Average Error: 41.7 → 0.4

Time: 3.8s

Precision: binary64

Cost: 19776

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

↓

\[\begin{array}{l} t_0 := e^{x \cdot 0.5}\\ t_0 \cdot \frac{t_0}{\mathsf{expm1}\left(x\right)} \end{array} \]

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* x 0.5)))) (* t_0 (/ t_0 (expm1 x)))))

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

↓

double code(double x) {
	double t_0 = exp((x * 0.5));
	return t_0 * (t_0 / expm1(x));
}

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

↓

public static double code(double x) {
	double t_0 = Math.exp((x * 0.5));
	return t_0 * (t_0 / Math.expm1(x));
}

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

↓

def code(x):
	t_0 = math.exp((x * 0.5))
	return t_0 * (t_0 / math.expm1(x))

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

↓

function code(x)
	t_0 = exp(Float64(x * 0.5))
	return Float64(t_0 * Float64(t_0 / expm1(x)))
end

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

↓

code[x_] := Block[{t$95$0 = N[Exp[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(t$95$0 / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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

↓

\begin{array}{l}
t_0 := e^{x \cdot 0.5}\\
t_0 \cdot \frac{t_0}{\mathsf{expm1}\left(x\right)}
\end{array}

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	41.7
Target	41.3
Herbie	0.4

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

Derivation

Initial program 41.7
\[\frac{e^{x}}{e^{x} - 1} \]
Simplified0.4
\[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
Proof
(/.f64 (exp.f64 x) (expm1.f64 x)): 0 points increase in error, 0 points decrease in error
(/.f64 (exp.f64 x) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 x) 1))): 176 points increase in error, 0 points decrease in error
Applied egg-rr0.4
\[\leadsto \color{blue}{\frac{\sqrt{e^{x}}}{1} \cdot \frac{\sqrt{e^{x}}}{\mathsf{expm1}\left(x\right)}} \]
Applied egg-rr0.4
\[\leadsto \frac{\sqrt{e^{x}}}{1} \cdot \frac{\color{blue}{e^{x \cdot 0.5}}}{\mathsf{expm1}\left(x\right)} \]
Applied egg-rr0.4
\[\leadsto \frac{\color{blue}{e^{x \cdot 0.5}}}{1} \cdot \frac{e^{x \cdot 0.5}}{\mathsf{expm1}\left(x\right)} \]
Final simplification0.4
\[\leadsto e^{x \cdot 0.5} \cdot \frac{e^{x \cdot 0.5}}{\mathsf{expm1}\left(x\right)} \]

Alternatives

Alternative 1
Error	0.4
Cost	12992

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

Alternative 2
Error	0.9
Cost	7104

\[e^{x \cdot 0.5} \cdot \left(x \cdot -0.041666666666666664 + \frac{1}{x}\right) \]

Alternative 3
Error	1.1
Cost	6848

\[e^{x \cdot 0.5} \cdot \frac{1}{x} \]

Alternative 4
Error	20.8
Cost	576

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

Alternative 5
Error	20.9
Cost	192

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

Reproduce

herbie shell --seed 2022306 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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