expax (section 3.5)

Average Error: 29.6 → 0.0

Time: 5.6s

Precision: binary64

Cost: 6592

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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));
}

Python

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

↓

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

Julia

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

↓

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

Wolfram

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]

Target

Original	29.6
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

1. Initial program 29.6

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]
   Proof

   [Start] 29.6	\[ e^{a \cdot x} - 1 \]
   expm1-def [=>] 0.0	\[ \color{blue}{\mathsf{expm1}\left(a \cdot x\right)} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	21.5
Cost	192

\[a \cdot x \]

Reproduce

herbie shell --seed 2023053 
(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))