expq2 (section 3.11)

Average Error: 41.4 → 0.4

Time: 4.5s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (* (/ 1.0 (expm1 x)) (exp x)))

C

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

function code(x)
	return Float64(Float64(1.0 / expm1(x)) * exp(x))
end

Wolfram

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

↓

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

Target

Original	41.4
Target	41.0
Herbie	0.4

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

Derivation

1. Initial program 41.4

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

2. Simplified 0.4

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

   [Start] 41.4	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 0.4	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 0.4

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

4. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	12992

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

Alternative 2
Error	1.6
Cost	6592

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

Alternative 3
Error	21.0
Cost	832

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

Alternative 4
Error	21.2
Cost	320

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

Alternative 5
Error	21.3
Cost	192

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

Reproduce

herbie shell --seed 2023060 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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