expq2 (section 3.11)

Percentage Accurate: 36.7% → 100.0%

Time: 7.4s

Alternatives: 8

Speedup: 68.3×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.7% 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, 2.0× 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 35.7%

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

   1. sub-neg 35.7%

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

   2. +-commutative 35.7%

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

   3. rgt-mult-inverse 3.7%

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

   4. exp-neg 3.8%

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

   5. distribute-rgt-neg-out 3.8%

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

   6. *-rgt-identity 3.8%

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

   7. distribute-lft-in 3.8%

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

   8. neg-sub0 3.8%

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

   9. associate-+l- 3.8%

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

   10. neg-sub0 3.2%

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

   11. associate-/r* 3.2%

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

   12. *-rgt-identity 3.2%

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

   13. associate-*r/ 3.2%

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

   14. rgt-mult-inverse 35.3%

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

   15. distribute-frac-neg2 35.3%

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

   16. distribute-neg-frac 35.3%

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

   17. metadata-eval 35.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 68.1% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x} + 0.5\right) - x \cdot -0.08333333333333333 \end{array} \]

FPCore

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

C

double code(double x) {
	return ((1.0 / x) + 0.5) - (x * -0.08333333333333333);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / x) + 0.5d0) - (x * (-0.08333333333333333d0))
end function

Java

public static double code(double x) {
	return ((1.0 / x) + 0.5) - (x * -0.08333333333333333);
}

Python

def code(x):
	return ((1.0 / x) + 0.5) - (x * -0.08333333333333333)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / x) + 0.5) - Float64(x * -0.08333333333333333))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / x) + 0.5) - (x * -0.08333333333333333);
end

Wolfram

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

Derivation

1. Initial program 35.7%

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

   1. sub-neg 35.7%

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

   2. +-commutative 35.7%

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

   3. rgt-mult-inverse 3.7%

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

   4. exp-neg 3.8%

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

   5. distribute-rgt-neg-out 3.8%

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

   6. *-rgt-identity 3.8%

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

   7. distribute-lft-in 3.8%

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

   8. neg-sub0 3.8%

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

   9. associate-+l- 3.8%

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

   10. neg-sub0 3.2%

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

   11. associate-/r* 3.2%

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

   12. *-rgt-identity 3.2%

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

   13. associate-*r/ 3.2%

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

   14. rgt-mult-inverse 35.3%

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

   15. distribute-frac-neg2 35.3%

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

   16. distribute-neg-frac 35.3%

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

   17. metadata-eval 35.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 68.6%

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

6. Taylor expanded in x around -inf 40.4%

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

   1. mul-1-neg 40.4%

      \[\leadsto \color{blue}{-x \cdot \left(-1 \cdot \frac{0.5 + \frac{1}{x}}{x} - 0.08333333333333333\right)} \]

   2. distribute-rgt-neg-in 40.4%

      \[\leadsto \color{blue}{x \cdot \left(-\left(-1 \cdot \frac{0.5 + \frac{1}{x}}{x} - 0.08333333333333333\right)\right)} \]

   3. sub-neg 40.4%

      \[\leadsto x \cdot \left(-\color{blue}{\left(-1 \cdot \frac{0.5 + \frac{1}{x}}{x} + \left(-0.08333333333333333\right)\right)}\right) \]

   4. metadata-eval 40.4%

      \[\leadsto x \cdot \left(-\left(-1 \cdot \frac{0.5 + \frac{1}{x}}{x} + \color{blue}{-0.08333333333333333}\right)\right) \]

   5. +-commutative 40.4%

      \[\leadsto x \cdot \left(-\color{blue}{\left(-0.08333333333333333 + -1 \cdot \frac{0.5 + \frac{1}{x}}{x}\right)}\right) \]

   6. distribute-neg-in 40.4%

      \[\leadsto x \cdot \color{blue}{\left(\left(--0.08333333333333333\right) + \left(--1 \cdot \frac{0.5 + \frac{1}{x}}{x}\right)\right)} \]

   7. metadata-eval 40.4%

      \[\leadsto x \cdot \left(\color{blue}{0.08333333333333333} + \left(--1 \cdot \frac{0.5 + \frac{1}{x}}{x}\right)\right) \]

   8. mul-1-neg 40.4%

      \[\leadsto x \cdot \left(0.08333333333333333 + \left(-\color{blue}{\left(-\frac{0.5 + \frac{1}{x}}{x}\right)}\right)\right) \]

