expq2 (section 3.11)

Percentage Accurate: 37.1% → 99.4%

Time: 4.4s

Alternatives: 6

Speedup: 68.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% 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 34.5%

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

   1. expm1-def 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. Final simplification 100.0%

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

Alternative 2: 67.0% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.5%

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

   1. expm1-def 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. Taylor expanded in x around 0 70.6%

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

   1. +-commutative 70.6%

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

   2. associate-+r+ 70.6%

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

   3. +-commutative 70.6%

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

   4. flip-+ 38.1%

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

   5. div-inv 38.0%

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

   6. fma-def 38.0%

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

   7. sub-neg 38.0%

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

   8. inv-pow 38.0%

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

   9. inv-pow 38.0%

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

   10. pow-prod-up 38.0%

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

   11. metadata-eval 38.0%

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

   12. metadata-eval 38.0%

       \[\leadsto \mathsf{fma}\left({x}^{-2} + \left(-\color{blue}{0.25}\right), \frac{1}{\frac{1}{x} - 0.5}, 0.08333333333333333 \cdot x\right) \]

   13. metadata-eval 38.0%

       \[\leadsto \mathsf{fma}\left({x}^{-2} + \color{blue}{-0.25}, \frac{1}{\frac{1}{x} - 0.5}, 0.08333333333333333 \cdot x\right) \]

   14. sub-neg 38.0%

       \[\leadsto \mathsf{fma}\left({x}^{-2} + -0.25, \frac{1}{\color{blue}{\frac{1}{x} + \left(-0.5\right)}}, 0.08333333333333333 \cdot x\right) \]

   15. metadata-eval 38.0%

       \[\leadsto \mathsf{fma}\left({x}^{-2} + -0.25, \frac{1}{\frac{1}{x} + \color{blue}{-0.5}}, 0.08333333333333333 \cdot x\right) \]

6. Applied egg-rr 38.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-2} + -0.25, \frac{1}{\frac{1}{x} + -0.5}, 0.08333333333333333 \cdot x\right)} \]
7. Step-by-step derivation

   1. fma-udef 38.0%

      \[\leadsto \color{blue}{\left({x}^{-2} + -0.25\right) \cdot \frac{1}{\frac{1}{x} + -0.5} + 0.08333333333333333 \cdot x} \]

   2. +-commutative 38.0%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot x + \left({x}^{-2} + -0.25\right) \cdot \frac{1}{\frac{1}{x} + -0.5}} \]

   3. *-commutative 38.0%

      \[\leadsto \color{blue}{x \cdot 0.08333333333333333} + \left({x}^{-2} + -0.25\right) \cdot \frac{1}{\frac{1}{x} + -0.5} \]

   4. associate-*r/ 37.9%

      \[\leadsto x \cdot 0.08333333333333333 + \color{blue}{\frac{\left({x}^{-2} + -0.25\right) \cdot 1}{\frac{1}{x} + -0.5}} \]

   5. *-rgt-identity 37.9%

      \[\leadsto x \cdot 0.08333333333333333 + \frac{\color{blue}{{x}^{-2} + -0.25}}{\frac{1}{x} + -0.5} \]

8. Simplified 37.9%

   \[\leadsto \color{blue}{x \cdot 0.08333333333333333 + \frac{{x}^{-2} + -0.25}{\frac{1}{x} + -0.5}} \]
9. Step-by-step derivation

   1. clear-num 38.0%

      \[\leadsto x \cdot 0.08333333333333333 + \color{blue}{\frac{1}{\frac{\frac{1}{x} + -0.5}{{x}^{-2} + -0.25}}} \]

   2. inv-pow 38.0%

      \[\leadsto x \cdot 0.08333333333333333 + \color{blue}{{\left(\frac{\frac{1}{x} + -0.5}{{x}^{-2} + -0.25}\right)}^{-1}} \]

   3. clear-num 37.9%

      \[\leadsto x \cdot 0.08333333333333333 + {\color{blue}{\left(\frac{1}{\frac{{x}^{-2} + -0.25}{\frac{1}{x} + -0.5}}\right)}}^{-1} \]

   4. metadata-eval 37.9%

      \[\leadsto x \cdot 0.08333333333333333 + {\left(\frac{1}{\frac{{x}^{-2} + \color{blue}{\left(-0.25\right)}}{\frac{1}{x} + -0.5}}\right)}^{-1} \]

   5. metadata-eval 37.9%

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

   6. sub-neg 37.9%

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

   7. metadata-eval 37.9%

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

   8. pow-sqr 38.1%

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

   9. inv-pow 38.1%

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

   10. inv-pow 38.1%

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

   11. flip-- 70.6%

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

   12. sub-neg 70.6%

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

   13. metadata-eval 70.6%

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

10. Applied egg-rr 70.6%

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

    1. unpow-1 70.6%

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

12. Applied egg-rr 70.6%

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

13. Final simplification 70.6%

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

Alternative 3: 67.0% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.5%

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

   1. expm1-def 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. Taylor expanded in x around 0 70.6%

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

5. Final simplification 70.6%

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

Alternative 4: 67.0% 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 34.5%

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

   1. expm1-def 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. Taylor expanded in x around 0 70.1%

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

   1. +-commutative 70.1%

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

6. Simplified 70.1%

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

7. Final simplification 70.1%

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

Alternative 5: 67.0% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.5%

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

   1. expm1-def 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. Taylor expanded in x around 0 69.9%

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

5. Final simplification 69.9%

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

Alternative 6: 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 34.5%

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

   1. expm1-def 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. Taylor expanded in x around 0 70.1%

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

   1. +-commutative 70.1%

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

6. Simplified 70.1%

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

7. Taylor expanded in x around inf 3.1%

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

8. Final simplification 3.1%

   \[\leadsto 0.5 \]

Developer target: 37.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023222 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

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