expq2 (section 3.11)

Percentage Accurate: 38.0% → 100.0%

Time: 6.0s

Alternatives: 6

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: 38.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. sub-neg 37.6%

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

   2. +-commutative 37.6%

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

   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 4.9%

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

   8. neg-sub0 4.9%

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

   9. associate-+l- 4.9%

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

   10. neg-sub0 5.2%

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

   11. associate-/r* 5.2%

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

   12. *-rgt-identity 5.2%

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

   13. associate-*r/ 5.2%

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

   14. rgt-mult-inverse 38.0%

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

   15. distribute-frac-neg2 38.0%

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

   16. distribute-neg-frac 38.0%

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

   17. metadata-eval 38.0%

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

   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. Final simplification 100.0%

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

Alternative 2: 66.3% accurate, 20.5× speedup

Math

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

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 * Float64(-(-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 37.6%

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

   1. sub-neg 37.6%

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

   2. +-commutative 37.6%

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

   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 4.9%

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

   8. neg-sub0 4.9%

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

   9. associate-+l- 4.9%

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

   10. neg-sub0 5.2%

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

   11. associate-/r* 5.2%

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

   12. *-rgt-identity 5.2%

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

   13. associate-*r/ 5.2%

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

   14. rgt-mult-inverse 38.0%

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

   15. distribute-frac-neg2 38.0%

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

   16. distribute-neg-frac 38.0%

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

   17. metadata-eval 38.0%

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

   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 66.9%

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

   1. *-commutative 66.9%

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

7. Simplified 66.9%

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

8. Taylor expanded in x around -inf 34.4%

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

9. Taylor expanded in x around -inf 34.2%

   \[\leadsto -1 \cdot \color{blue}{\left(-1 \cdot \left(x \cdot \left(0.08333333333333333 + \left(0.5 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)\right)} \]
10. Step-by-step derivation

    1. mul-1-neg 34.2%

       \[\leadsto -1 \cdot \color{blue}{\left(-x \cdot \left(0.08333333333333333 + \left(0.5 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)\right)} \]

    2. distribute-rgt-in 34.3%

       \[\leadsto -1 \cdot \left(-\color{blue}{\left(0.08333333333333333 \cdot x + \left(0.5 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right) \cdot x\right)}\right) \]

    3. distribute-neg-in 34.3%

       \[\leadsto -1 \cdot \color{blue}{\left(\left(-0.08333333333333333 \cdot x\right) + \left(-\left(0.5 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right) \cdot x\right)\right)} \]

    4. distribute-lft-neg-in 34.3%

       \[\leadsto -1 \cdot \left(\color{blue}{\left(-0.08333333333333333\right) \cdot x} + \left(-\left(0.5 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right) \cdot x\right)\right) \]

    5. metadata-eval 34.3%

       \[\leadsto -1 \cdot \left(\color{blue}{-0.08333333333333333} \cdot x + \left(-\left(0.5 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right) \cdot x\right)\right) \]

    6. *-commutative 34.3%

       \[\leadsto -1 \cdot \left(\color{blue}{x \cdot -0.08333333333333333} + \left(-\left(0.5 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right) \cdot x\right)\right) \]

    7. *-commutative 34.3%

       \[\leadsto -1 \cdot \left(x \cdot -0.08333333333333333 + \left(-\color{blue}{x \cdot \left(0.5 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)}\right)\right) \]

    8. distribute-rgt-in 34.3%

       \[\leadsto -1 \cdot \left(x \cdot -0.08333333333333333 + \left(-\color{blue}{\left(\left(0.5 \cdot \frac{1}{x}\right) \cdot x + \frac{1}{{x}^{2}} \cdot x\right)}\right)\right) \]

    9. unpow2 34.3%

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

    10. associate-/r* 34.4%

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

    11. *-rgt-identity 34.4%

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

    12. associate-*r/ 34.3%

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

    13. distribute-rgt-in 34.3%

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

    14. distribute-rgt-in 34.3%

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

    15. associate-*r* 66.8%

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

    16. rgt-mult-inverse 67.0%

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

    17. *-lft-identity 67.0%

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

11. Simplified 67.0%

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

12. Final simplification 67.0%

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

Alternative 3: 66.3% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. sub-neg 37.6%

