expq2 (section 3.11)

Percentage Accurate: 38.2% → 100.0%

Time: 3.3s

Alternatives: 3

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: 38.2% 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 40.1%

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

5. Taylor expanded in x around inf 40.1%

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

   1. expm1-def 100.0%

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

   2. *-lft-identity 100.0%

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

   3. metadata-eval 100.0%

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

   4. times-frac 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/l* 100.0%

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

   7. neg-sub0 100.0%

      \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{expm1}\left(x\right)}}{e^{x}}} \]

   8. expm1-def 39.8%

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

   9. associate--r- 39.8%

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

   10. neg-sub0 39.8%

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

   11. +-commutative 39.8%

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

   12. sub-neg 39.8%

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

   13. div-sub 2.3%

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

   14. exp-neg 2.3%

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

   15. *-inverses 39.8%

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

   16. expm1-def 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 66.3% 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 40.1%

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

5. Taylor expanded in x around 0 64.2%

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

6. Final simplification 64.2%

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

Alternative 3: 3.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 40.1%

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

5. Taylor expanded in x around inf 40.1%

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

   1. expm1-def 100.0%

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

   2. *-lft-identity 100.0%

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

   3. metadata-eval 100.0%

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

   4. times-frac 100.0%

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

   5. neg-mul-1 100.0%

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

   6. associate-/l* 100.0%

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

   7. neg-sub0 100.0%

      \[\leadsto \frac{-1}{\frac{\color{blue}{0 - \mathsf{expm1}\left(x\right)}}{e^{x}}} \]

   8. expm1-def 39.8%

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

   9. associate--r- 39.8%

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

   10. neg-sub0 39.8%

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

   11. +-commutative 39.8%

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

   12. sub-neg 39.8%

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

   13. div-sub 2.3%

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

   14. exp-neg 2.3%

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

   15. *-inverses 39.8%

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

   16. expm1-def 100.0%

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

7. Simplified 100.0%

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

9. Applied egg-rr 0.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{expm1}\left(x\right)} \cdot {\left(-\sqrt{\mathsf{expm1}\left(x\right)}\right)}^{-1}} \]
10. Step-by-step derivation

    1. *-commutative 0.4%

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

    2. unpow-1 0.4%

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

    3. associate-*l/ 0.4%

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

    4. neg-mul-1 0.4%

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

    5. times-frac 0.4%

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

    6. metadata-eval 0.4%

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

    7. *-inverses 3.1%

       \[\leadsto -1 \cdot \color{blue}{1} \]

    8. metadata-eval 3.1%

       \[\leadsto \color{blue}{-1} \]

11. Simplified 3.1%

    \[\leadsto \color{blue}{-1} \]

12. Final simplification 3.1%

    \[\leadsto -1 \]
13. 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 2024027 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64
  :pre (> 710.0 x)

  :herbie-target
  (/ (- 1.0) (expm1 (- x)))

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