expq2 (section 3.11)

Percentage Accurate: 37.9% → 100.0%

Time: 8.4s

Alternatives: 15

Speedup: 17.9×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.9% 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 36.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. neg-sub0 N/A

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

   14. lift--.f64 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. div-sub N/A

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

   20. *-inverses N/A

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

   21. lift-exp.f64 N/A

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

   22. rec-exp N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 93.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{\frac{0.16666666666666666}{\mathsf{fma}\left(x \cdot x, 0.001736111111111111, -0.027777777777777776\right)}}, 0.5\right), -1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (*
   x
   (fma
    x
    (fma
     x
     (/
      1.0
      (/
       0.16666666666666666
       (fma (* x x) 0.001736111111111111 -0.027777777777777776)))
     0.5)
    -1.0))))

C

double code(double x) {
	return -1.0 / (x * fma(x, fma(x, (1.0 / (0.16666666666666666 / fma((x * x), 0.001736111111111111, -0.027777777777777776))), 0.5), -1.0));
}

Julia

function code(x)
	return Float64(-1.0 / Float64(x * fma(x, fma(x, Float64(1.0 / Float64(0.16666666666666666 / fma(Float64(x * x), 0.001736111111111111, -0.027777777777777776))), 0.5), -1.0)))
end

Wolfram

code[x_] := N[(-1.0 / N[(x * N[(x * N[(x * N[(1.0 / N[(0.16666666666666666 / N[(N[(x * x), $MachinePrecision] * 0.001736111111111111 + -0.027777777777777776), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. neg-sub0 N/A

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

   14. lift--.f64 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. div-sub N/A

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

   20. *-inverses N/A

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

   21. lift-exp.f64 N/A

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

   22. rec-exp N/A

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

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 92.8

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

7. Applied rewrites 92.8%

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

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. swap-sqr N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{\frac{\mathsf{fma}\left(x, \frac{1}{24}, \frac{1}{6}\right)}{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{576}}, \mathsf{neg}\left(\frac{-1}{6} \cdot \frac{-1}{6}\right)\right)}}, \frac{1}{2}\right), -1\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{-1}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{\frac{\mathsf{fma}\left(x, \frac{1}{24}, \frac{1}{6}\right)}{\mathsf{fma}\left(x \cdot x, \frac{1}{576}, \mathsf{neg}\left(\color{blue}{\frac{1}{36}}\right)\right)}}, \frac{1}{2}\right), -1\right)} \]

   14. metadata-eval 92.8

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

9. Applied rewrites 92.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 94.3%

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

Alternative 3: 91.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;\frac{-1}{x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.7)
   (/ -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 <= -3.7) {
		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 <= -3.7)
		tmp = Float64(-1.0 / Float64(x * Float64(x * Float64(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, -3.7], N[(-1.0 / N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + -0.16666666666666666), $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 < -3.7000000000000002

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. clear-num N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. div-sub N/A

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

      20. *-inverses N/A

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

      21. lift-exp.f64 N/A

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

      22. rec-exp N/A

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

   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 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 77.3%

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

   if -3.7000000000000002 < x

   1. Initial program 7.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7:\\ \;\;\;\;\frac{-1}{x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.041666666666666664, -0.16666666666666666\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.75)
   (/ -24.0 (* x (* x (* x x))))
   (fma
    x
    (fma x (* x -0.001388888888888889) 0.08333333333333333)
    (+ 0.5 (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.75) {
		tmp = -24.0 / (x * (x * (x * x)));
	} else {
		tmp = fma(x, fma(x, (x * -0.001388888888888889), 0.08333333333333333), (0.5 + (1.0 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.75)
		tmp = Float64(-24.0 / Float64(x * Float64(x * Float64(x * x))));
	else
		tmp = fma(x, fma(x, Float64(x * -0.001388888888888889), 0.08333333333333333), Float64(0.5 + Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -3.75], N[(-24.0 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] + N[(0.5 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.75

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. clear-num N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. div-sub N/A

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

      20. *-inverses N/A

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

      21. lift-exp.f64 N/A

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

      22. rec-exp N/A

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

   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 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 77.3

