expq2 (section 3.11)

Percentage Accurate: 37.9% → 100.0%

Time: 5.2s

Alternatives: 5

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 42.8%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. lift-exp.f64 N/A

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

   21. rec-exp N/A

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

   22. *-inverses N/A

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

   23. lower-expm1.f64 N/A

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

4. Applied rewrites 100.0%

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

Alternative 2: 91.3% 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 42.8%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. lift-exp.f64 N/A

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

   21. rec-exp N/A

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

   22. *-inverses N/A

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

   23. lower-expm1.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 88.4

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

7. Applied rewrites 88.4%

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

Alternative 3: 88.4% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.8%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. lift-exp.f64 N/A

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

   21. rec-exp N/A

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

   22. *-inverses N/A

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

   23. lower-expm1.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 85.7

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

7. Applied rewrites 85.7%

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

Alternative 4: 83.2% 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 42.8%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. lift-exp.f64 N/A

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

   21. rec-exp N/A

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

   22. *-inverses N/A

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

   23. lower-expm1.f64 N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 80.3

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

7. Applied rewrites 80.3%

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

Alternative 5: 66.5% 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 42.8%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 61.9

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

5. Applied rewrites 61.9%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. 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 2024212 
(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)))