   9. remove-double-neg 40.4%

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

   10. +-commutative 40.4%

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

8. Simplified 40.4%

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

   1. distribute-lft-in 40.4%

      \[\leadsto \color{blue}{x \cdot 0.08333333333333333 + x \cdot \frac{\frac{1}{x} + 0.5}{x}} \]

   2. flip-+ 38.1%

      \[\leadsto \color{blue}{\frac{\left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right) - \left(x \cdot \frac{\frac{1}{x} + 0.5}{x}\right) \cdot \left(x \cdot \frac{\frac{1}{x} + 0.5}{x}\right)}{x \cdot 0.08333333333333333 - x \cdot \frac{\frac{1}{x} + 0.5}{x}}} \]

10. Applied egg-rr 38.1%

    \[\leadsto \color{blue}{\frac{\left(x \cdot 0.08333333333333333\right) \cdot \left(x \cdot 0.08333333333333333\right) - \left(x \cdot \frac{\frac{1}{x} + 0.5}{x}\right) \cdot \left(x \cdot \frac{\frac{1}{x} + 0.5}{x}\right)}{x \cdot 0.08333333333333333 - x \cdot \frac{\frac{1}{x} + 0.5}{x}}} \]

11. Taylor expanded in x around -inf 40.4%

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

    1. associate-*r* 40.4%

       \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{0.5 + \frac{1}{x}}{x} - 0.08333333333333333\right)} \]

    2. neg-mul-1 40.4%

       \[\leadsto \color{blue}{\left(-x\right)} \cdot \left(-1 \cdot \frac{0.5 + \frac{1}{x}}{x} - 0.08333333333333333\right) \]

    3. sub-neg 40.4%

       \[\leadsto \left(-x\right) \cdot \color{blue}{\left(-1 \cdot \frac{0.5 + \frac{1}{x}}{x} + \left(-0.08333333333333333\right)\right)} \]

    4. distribute-lft-in 40.4%

       \[\leadsto \color{blue}{\left(-x\right) \cdot \left(-1 \cdot \frac{0.5 + \frac{1}{x}}{x}\right) + \left(-x\right) \cdot \left(-0.08333333333333333\right)} \]

13. Simplified 68.7%

    \[\leadsto \color{blue}{\left(\frac{1}{x} + 0.5\right) + \left(-x\right) \cdot -0.08333333333333333} \]

14. Final simplification 68.7%

    \[\leadsto \left(\frac{1}{x} + 0.5\right) - x \cdot -0.08333333333333333 \]
15. Add Preprocessing

Alternative 3: 68.0% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

   1. sub-neg 35.7%

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

   2. +-commutative 35.7%

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

   3. rgt-mult-inverse 3.7%

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

   4. exp-neg 3.8%

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

   5. distribute-rgt-neg-out 3.8%

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

   6. *-rgt-identity 3.8%

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

   7. distribute-lft-in 3.8%

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

   8. neg-sub0 3.8%

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

   9. associate-+l- 3.8%

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

   10. neg-sub0 3.2%

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

   11. associate-/r* 3.2%

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

   12. *-rgt-identity 3.2%

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

   13. associate-*r/ 3.2%

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

   14. rgt-mult-inverse 35.3%

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

   15. distribute-frac-neg2 35.3%

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

   16. distribute-neg-frac 35.3%

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

   17. metadata-eval 35.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 68.6%

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

6. Taylor expanded in x around 0 68.6%

   \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot x}{x}} \]
7. Step-by-step derivation

   1. +-commutative 68.6%

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

   2. *-commutative 68.6%

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

   3. fma-undefine 68.6%

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

   4. *-lft-identity 68.6%

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

   5. associate-*l/ 68.6%

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

   6. fma-undefine 68.6%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(x \cdot 0.5 + 1\right)} \]

   7. distribute-lft-in 68.6%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot 0.5\right) + \frac{1}{x} \cdot 1} \]

   8. *-commutative 68.6%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 \cdot x\right)} + \frac{1}{x} \cdot 1 \]

   9. associate-*l* 68.6%

      \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot 0.5\right) \cdot x} + \frac{1}{x} \cdot 1 \]

   10. *-commutative 68.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x}\right)} \cdot x + \frac{1}{x} \cdot 1 \]

   11. associate-*l* 68.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{x} \cdot x\right)} + \frac{1}{x} \cdot 1 \]

   12. lft-mult-inverse 68.6%

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

   13. metadata-eval 68.6%

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

   14. *-rgt-identity 68.6%

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

   15. +-commutative 68.6%

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

8. Simplified 68.6%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
9. Step-by-step derivation

   1. +-commutative 68.6%

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

   2. flip-+ 40.3%

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

   3. metadata-eval 40.3%

      \[\leadsto \frac{\color{blue}{0.25} - \frac{1}{x} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}} \]