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

   10. Applied rewrites 77.3%

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

   if -3.75 < x

   1. Initial program 7.5%

      \[\frac{e^{x}}{e^{x} - 1} \]
   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 100.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75:\\ \;\;\;\;\frac{-24}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot -0.001388888888888889, 0.08333333333333333\right), 0.5 + \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. neg-sub0 N/A

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

   14. lift--.f64 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. div-sub N/A

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

   20. *-inverses N/A

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

   21. lift-exp.f64 N/A

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

   22. rec-exp N/A

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

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 92.8

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

7. Applied rewrites 92.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. neg-mul-1 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 92.8

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

9. Applied rewrites 92.8%

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

Alternative 6: 91.6% 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 36.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. neg-sub0 N/A

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

   14. lift--.f64 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. div-sub N/A

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

   20. *-inverses N/A

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

   21. lift-exp.f64 N/A

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

   22. rec-exp N/A

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

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 92.8

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

7. Applied rewrites 92.8%

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

Alternative 7: 91.7% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. clear-num N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. div-sub N/A

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

      20. *-inverses N/A

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

      21. lift-exp.f64 N/A

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

      22. rec-exp N/A

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

   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 77.3

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

   7. Applied rewrites 77.3%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. lower-*.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 77.3

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

   10. Applied rewrites 77.3%

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

   if -4.20000000000000018 < x

   1. Initial program 7.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. clear-num N/A

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

      9. frac-2neg N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. neg-sub0 N/A

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

      14. lift--.f64 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. +-commutative N/A

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

      18. sub-neg N/A

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

      19. div-sub N/A

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

      20. *-inverses N/A

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

      21. lift-exp.f64 N/A

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

      22. rec-exp N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.8%

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

4. Final simplification 92.9%

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

Alternative 8: 83.6% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. neg-sub0 N/A

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

   14. lift--.f64 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. div-sub N/A

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

   20. *-inverses N/A

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

   21. lift-exp.f64 N/A

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

   22. rec-exp N/A

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

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 92.8

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

7. Applied rewrites 92.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. distribute-rgt-in N/A

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

   4. neg-mul-1 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 92.8

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

9. Applied rewrites 92.8%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 85.0%

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

Alternative 9: 83.6% 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 36.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. neg-sub0 N/A

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

   14. lift--.f64 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. div-sub N/A

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

   20. *-inverses N/A

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

   21. lift-exp.f64 N/A

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

   22. rec-exp N/A

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

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 85.0

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

7. Applied rewrites 85.0%

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

Alternative 10: 66.8% 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 36.0%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. +-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 69.7%

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

6. Final simplification 69.7%

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

Alternative 11: 66.7% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. clear-num N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. neg-sub0 N/A

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

   14. lift--.f64 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. +-commutative N/A

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

   18. sub-neg N/A

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

   19. div-sub N/A

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

   20. *-inverses N/A

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

   21. lift-exp.f64 N/A

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

   22. rec-exp N/A

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

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 92.8

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

7. Applied rewrites 92.8%

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

8. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 69.2

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

10. Applied rewrites 69.2%

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

Alternative 12: 66.7% 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 36.0%

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

3. Taylor expanded in x around 0

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

   1. *-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 69.2

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

5. Applied rewrites 69.2%

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

6. Final simplification 69.2%

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

Alternative 13: 66.6% 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 36.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 68.5

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

5. Applied rewrites 68.5%

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

Alternative 14: 3.3% 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 36.0%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-/l* N/A

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

   3. associate-*l/ N/A

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

   4. +-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 69.7%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. lft-mult-inverse N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lft-mult-inverse N/A

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

   8. metadata-eval 3.6

      \[\leadsto x \cdot \color{blue}{0.08333333333333333} \]

8. Applied rewrites 3.6%

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

Alternative 15: 3.2% accurate, 215.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 36.0%

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

3. Taylor expanded in x around 0

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

   1. *-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 69.2

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

5. Applied rewrites 69.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 3.3%

   \[\leadsto \color{blue}{0.5} \]
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 2024219 
(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)))