expq2 (section 3.11)

Percentage Accurate: 36.7% → 100.0%

Time: 4.9s

Alternatives: 7

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 37.5%

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

   1. sub-neg 37.5%

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

   2. +-commutative 37.5%

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

   3. rgt-mult-inverse 3.5%

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

   4. exp-neg 3.5%

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

   5. distribute-rgt-neg-out 3.5%

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

   6. *-rgt-identity 3.5%

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

   7. distribute-lft-in 3.5%

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

   8. neg-sub0 3.5%

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

   9. associate-+l- 3.5%

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

   10. neg-sub0 3.7%

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

   11. associate-/r* 3.7%

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

   12. *-rgt-identity 3.7%

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

   13. associate-*r/ 3.7%

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

   14. rgt-mult-inverse 37.7%

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

   15. distribute-frac-neg2 37.7%

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

   16. distribute-neg-frac 37.7%

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

   17. metadata-eval 37.7%

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

   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: 91.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((x * (0.5d0 + (x * ((0.041666666666666664d0 * x) - 0.16666666666666666d0)))) - 1.0d0))
end function

Java

public static double code(double x) {
	return -1.0 / (x * ((x * (0.5 + (x * ((0.041666666666666664 * x) - 0.16666666666666666)))) - 1.0));
}

Python

def code(x):
	return -1.0 / (x * ((x * (0.5 + (x * ((0.041666666666666664 * x) - 0.16666666666666666)))) - 1.0))

Julia

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

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * ((x * (0.5 + (x * ((0.041666666666666664 * x) - 0.16666666666666666)))) - 1.0));
end

Wolfram

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

Derivation

1. Initial program 37.5%

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

   1. sub-neg 37.5%

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

   2. +-commutative 37.5%

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

   3. rgt-mult-inverse 3.5%

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

   4. exp-neg 3.5%

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

   5. distribute-rgt-neg-out 3.5%

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

   6. *-rgt-identity 3.5%

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

   7. distribute-lft-in 3.5%

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

   8. neg-sub0 3.5%

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

   9. associate-+l- 3.5%

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

   10. neg-sub0 3.7%

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

   11. associate-/r* 3.7%

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

   12. *-rgt-identity 3.7%

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

   13. associate-*r/ 3.7%

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

   14. rgt-mult-inverse 37.7%

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

   15. distribute-frac-neg2 37.7%

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

   16. distribute-neg-frac 37.7%

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

   17. metadata-eval 37.7%

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

   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 91.7%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}} \]
6. Add Preprocessing

Alternative 3: 88.9% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / (x * ((x * (0.5d0 + ((-0.16666666666666666d0) * x))) - 1.0d0))
end function

Java

public static double code(double x) {
	return -1.0 / (x * ((x * (0.5 + (-0.16666666666666666 * x))) - 1.0));
}

Python

def code(x):
	return -1.0 / (x * ((x * (0.5 + (-0.16666666666666666 * x))) - 1.0))

Julia

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

MATLAB

function tmp = code(x)
	tmp = -1.0 / (x * ((x * (0.5 + (-0.16666666666666666 * x))) - 1.0));
end

Wolfram

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

Derivation

1. Initial program 37.5%

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

   1. sub-neg 37.5%

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

   2. +-commutative 37.5%

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

   3. rgt-mult-inverse 3.5%

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

   4. exp-neg 3.5%

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

   5. distribute-rgt-neg-out 3.5%

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

   6. *-rgt-identity 3.5%

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

   7. distribute-lft-in 3.5%

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

   8. neg-sub0 3.5%

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

   9. associate-+l- 3.5%

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

   10. neg-sub0 3.7%

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

   11. associate-/r* 3.7%

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

   12. *-rgt-identity 3.7%

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

   13. associate-*r/ 3.7%

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

   14. rgt-mult-inverse 37.7%

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

   15. distribute-frac-neg2 37.7%

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

   16. distribute-neg-frac 37.7%

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

   17. metadata-eval 37.7%

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

   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 88.2%

   \[\leadsto \frac{-1}{\color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}} \]
6. Add Preprocessing

