Average Error: 29.3 → 0.0

Time: 7.3s

Precision: binary64

Cost: 6592

\[e^{a \cdot x} - 1 \]

↓

\[\mathsf{expm1}\left(a \cdot x\right) \]

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

↓

(FPCore (a x) :precision binary64 (expm1 (* a x)))

double code(double a, double x) {
	return exp((a * x)) - 1.0;
}

↓

double code(double a, double x) {
	return expm1((a * x));
}

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

↓

public static double code(double a, double x) {
	return Math.expm1((a * x));
}

def code(a, x):
	return math.exp((a * x)) - 1.0

↓

def code(a, x):
	return math.expm1((a * x))

function code(a, x)
	return Float64(exp(Float64(a * x)) - 1.0)
end

↓

function code(a, x)
	return expm1(Float64(a * x))
end

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

↓

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

e^{a \cdot x} - 1

↓

\mathsf{expm1}\left(a \cdot x\right)

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	29.3
Target	0.2
Herbie	0.0

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array} \]

Derivation

Initial program 29.3
\[e^{a \cdot x} - 1 \]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
Proof
(expm1.f64 (*.f64 a x)): 0 points increase in error, 0 points decrease in error
(Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (*.f64 a x)) 1)): 127 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto \mathsf{expm1}\left(a \cdot x\right) \]

Alternatives

Alternative 1
Error	18.1
Cost	964

\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -5:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + 0.5 \cdot \left(a \cdot \left(a \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	18.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -5:\\ \;\;\;\;-2 + \frac{-4}{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \]

Alternative 3
Error	18.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -5:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array} \]

Alternative 4
Error	58.4
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1.0 (+ (/ (* a x) 2.0) (/ (pow (* a x) 2.0) 6.0)))) (- (exp (* a x)) 1.0))

  (- (exp (* a x)) 1.0))