expq2 (section 3.11)

Percentage Accurate: 36.9% → 100.0%

Time: 7.8s

Alternatives: 12

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: 36.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (expm1 x)))

C

double code(double x) {
	return exp(x) / expm1(x);
}

Java

public static double code(double x) {
	return Math.exp(x) / Math.expm1(x);
}

Python

def code(x):
	return math.exp(x) / math.expm1(x)

Julia

function code(x)
	return Float64(exp(x) / expm1(x))
end

Wolfram

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

Derivation

1. Initial program 39.2%

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

Alternative 2: 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 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 3: 98.2% 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 39.2%

   \[\frac{e^{x}}{e^{x} - 1} \]
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 98.0%

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

Alternative 4: 92.3% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 90.7%

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

6. Final simplification 90.7%

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

Alternative 5: 91.9% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 90.7%

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

6. Taylor expanded in x around inf 90.2%

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

   1. *-commutative 90.2%

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

8. Simplified 90.2%

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

9. Final simplification 90.2%

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

Alternative 6: 89.2% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 89.0%

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

6. Final simplification 89.0%

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

Alternative 7: 84.0% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 82.4%

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

   1. sub-neg 82.4%

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

   2. metadata-eval 82.4%

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

   3. distribute-rgt-in 82.4%

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

   4. *-commutative 82.4%

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

   5. neg-mul-1 82.4%

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

7. Applied egg-rr 82.4%

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

8. Final simplification 82.4%

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

Alternative 8: 84.0% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 82.4%

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

6. Final simplification 82.4%

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

Alternative 9: 67.7% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 65.6%

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

   1. *-commutative 65.6%

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

7. Simplified 65.6%

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

8. Taylor expanded in x around -inf 35.5%

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

   1. mul-1-neg 35.5%

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

   2. distribute-rgt-neg-in 35.5%

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

   3. sub-neg 35.5%

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

   4. associate-*r/ 35.5%

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

   5. distribute-lft-in 35.5%

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

   6. metadata-eval 35.5%

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

   7. associate-*r/ 35.5%

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

   8. metadata-eval 35.5%

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

   9. metadata-eval 35.5%

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

10. Simplified 35.5%

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

    1. distribute-neg-in 35.5%

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

    2. distribute-neg-frac2 35.5%

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

    3. add-sqr-sqrt 19.0%

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

    4. sqrt-unprod 19.3%

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

    5. sqr-neg 19.3%

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

    6. sqrt-unprod 0.2%

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

    7. add-sqr-sqrt 1.2%

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

    8. sub-neg 1.2%

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

    9. frac-2neg 1.2%

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

    10. distribute-neg-in 1.2%

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

    11. metadata-eval 1.2%

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

    12. distribute-frac-neg2 1.2%

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

    13. metadata-eval 1.2%

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

    14. frac-2neg 1.2%

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

    15. add-sqr-sqrt 1.0%

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

    16. sqrt-unprod 17.4%

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

    17. sqr-neg 17.4%

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

    18. sqrt-unprod 16.3%

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

    19. add-sqr-sqrt 35.5%

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

12. Applied egg-rr 35.5%

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

    1. pow1 35.5%

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

    2. sub-neg 35.5%

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

    3. metadata-eval 35.5%

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

14. Applied egg-rr 35.5%

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

    1. unpow1 35.5%

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

    2. distribute-lft-in 35.5%

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

    3. *-commutative 35.5%

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

    4. associate-*l/ 65.7%

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

    5. associate-/l* 65.7%

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

    6. *-inverses 65.7%

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

    7. *-rgt-identity 65.7%

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

    8. +-commutative 65.7%

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

16. Simplified 65.7%

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

17. Final simplification 65.7%

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

Alternative 10: 67.6% 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 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 65.4%

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

   1. *-commutative 65.4%

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

7. Simplified 65.4%

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

8. Taylor expanded in x around 0 65.4%

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

   1. +-commutative 65.4%

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

   2. *-commutative 65.4%

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

   3. fma-undefine 65.4%

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

   4. *-lft-identity 65.4%

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

   5. associate-*l/ 65.4%

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

   6. fma-undefine 65.4%

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

   7. distribute-rgt-in 65.4%

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

   8. associate-*r* 65.4%

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

   9. *-commutative 65.4%

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

   10. associate-*r* 65.4%

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

   11. rgt-mult-inverse 65.4%

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

   12. metadata-eval 65.4%

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

   13. *-lft-identity 65.4%

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

   14. +-commutative 65.4%

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

10. Simplified 65.4%

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

11. Final simplification 65.4%

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

Alternative 11: 67.6% 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 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 65.4%

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

Alternative 12: 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 39.2%

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

   1. sub-neg 39.2%

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

   2. +-commutative 39.2%

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

   3. rgt-mult-inverse 4.8%

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

   4. exp-neg 4.8%

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

   5. distribute-rgt-neg-out 4.8%

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

   6. *-rgt-identity 4.8%

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

   7. distribute-lft-in 4.7%

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

   8. neg-sub0 4.7%

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

   9. associate-+l- 4.7%

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

   10. neg-sub0 4.7%

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

   11. associate-/r* 4.7%

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

   12. *-rgt-identity 4.7%

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

   13. associate-*r/ 4.7%

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

   14. rgt-mult-inverse 39.0%

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

   15. distribute-frac-neg2 39.0%

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

   16. distribute-neg-frac 39.0%

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

   17. metadata-eval 39.0%

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

   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 65.4%

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

   1. *-commutative 65.4%

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

7. Simplified 65.4%

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

8. Taylor expanded in x around inf 3.2%

   \[\leadsto \color{blue}{0.5} \]
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 2024169 
(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)))