expq2 (section 3.11)

Percentage Accurate: 28.1% → 79.5%

Time: 3.8s

Precision: binary64

Cost: 26180

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} + -1} \leq 10000:\\ \;\;\;\;\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= (/ (exp x) (+ (exp x) -1.0)) 10000.0)
   (/ (exp x) (expm1 x))
   (+ (/ 1.0 x) 0.5)))

C

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

↓

double code(double x) {
	double tmp;
	if ((exp(x) / (exp(x) + -1.0)) <= 10000.0) {
		tmp = exp(x) / expm1(x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

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 ((Math.exp(x) / (Math.exp(x) + -1.0)) <= 10000.0) {
		tmp = Math.exp(x) / Math.expm1(x);
	} else {
		tmp = (1.0 / x) + 0.5;
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if (math.exp(x) / (math.exp(x) + -1.0)) <= 10000.0:
		tmp = math.exp(x) / math.expm1(x)
	else:
		tmp = (1.0 / x) + 0.5
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (Float64(exp(x) / Float64(exp(x) + -1.0)) <= 10000.0)
		tmp = Float64(exp(x) / expm1(x));
	else
		tmp = Float64(Float64(1.0 / x) + 0.5);
	end
	return tmp
end

Wolfram

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

↓

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 10000.0], N[(N[Exp[x], $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] + 0.5), $MachinePrecision]]

Target

Original	28.1%
Target	53.3%
Herbie	79.5%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 x) (-.f64 (exp.f64 x) 1)) < 1e4

   1. Initial program 95.5%

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

   2. Simplified 100.0%

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

      [Start] 95.5%	\[ \frac{e^{x}}{e^{x} - 1} \]
      expm1-def [=>] 100.0%	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

   if 1e4 < (/.f64 (exp.f64 x) (-.f64 (exp.f64 x) 1))

   1. Initial program 2.9%

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

   2. Simplified 70.3%

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

      [Start] 2.9%	\[ \frac{e^{x}}{e^{x} - 1} \]
      expm1-def [=>] 70.3%	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

   3. Taylor expanded in x around 0 75.8%

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

   4. Simplified 75.8%

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

      [Start] 75.8%	\[ 0.5 + \frac{1}{x} \]
      +-commutative [=>] 75.8%	\[ \color{blue}{\frac{1}{x} + 0.5} \]

3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} + -1} \leq 10000:\\ \;\;\;\;\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.5%
Cost	26180

\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} + -1} \leq 10000:\\ \;\;\;\;\frac{e^{x}}{\mathsf{expm1}\left(x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \]

Alternative 2
Accuracy	54.8%
Cost	320

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

Alternative 3
Accuracy	51.4%
Cost	192

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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