Average Error: 29.9 → 29.9
Time: 5.7s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\frac{-1 + {\left(e^{a \cdot x}\right)}^{3}}{1 + \left(\sqrt[3]{e^{a \cdot x}} \cdot \left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right)\right) \cdot \left(e^{a \cdot x} + 1\right)}\]
e^{a \cdot x} - 1
↓
\frac{-1 + {\left(e^{a \cdot x}\right)}^{3}}{1 + \left(\sqrt[3]{e^{a \cdot x}} \cdot \left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right)\right) \cdot \left(e^{a \cdot x} + 1\right)}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (/
  (+ -1.0 (pow (exp (* a x)) 3.0))
  (+
   1.0
   (*
    (* (cbrt (exp (* a x))) (* (cbrt (exp (* a x))) (cbrt (exp (* a x)))))
    (+ (exp (* a x)) 1.0)))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	return (-1.0 + pow(exp(a * x), 3.0)) / (1.0 + ((cbrt(exp(a * x)) * (cbrt(exp(a * x)) * cbrt(exp(a * x)))) * (exp(a * x) + 1.0)));
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.9
Target0.2
Herbie29.9
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Initial program 29.9

    \[e^{a \cdot x} - 1\]
  2. Using strategy rm

  3. