expq2 (section 3.11)

Percentage Accurate: 36.6% → 100.0%

Time: 8.8s

Alternatives: 8

Speedup: 68.3×

Specification

\[710 > x\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.6% 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 34.0%

   \[\frac{e^{x}}{e^{x} - 1} \]
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. Final simplification 100.0%

   \[\leadsto \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]
6. 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 34.0%

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

   1. sub-neg 34.0%

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

   2. +-commutative 34.0%

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

   3. rgt-mult-inverse 5.5%

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

   4. exp-neg 5.5%

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

   5. distribute-rgt-neg-out 5.5%

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

   6. *-rgt-identity 5.5%

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

   7. distribute-lft-in 5.5%

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

   8. neg-sub0 5.5%

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

   9. associate-+l- 5.5%

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

   10. neg-sub0 6.0%

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

   11. associate-/r* 6.0%

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

   12. *-rgt-identity 6.0%

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

   13. associate-*r/ 6.0%

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

   14. rgt-mult-inverse 34.6%

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

   15. distribute-frac-neg2 34.6%

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

   16. distribute-neg-frac 34.6%

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

   17. metadata-eval 34.6%

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

   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. Final simplification 100.0%

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

Alternative 3: 91.8% 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 34.0%

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

   1. sub-neg 34.0%

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

   2. +-commutative 34.0%

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

   3. rgt-mult-inverse 5.5%

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

   4. exp-neg 5.5%

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

   5. distribute-rgt-neg-out 5.5%

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

   6. *-rgt-identity 5.5%

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

   7. distribute-lft-in 5.5%

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

   8. neg-sub0 5.5%

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

   9. associate-+l- 5.5%

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

   10. neg-sub0 6.0%

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

   11. associate-/r* 6.0%

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

   12. *-rgt-identity 6.0%

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

   13. associate-*r/ 6.0%

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

   14. rgt-mult-inverse 34.6%

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

   15. distribute-frac-neg2 34.6%

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

   16. distribute-neg-frac 34.6%

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

   17. metadata-eval 34.6%

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

   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 93.2%

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

6. Final simplification 93.2%

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

Alternative 4: 89.0% 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 34.0%

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

   1. sub-neg 34.0%

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

   2. +-commutative 34.0%

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

   3. rgt-mult-inverse 5.5%

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

   4. exp-neg 5.5%

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

   5. distribute-rgt-neg-out 5.5%

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

   6. *-rgt-identity 5.5%

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

   7. distribute-lft-in 5.5%

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

   8. neg-sub0 5.5%

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

   9. associate-+l- 5.5%

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

   10. neg-sub0 6.0%

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

   11. associate-/r* 6.0%

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

   12. *-rgt-identity 6.0%

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

   13. associate-*r/ 6.0%

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

   14. rgt-mult-inverse 34.6%

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

   15. distribute-frac-neg2 34.6%

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

   16. distribute-neg-frac 34.6%

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

   17. metadata-eval 34.6%

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

   18. expm1-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 91.2%

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

6. Final simplification 91.2%

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

Alternative 5: 83.9% 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 34.0%

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

   1. sub-neg 34.0%

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

   2. +-commutative 34.0%

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

   3. rgt-mult-inverse 5.5%

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

   4. exp-neg 5.5%

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

   5. distribute-rgt-neg-out 5.5%

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

   6. *-rgt-identity 5.5%

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

   7. distribute-lft-in 5.5%

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

   8. neg-sub0 5.5%

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

   9. associate-+l- 5.5%

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

   10. neg-sub0 6.0%

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

   11. associate-/r* 6.0%

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

   12. *-rgt-identity 6.0%

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

   13. associate-*r/ 6.0%

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

   14. rgt-mult-inverse 34.6%

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

   15. distribute-frac-neg2 34.6%

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

   16. distribute-neg-frac 34.6%

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

   17. metadata-eval 34.6%

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

   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 86.5%

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

6. Final simplification 86.5%

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

Alternative 6: 67.8% 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 34.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. associate-/l* 100.0%