   4. inv-pow 40.3%

      \[\leadsto \frac{0.25 - \color{blue}{{x}^{-1}} \cdot \frac{1}{x}}{0.5 - \frac{1}{x}} \]

   5. inv-pow 40.3%

      \[\leadsto \frac{0.25 - {x}^{-1} \cdot \color{blue}{{x}^{-1}}}{0.5 - \frac{1}{x}} \]

   6. pow-prod-up 40.1%

      \[\leadsto \frac{0.25 - \color{blue}{{x}^{\left(-1 + -1\right)}}}{0.5 - \frac{1}{x}} \]

   7. metadata-eval 40.1%

      \[\leadsto \frac{0.25 - {x}^{\color{blue}{-2}}}{0.5 - \frac{1}{x}} \]

10. Applied egg-rr 40.1%

    \[\leadsto \color{blue}{\frac{0.25 - {x}^{-2}}{0.5 - \frac{1}{x}}} \]
11. Step-by-step derivation

    1. clear-num 40.2%

       \[\leadsto \color{blue}{\frac{1}{\frac{0.5 - \frac{1}{x}}{0.25 - {x}^{-2}}}} \]

    2. clear-num 40.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{0.25 - {x}^{-2}}{0.5 - \frac{1}{x}}}}} \]

    3. metadata-eval 40.1%

       \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{0.5 \cdot 0.5} - {x}^{-2}}{0.5 - \frac{1}{x}}}} \]

    4. metadata-eval 40.1%

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

    5. pow-prod-up 40.3%

       \[\leadsto \frac{1}{\frac{1}{\frac{0.5 \cdot 0.5 - \color{blue}{{x}^{-1} \cdot {x}^{-1}}}{0.5 - \frac{1}{x}}}} \]

    6. inv-pow 40.3%

       \[\leadsto \frac{1}{\frac{1}{\frac{0.5 \cdot 0.5 - \color{blue}{\frac{1}{x}} \cdot {x}^{-1}}{0.5 - \frac{1}{x}}}} \]

    7. inv-pow 40.3%

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

    8. flip-+ 68.6%

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

12. Applied egg-rr 68.6%

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

13. Final simplification 68.6%

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

Alternative 4: 68.0% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.7%

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

   1. sub-neg 35.7%

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

   2. +-commutative 35.7%

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

   3. rgt-mult-inverse 3.7%

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

   4. exp-neg 3.8%

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

   5. distribute-rgt-neg-out 3.8%

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

   6. *-rgt-identity 3.8%

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

   7. distribute-lft-in 3.8%

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

   8. neg-sub0 3.8%

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

   9. associate-+l- 3.8%

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

   10. neg-sub0 3.2%

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

   11. associate-/r* 3.2%

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

   12. *-rgt-identity 3.2%

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

   13. associate-*r/ 3.2%

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

   14. rgt-mult-inverse 35.3%

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

   15. distribute-frac-neg2 35.3%

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

   16. distribute-neg-frac 35.3%

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

   17. metadata-eval 35.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 68.6%

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

6. Taylor expanded in x around 0 68.6%

   \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot x}{x}} \]
7. Step-by-step derivation

   1. +-commutative 68.6%

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

   2. *-commutative 68.6%

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

   3. fma-undefine 68.6%

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

   4. *-lft-identity 68.6%

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

   5. associate-*l/ 68.6%

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

   6. fma-undefine 68.6%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(x \cdot 0.5 + 1\right)} \]

   7. distribute-lft-in 68.6%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(x \cdot 0.5\right) + \frac{1}{x} \cdot 1} \]

   8. *-commutative 68.6%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(0.5 \cdot x\right)} + \frac{1}{x} \cdot 1 \]

   9. associate-*l* 68.6%

      \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot 0.5\right) \cdot x} + \frac{1}{x} \cdot 1 \]

   10. *-commutative 68.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{1}{x}\right)} \cdot x + \frac{1}{x} \cdot 1 \]

   11. associate-*l* 68.6%

       \[\leadsto \color{blue}{0.5 \cdot \left(\frac{1}{x} \cdot x\right)} + \frac{1}{x} \cdot 1 \]

   12. lft-mult-inverse 68.6%

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

   13. metadata-eval 68.6%

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

   14. *-rgt-identity 68.6%

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

   15. +-commutative 68.6%

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

8. Simplified 68.6%

   \[\leadsto \color{blue}{\frac{1}{x} + 0.5} \]
9. Add Preprocessing

Alternative 5: 67.9% accurate, 68.3× 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 35.7%

