expq2 (section 3.11)

Percentage Accurate: 37.5% → 100.0%

Time: 7.6s

Alternatives: 10

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: 37.5% 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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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: 98.0% 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 35.7%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 98.3%

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

Alternative 3: 91.2% 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 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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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 90.6%

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

6. Final simplification 90.6%

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

Alternative 4: 90.8% 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 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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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 90.6%

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

6. Taylor expanded in x around inf 90.4%

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

   1. *-commutative 90.4%

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

8. Simplified 90.4%

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

9. Final simplification 90.4%

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

Alternative 5: 88.6% 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 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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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 88.3%

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

6. Final simplification 88.3%

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

Alternative 6: 83.1% 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 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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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 82.2%

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

6. Final simplification 82.2%

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

Alternative 7: 67.2% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(0.5 + N[(1.0 / x), $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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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. Step-by-step derivation

   1. *-commutative 68.6%

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

7. Simplified 68.6%

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

8. Taylor expanded in x around -inf 37.2%

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

   1. mul-1-neg 37.2%

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

   2. distribute-rgt-neg-in 37.2%

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

   3. sub-neg 37.2%

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

   4. associate-*r/ 37.2%

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

   5. distribute-lft-in 37.2%

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

   6. metadata-eval 37.2%

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

   7. associate-*r/ 37.2%

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

   8. metadata-eval 37.2%

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

   9. metadata-eval 37.2%

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

10. Simplified 37.2%

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

    1. distribute-neg-in 37.2%

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

    2. div-inv 37.1%

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

    3. distribute-rgt-neg-in 37.1%

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

    4. mul-1-neg 37.1%

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

    5. div-inv 37.1%

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

    6. add-sqr-sqrt 16.5%

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

    7. sqrt-unprod 16.8%

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

    8. frac-times 16.7%

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

    9. metadata-eval 16.7%

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

    10. metadata-eval 16.7%

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

    11. frac-times 16.8%

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

    12. sqrt-unprod 0.2%

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

    13. add-sqr-sqrt 1.2%

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

    14. div-inv 1.2%

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

    15. sub-neg 1.2%

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

12. Applied egg-rr 37.2%

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

    1. pow1 37.2%

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

    2. sub-neg 37.2%

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

    3. metadata-eval 37.2%

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

14. Applied egg-rr 37.2%

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

    1. unpow1 37.2%

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

    2. +-commutative 37.2%

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

    3. distribute-lft-in 37.2%

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

    4. *-lft-identity 37.2%

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

    5. associate-*l/ 37.1%

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

    6. associate-*r* 68.4%

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

    7. rgt-mult-inverse 68.6%

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

    8. *-lft-identity 68.6%

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

16. Simplified 68.6%

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

Alternative 8: 67.0% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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. Step-by-step derivation

   1. *-commutative 68.6%

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

7. Simplified 68.6%

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

8. Taylor expanded in x around 0 68.6%

   \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot x}{x}} \]
9. 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-rgt-in 68.6%

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

   8. associate-*r* 68.6%

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

   9. *-commutative 68.6%

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

   10. associate-*r* 68.6%

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

   11. rgt-mult-inverse 68.6%

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

   12. metadata-eval 68.6%

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

   13. *-lft-identity 68.6%

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

   14. +-commutative 68.6%

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

10. Simplified 68.6%

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

11. Final simplification 68.6%

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

Alternative 9: 67.0% 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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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 10: 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 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 4.5%

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

   4. exp-neg 4.5%

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

   5. distribute-rgt-neg-out 4.5%

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

   6. *-rgt-identity 4.5%

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

   7. distribute-lft-in 4.5%

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

   8. neg-sub0 4.5%

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

   9. associate-+l- 4.5%

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

   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 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. Step-by-step derivation

   1. *-commutative 68.6%

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

7. Simplified 68.6%

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

8. Taylor expanded in x around inf 3.3%

   \[\leadsto \color{blue}{0.5} \]
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 2024136 
(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)))