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

6. Applied egg-rr 100.0%

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

   1. log1p-expm1-u 34.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{e^{x}} \cdot \frac{\sqrt{e^{x}}}{\mathsf{expm1}\left(x\right)}\right)\right)} \]

   2. associate-*r/ 34.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}{\mathsf{expm1}\left(x\right)}}\right)\right) \]

   3. add-sqr-sqrt 34.5%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{e^{x}}}{\mathsf{expm1}\left(x\right)}\right)\right) \]

8. Applied egg-rr 34.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\right)\right)} \]

9. Taylor expanded in x around 0 70.6%

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

    1. +-commutative 70.6%

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

    2. *-commutative 70.6%

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

    3. fma-undefine 70.6%

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

    4. *-lft-identity 70.6%

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

    5. associate-*l/ 70.6%

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

    6. fma-undefine 70.6%

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

    7. distribute-lft-in 70.3%

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

    8. *-commutative 70.3%

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

    9. associate-*l* 70.3%

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

    10. *-commutative 70.3%

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

    11. associate-*l* 70.3%

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

    12. lft-mult-inverse 70.7%

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

    13. metadata-eval 70.7%

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

    14. *-rgt-identity 70.7%

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

    15. +-commutative 70.7%

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

11. Simplified 70.7%

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

12. Final simplification 70.7%

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

Alternative 7: 67.8% 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 34.0%

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

   1. sub-neg 34.0%

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

   2. +-commutative 34.0%

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

   3. rgt-mult-inverse 5.5%

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

   4. exp-neg 5.5%

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

   5. distribute-rgt-neg-out 5.5%

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

   6. *-rgt-identity 5.5%

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

   7. distribute-lft-in 5.5%

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

   8. neg-sub0 5.5%

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

   9. associate-+l- 5.5%

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

   10. neg-sub0 6.0%

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

   11. associate-/r* 6.0%

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

   12. *-rgt-identity 6.0%

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

   13. associate-*r/ 6.0%

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

   14. rgt-mult-inverse 34.6%

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

   15. distribute-frac-neg2 34.6%

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

   16. distribute-neg-frac 34.6%

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

   17. metadata-eval 34.6%

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

   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 70.6%

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

6. Final simplification 70.6%

   \[\leadsto \frac{1}{x} \]
7. Add Preprocessing

Alternative 8: 3.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 34.0%

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

   1. sub-neg 34.0%

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

   2. +-commutative 34.0%

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

   3. rgt-mult-inverse 5.5%

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

   4. exp-neg 5.5%

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

   5. distribute-rgt-neg-out 5.5%

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

   6. *-rgt-identity 5.5%

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

   7. distribute-lft-in 5.5%

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

   8. neg-sub0 5.5%

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

   9. associate-+l- 5.5%

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

   10. neg-sub0 6.0%

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

   11. associate-/r* 6.0%

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

   12. *-rgt-identity 6.0%

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

   13. associate-*r/ 6.0%

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

   14. rgt-mult-inverse 34.6%

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

   15. distribute-frac-neg2 34.6%

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

   16. distribute-neg-frac 34.6%

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

   17. metadata-eval 34.6%

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

   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 70.6%

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

   1. *-commutative 70.6%

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

7. Simplified 70.6%

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

8. Taylor expanded in x around inf 3.2%

   \[\leadsto \color{blue}{0.5} \]

9. Final simplification 3.2%

   \[\leadsto 0.5 \]
10. Add Preprocessing

Developer target: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{expm1}\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0) (expm1 (- x))))

C

double code(double x) {
	return -1.0 / expm1(-x);
}

Java

public static double code(double x) {
	return -1.0 / Math.expm1(-x);
}

Python

def code(x):
	return -1.0 / math.expm1(-x)

Julia

function code(x)
	return Float64(Float64(-1.0) / expm1(Float64(-x)))
end

Wolfram

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

Reproduce

herbie shell --seed 2024089 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

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

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