Alternative 4: 83.5% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

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

   1. sub-neg 37.5%

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

   2. +-commutative 37.5%

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

   3. rgt-mult-inverse 3.5%

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

   4. exp-neg 3.5%

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

   5. distribute-rgt-neg-out 3.5%

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

   6. *-rgt-identity 3.5%

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

   7. distribute-lft-in 3.5%

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

   8. neg-sub0 3.5%

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

   9. associate-+l- 3.5%

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

   10. neg-sub0 3.7%

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

   11. associate-/r* 3.7%

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

   12. *-rgt-identity 3.7%

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

   13. associate-*r/ 3.7%

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

   14. rgt-mult-inverse 37.7%

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

   15. distribute-frac-neg2 37.7%

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

   16. distribute-neg-frac 37.7%

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

   17. metadata-eval 37.7%

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

   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 82.1%

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

Alternative 5: 3.3% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 0.08333333333333333 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.5%

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

   1. sub-neg 37.5%

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

   2. +-commutative 37.5%

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

   3. rgt-mult-inverse 3.5%

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

   4. exp-neg 3.5%

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

   5. distribute-rgt-neg-out 3.5%

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

   6. *-rgt-identity 3.5%

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

   7. distribute-lft-in 3.5%

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

   8. neg-sub0 3.5%

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

   9. associate-+l- 3.5%

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

   10. neg-sub0 3.7%

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

   11. associate-/r* 3.7%

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

   12. *-rgt-identity 3.7%

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

   13. associate-*r/ 3.7%

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

   14. rgt-mult-inverse 37.7%

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

   15. distribute-frac-neg2 37.7%

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

   16. distribute-neg-frac 37.7%

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

   17. metadata-eval 37.7%

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

   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 66.2%

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

6. Taylor expanded in x around inf 3.2%

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

Alternative 6: 67.7% 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 37.5%

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

   1. sub-neg 37.5%

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

   2. +-commutative 37.5%

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

   3. rgt-mult-inverse 3.5%

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

   4. exp-neg 3.5%

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

   5. distribute-rgt-neg-out 3.5%

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

   6. *-rgt-identity 3.5%

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

   7. distribute-lft-in 3.5%

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

   8. neg-sub0 3.5%

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

   9. associate-+l- 3.5%

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

   10. neg-sub0 3.7%

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

   11. associate-/r* 3.7%

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

   12. *-rgt-identity 3.7%

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

   13. associate-*r/ 3.7%

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

   14. rgt-mult-inverse 37.7%

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

   15. distribute-frac-neg2 37.7%

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

   16. distribute-neg-frac 37.7%

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

   17. metadata-eval 37.7%

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

   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 66.7%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
6. 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 37.5%

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

   1. sub-neg 37.5%

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

   2. +-commutative 37.5%

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

   3. rgt-mult-inverse 3.5%

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

   4. exp-neg 3.5%

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

   5. distribute-rgt-neg-out 3.5%

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

   6. *-rgt-identity 3.5%

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

   7. distribute-lft-in 3.5%

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

   8. neg-sub0 3.5%

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

   9. associate-+l- 3.5%

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

   10. neg-sub0 3.7%

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

   11. associate-/r* 3.7%

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

   12. *-rgt-identity 3.7%

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

   13. associate-*r/ 3.7%

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

   14. rgt-mult-inverse 37.7%

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

   15. distribute-frac-neg2 37.7%

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

   16. distribute-neg-frac 37.7%

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

   17. metadata-eval 37.7%

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

   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 66.2%

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

6. Taylor expanded in x around inf 3.1%

   \[\leadsto \color{blue}{0.5} \]
7. 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 2024050 -o generate:simplify
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

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

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