expq2 (section 3.11)

Average Accuracy: 35.1% → 98.9%

Time: 5.2s

Precision: binary64

Cost: 6916

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00155:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x \cdot \left(1 + -0.08333333333333333 \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.00155)
   (/ 1.0 (- 1.0 (exp (- x))))
   (+ 0.5 (/ 1.0 (* x (+ 1.0 (* -0.08333333333333333 (* x x))))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.00155) {
		tmp = 1.0 / (1.0 - exp(-x));
	} else {
		tmp = 0.5 + (1.0 / (x * (1.0 + (-0.08333333333333333 * (x * x)))));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.00155d0)) then
        tmp = 1.0d0 / (1.0d0 - exp(-x))
    else
        tmp = 0.5d0 + (1.0d0 / (x * (1.0d0 + ((-0.08333333333333333d0) * (x * x)))))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -0.00155) {
		tmp = 1.0 / (1.0 - Math.exp(-x));
	} else {
		tmp = 0.5 + (1.0 / (x * (1.0 + (-0.08333333333333333 * (x * x)))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.00155:
		tmp = 1.0 / (1.0 - math.exp(-x))
	else:
		tmp = 0.5 + (1.0 / (x * (1.0 + (-0.08333333333333333 * (x * x)))))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -0.00155)
		tmp = Float64(1.0 / Float64(1.0 - exp(Float64(-x))));
	else
		tmp = Float64(0.5 + Float64(1.0 / Float64(x * Float64(1.0 + Float64(-0.08333333333333333 * Float64(x * x))))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.00155)
		tmp = 1.0 / (1.0 - exp(-x));
	else
		tmp = 0.5 + (1.0 / (x * (1.0 + (-0.08333333333333333 * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_] := If[LessEqual[x, -0.00155], N[(1.0 / N[(1.0 - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 + N[(1.0 / N[(x * N[(1.0 + N[(-0.08333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	35.1%
Target	35.8%
Herbie	98.9%

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

Derivation

1. Split input into 2 regimes

2. if x < -0.00154999999999999995

   1. Initial program 99.9%

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

   2. Simplified 100.0%

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

      [Start] 99.9	\[ \frac{e^{x}}{e^{x} - 1} \]
      expm1-def [=>] 100.0	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

   3. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]
      clear-num [=>] 100.0	\[ \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}} \]
      inv-pow [=>] 100.0	\[ \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1}} \]

   4. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ {\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1} \]
      unpow-1 [=>] 100.0	\[ \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}} \]

   5. Taylor expanded in x around inf 99.9%

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

   6. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{1}{\frac{e^{x} - 1}{e^{x}}} \]
      div-sub [=>] 1.8	\[ \frac{1}{\color{blue}{\frac{e^{x}}{e^{x}} - \frac{1}{e^{x}}}} \]
      *-inverses [=>] 99.9	\[ \frac{1}{\color{blue}{1} - \frac{1}{e^{x}}} \]
      exp-neg [<=] 99.9	\[ \frac{1}{1 - \color{blue}{e^{-x}}} \]

   if -0.00154999999999999995 < x

   1. Initial program 3.4%

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

   2. Simplified 98.9%

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

      [Start] 3.4	\[ \frac{e^{x}}{e^{x} - 1} \]
      expm1-def [=>] 98.9	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

   3. Taylor expanded in x around 0 98.4%

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

   4. Applied egg-rr 48.6%

      \[\leadsto 0.5 + \color{blue}{\frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{0.006944444444444444 \cdot \left(x \cdot x\right) - {x}^{-2}}}} \]
      Proof

      [Start] 98.4	\[ 0.5 + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right) \]
      flip-+ [=>] 48.7	\[ 0.5 + \color{blue}{\frac{\left(0.08333333333333333 \cdot x\right) \cdot \left(0.08333333333333333 \cdot x\right) - \frac{1}{x} \cdot \frac{1}{x}}{0.08333333333333333 \cdot x - \frac{1}{x}}} \]
      clear-num [=>] 48.7	\[ 0.5 + \color{blue}{\frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{\left(0.08333333333333333 \cdot x\right) \cdot \left(0.08333333333333333 \cdot x\right) - \frac{1}{x} \cdot \frac{1}{x}}}} \]
      swap-sqr [=>] 48.7	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{\color{blue}{\left(0.08333333333333333 \cdot 0.08333333333333333\right) \cdot \left(x \cdot x\right)} - \frac{1}{x} \cdot \frac{1}{x}}} \]
      metadata-eval [=>] 48.7	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{\color{blue}{0.006944444444444444} \cdot \left(x \cdot x\right) - \frac{1}{x} \cdot \frac{1}{x}}} \]
      inv-pow [=>] 48.7	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{0.006944444444444444 \cdot \left(x \cdot x\right) - \color{blue}{{x}^{-1}} \cdot \frac{1}{x}}} \]
      metadata-eval [<=] 48.7	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{0.006944444444444444 \cdot \left(x \cdot x\right) - {x}^{\color{blue}{\left(-1\right)}} \cdot \frac{1}{x}}} \]
      inv-pow [=>] 48.7	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{0.006944444444444444 \cdot \left(x \cdot x\right) - {x}^{\left(-1\right)} \cdot \color{blue}{{x}^{-1}}}} \]
      metadata-eval [<=] 48.7	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{0.006944444444444444 \cdot \left(x \cdot x\right) - {x}^{\left(-1\right)} \cdot {x}^{\color{blue}{\left(-1\right)}}}} \]
      pow-sqr [=>] 48.6	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{0.006944444444444444 \cdot \left(x \cdot x\right) - \color{blue}{{x}^{\left(2 \cdot \left(-1\right)\right)}}}} \]
      metadata-eval [=>] 48.6	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{0.006944444444444444 \cdot \left(x \cdot x\right) - {x}^{\left(2 \cdot \color{blue}{-1}\right)}}} \]
      metadata-eval [=>] 48.6	\[ 0.5 + \frac{1}{\frac{0.08333333333333333 \cdot x - \frac{1}{x}}{0.006944444444444444 \cdot \left(x \cdot x\right) - {x}^{\color{blue}{-2}}}} \]

   5. Taylor expanded in x around 0 98.4%

      \[\leadsto 0.5 + \frac{1}{\color{blue}{-0.08333333333333333 \cdot {x}^{3} + x}} \]

   6. Applied egg-rr 98.4%

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

      [Start] 98.4	\[ 0.5 + \frac{1}{-0.08333333333333333 \cdot {x}^{3} + x} \]
      unpow3 [=>] 98.4	\[ 0.5 + \frac{1}{-0.08333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + x} \]
      associate-*r* [=>] 98.4	\[ 0.5 + \frac{1}{\color{blue}{\left(-0.08333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x} + x} \]
      distribute-lft1-in [=>] 98.4	\[ 0.5 + \frac{1}{\color{blue}{\left(-0.08333333333333333 \cdot \left(x \cdot x\right) + 1\right) \cdot x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00155:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x \cdot \left(1 + -0.08333333333333333 \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	13120

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

Alternative 2
Accuracy	99.2%
Cost	12992

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

Alternative 3
Accuracy	67.3%
Cost	832

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

Alternative 4
Accuracy	66.9%
Cost	192

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

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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