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

   1. sub-neg 35.7%

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

   2. +-commutative 35.7%

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

   3. rgt-mult-inverse 3.7%

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

   4. exp-neg 3.8%

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

   5. distribute-rgt-neg-out 3.8%

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

   6. *-rgt-identity 3.8%

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

   7. distribute-lft-in 3.8%

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

   8. neg-sub0 3.8%

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

   9. associate-+l- 3.8%

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

   10. neg-sub0 3.2%

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

   11. associate-/r* 3.2%

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

   12. *-rgt-identity 3.2%

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

   13. associate-*r/ 3.2%

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

   14. rgt-mult-inverse 35.3%

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

   15. distribute-frac-neg2 35.3%

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

   16. distribute-neg-frac 35.3%

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

   17. metadata-eval 35.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 68.6%

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

Alternative 6: 3.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 35.7%

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

   1. sub-neg 35.7%

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

   2. +-commutative 35.7%

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

   3. rgt-mult-inverse 3.7%

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

   4. exp-neg 3.8%

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

   5. distribute-rgt-neg-out 3.8%

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

   6. *-rgt-identity 3.8%

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

   7. distribute-lft-in 3.8%

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

   8. neg-sub0 3.8%

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

   9. associate-+l- 3.8%

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

   10. neg-sub0 3.2%

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

   11. associate-/r* 3.2%

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

   12. *-rgt-identity 3.2%

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

   13. associate-*r/ 3.2%

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

   14. rgt-mult-inverse 35.3%

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

   15. distribute-frac-neg2 35.3%

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

   16. distribute-neg-frac 35.3%

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

   17. metadata-eval 35.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 2.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{expm1}\left(x\right)}}{\sqrt{\mathsf{expm1}\left(x\right)}}} \]
7. Step-by-step derivation

   1. *-inverses 3.3%

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

8. Simplified 3.3%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Alternative 7: 3.2% accurate, 205.0× 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 35.7%

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

   1. sub-neg 35.7%

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

   2. +-commutative 35.7%

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

   3. rgt-mult-inverse 3.7%

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

   4. exp-neg 3.8%

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

   5. distribute-rgt-neg-out 3.8%

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

   6. *-rgt-identity 3.8%

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

   7. distribute-lft-in 3.8%

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

   8. neg-sub0 3.8%

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

   9. associate-+l- 3.8%

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

   10. neg-sub0 3.2%

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

   11. associate-/r* 3.2%

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

   12. *-rgt-identity 3.2%

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

   13. associate-*r/ 3.2%

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

   14. rgt-mult-inverse 35.3%

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

   15. distribute-frac-neg2 35.3%

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

   16. distribute-neg-frac 35.3%

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

   17. metadata-eval 35.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 68.6%

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

6. Taylor expanded in x around inf 3.3%

   \[\leadsto \color{blue}{0.5} \]
7. Add Preprocessing

Alternative 8: 3.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 35.7%

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

   1. sub-neg 35.7%

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

   2. +-commutative 35.7%

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

   3. rgt-mult-inverse 3.7%

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

   4. exp-neg 3.8%

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

   5. distribute-rgt-neg-out 3.8%

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

   6. *-rgt-identity 3.8%

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

   7. distribute-lft-in 3.8%

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

   8. neg-sub0 3.8%

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

   9. associate-+l- 3.8%

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

   10. neg-sub0 3.2%

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

   11. associate-/r* 3.2%

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

   12. *-rgt-identity 3.2%

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

   13. associate-*r/ 3.2%

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

   14. rgt-mult-inverse 35.3%

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

   15. distribute-frac-neg2 35.3%

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

   16. distribute-neg-frac 35.3%

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

   17. metadata-eval 35.3%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

6. Applied egg-rr 0.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{expm1}\left(x\right)} \cdot {\left(-\sqrt{\mathsf{expm1}\left(x\right)}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 0.4%

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

   2. distribute-frac-neg2 0.4%

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

   3. distribute-rgt-neg-out 0.4%

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

   4. rgt-mult-inverse 3.1%

      \[\leadsto -\color{blue}{1} \]

   5. metadata-eval 3.1%

      \[\leadsto \color{blue}{-1} \]

8. Simplified 3.1%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Developer target: 100.0% accurate, 2.0× 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 2024095 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :alt
  (/ (- 1.0) (expm1 (- x)))

  (/ (exp x) (- (exp x) 1.0)))