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

   2. +-commutative 37.6%

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

   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 4.9%

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

   8. neg-sub0 4.9%

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

   9. associate-+l- 4.9%

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

   10. neg-sub0 5.2%

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

   11. associate-/r* 5.2%

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

   12. *-rgt-identity 5.2%

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

   13. associate-*r/ 5.2%

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

   14. rgt-mult-inverse 38.0%

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

   15. distribute-frac-neg2 38.0%

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

   16. distribute-neg-frac 38.0%

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

   17. metadata-eval 38.0%

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

   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 66.9%

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

   1. *-commutative 66.9%

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

7. Simplified 66.9%

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

8. Taylor expanded in x around 0 66.9%

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

   1. +-commutative 66.9%

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

   2. *-commutative 66.9%

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

   3. fma-undefine 66.9%

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

   4. *-lft-identity 66.9%

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

   5. associate-*l/ 66.9%

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

   6. fma-undefine 66.9%

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

   7. distribute-rgt-in 66.9%

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

   8. associate-*r* 66.9%

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

   9. *-commutative 66.9%

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

   10. associate-*l* 66.9%

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

   11. lft-mult-inverse 66.9%

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

   12. metadata-eval 66.9%

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

   13. *-lft-identity 66.9%

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

   14. +-commutative 66.9%

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

10. Simplified 66.9%

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

11. Final simplification 66.9%

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

Alternative 4: 3.3% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.08333333333333333

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. sub-neg 37.6%

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

   2. +-commutative 37.6%

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

   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 4.9%

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

   8. neg-sub0 4.9%

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

   9. associate-+l- 4.9%

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

   10. neg-sub0 5.2%

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

   11. associate-/r* 5.2%

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

   12. *-rgt-identity 5.2%

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

   13. associate-*r/ 5.2%

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

   14. rgt-mult-inverse 38.0%

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

   15. distribute-frac-neg2 38.0%

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

   16. distribute-neg-frac 38.0%

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

   17. metadata-eval 38.0%

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

   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 66.9%

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

   1. *-commutative 66.9%

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

7. Simplified 66.9%

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

8. Taylor expanded in x around inf 3.4%

   \[\leadsto \color{blue}{0.08333333333333333 \cdot x} \]
9. Step-by-step derivation

   1. *-commutative 3.4%

      \[\leadsto \color{blue}{x \cdot 0.08333333333333333} \]

10. Simplified 3.4%

    \[\leadsto \color{blue}{x \cdot 0.08333333333333333} \]

11. Final simplification 3.4%

    \[\leadsto x \cdot 0.08333333333333333 \]
12. Add Preprocessing

Alternative 5: 66.5% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.6%

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

   1. sub-neg 37.6%

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

   2. +-commutative 37.6%

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

   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 4.9%

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

   8. neg-sub0 4.9%

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

   9. associate-+l- 4.9%

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

   10. neg-sub0 5.2%

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

   11. associate-/r* 5.2%

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

   12. *-rgt-identity 5.2%

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

   13. associate-*r/ 5.2%

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

   14. rgt-mult-inverse 38.0%

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

   15. distribute-frac-neg2 38.0%

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

   16. distribute-neg-frac 38.0%

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

   17. metadata-eval 38.0%

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

   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 66.9%

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

6. Final simplification 66.9%

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

Alternative 6: 3.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 37.6%

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

   1. sub-neg 37.6%

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

   2. +-commutative 37.6%

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

   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 4.9%

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

   8. neg-sub0 4.9%

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

   9. associate-+l- 4.9%

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

   10. neg-sub0 5.2%

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

   11. associate-/r* 5.2%

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

   12. *-rgt-identity 5.2%

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

   13. associate-*r/ 5.2%

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

   14. rgt-mult-inverse 38.0%

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

   15. distribute-frac-neg2 38.0%

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

   16. distribute-neg-frac 38.0%

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

   17. metadata-eval 38.0%

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

   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 66.9%

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

   1. *-commutative 66.9%

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

7. Simplified 66.9%

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

8. Taylor expanded in x around inf 3.1%

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

9. Final simplification 3.1%

   \[\leadsto 0.5 \]
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 2024074 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

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

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