expq2 (section 3.11)

Percentage Accurate: 37.1% → 100.0%

Time: 6.8s

Alternatives: 9

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: 37.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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 40.9%

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

   1. sub-neg 40.9%

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

   2. +-commutative 40.9%

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

   3. rgt-mult-inverse 4.1%

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

   4. exp-neg 4.2%

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

   5. distribute-rgt-neg-out 4.2%

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

   6. *-rgt-identity 4.2%

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

   7. distribute-lft-in 4.2%

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

   8. neg-sub0 4.2%

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

   9. associate-+l- 4.2%

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

   10. neg-sub0 4.1%

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

   11. associate-/r* 4.1%

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

   12. *-rgt-identity 4.1%

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

   13. associate-*r/ 4.1%

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

   14. rgt-mult-inverse 40.8%

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

   15. distribute-frac-neg2 40.8%

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

   16. distribute-neg-frac 40.8%

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

   17. metadata-eval 40.8%

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

   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: 98.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 98.2%

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

Alternative 3: 92.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

   1. sub-neg 40.9%

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

   2. +-commutative 40.9%

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

   3. rgt-mult-inverse 4.1%

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

   4. exp-neg 4.2%

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

   5. distribute-rgt-neg-out 4.2%

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

   6. *-rgt-identity 4.2%

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

   7. distribute-lft-in 4.2%

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

   8. neg-sub0 4.2%

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

   9. associate-+l- 4.2%

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

   10. neg-sub0 4.1%

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

   11. associate-/r* 4.1%

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

   12. *-rgt-identity 4.1%

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

   13. associate-*r/ 4.1%

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

   14. rgt-mult-inverse 40.8%

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

   15. distribute-frac-neg2 40.8%

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

   16. distribute-neg-frac 40.8%

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

   17. metadata-eval 40.8%

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

   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 89.8%

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

6. Final simplification 89.8%

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

Alternative 4: 91.8% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

   1. sub-neg 40.9%

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

   2. +-commutative 40.9%

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

   3. rgt-mult-inverse 4.1%

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

   4. exp-neg 4.2%

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

   5. distribute-rgt-neg-out 4.2%

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

   6. *-rgt-identity 4.2%

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

   7. distribute-lft-in 4.2%

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

   8. neg-sub0 4.2%

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

   9. associate-+l- 4.2%

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

   10. neg-sub0 4.1%

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

   11. associate-/r* 4.1%

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

   12. *-rgt-identity 4.1%

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

   13. associate-*r/ 4.1%

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

   14. rgt-mult-inverse 40.8%

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

   15. distribute-frac-neg2 40.8%

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

   16. distribute-neg-frac 40.8%

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

   17. metadata-eval 40.8%

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

   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 89.8%

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

6. Taylor expanded in x around inf 89.6%

   \[\leadsto \frac{-1}{x \cdot \left(x \cdot \left(0.5 + x \cdot \color{blue}{\left(0.041666666666666664 \cdot x\right)}\right) - 1\right)} \]
7. Step-by-step derivation

   1. *-commutative 89.6%

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

8. Simplified 89.6%

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

9. Final simplification 89.6%

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

Alternative 5: 89.6% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

   1. sub-neg 40.9%

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

   2. +-commutative 40.9%

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

   3. rgt-mult-inverse 4.1%

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

   4. exp-neg 4.2%

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

   5. distribute-rgt-neg-out 4.2%

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

   6. *-rgt-identity 4.2%

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

   7. distribute-lft-in 4.2%

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

   8. neg-sub0 4.2%

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

   9. associate-+l- 4.2%

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

   10. neg-sub0 4.1%

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

   11. associate-/r* 4.1%

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

   12. *-rgt-identity 4.1%

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

   13. associate-*r/ 4.1%

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

   14. rgt-mult-inverse 40.8%

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

   15. distribute-frac-neg2 40.8%

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

   16. distribute-neg-frac 40.8%

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

   17. metadata-eval 40.8%

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

   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 87.1%

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

6. Final simplification 87.1%

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

Alternative 6: 83.9% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.9%

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

   1. sub-neg 40.9%

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

   2. +-commutative 40.9%

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

   3. rgt-mult-inverse 4.1%

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

   4. exp-neg 4.2%

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

   5. distribute-rgt-neg-out 4.2%

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

   6. *-rgt-identity 4.2%

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

   7. distribute-lft-in 4.2%

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

   8. neg-sub0 4.2%

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

   9. associate-+l- 4.2%

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

   10. neg-sub0 4.1%

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

   11. associate-/r* 4.1%

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

   12. *-rgt-identity 4.1%

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

   13. associate-*r/ 4.1%

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

   14. rgt-mult-inverse 40.8%

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

   15. distribute-frac-neg2 40.8%

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

   16. distribute-neg-frac 40.8%

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

   17. metadata-eval 40.8%

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

   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 79.6%

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

6. Final simplification 79.6%

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

Alternative 7: 67.3% 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 40.9%

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

   1. sub-neg 40.9%

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

   2. +-commutative 40.9%

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

   3. rgt-mult-inverse 4.1%

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

   4. exp-neg 4.2%

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

   5. distribute-rgt-neg-out 4.2%

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

   6. *-rgt-identity 4.2%

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

   7. distribute-lft-in 4.2%

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

   8. neg-sub0 4.2%

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

   9. associate-+l- 4.2%

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

   10. neg-sub0 4.1%

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

   11. associate-/r* 4.1%

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

   12. *-rgt-identity 4.1%

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

   13. associate-*r/ 4.1%

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

   14. rgt-mult-inverse 40.8%

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

   15. distribute-frac-neg2 40.8%

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

   16. distribute-neg-frac 40.8%

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

   17. metadata-eval 40.8%

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

   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 63.4%

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

Alternative 8: 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 40.9%

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

   1. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 98.2%

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

6. Taylor expanded in x around 0 62.7%

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

7. Taylor expanded in x around inf 3.8%

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

Alternative 9: 3.3% 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 40.9%

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

   1. sub-neg 40.9%

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

   2. +-commutative 40.9%

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

   3. rgt-mult-inverse 4.1%

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

   4. exp-neg 4.2%

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

   5. distribute-rgt-neg-out 4.2%

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

   6. *-rgt-identity 4.2%

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

   7. distribute-lft-in 4.2%

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

   8. neg-sub0 4.2%

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

   9. associate-+l- 4.2%

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

   10. neg-sub0 4.1%

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

   11. associate-/r* 4.1%

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

   12. *-rgt-identity 4.1%

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

   13. associate-*r/ 4.1%

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

   14. rgt-mult-inverse 40.8%

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

   15. distribute-frac-neg2 40.8%

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

   16. distribute-neg-frac 40.8%

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

   17. metadata-eval 40.8%

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

   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 63.3%

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

   1. *-commutative 63.3%

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

7. Simplified 63.3%

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

8. Taylor expanded in x around inf 3.4%

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

Developer Target 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(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(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)))