rewrites 60.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 3.2%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024237 
(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)))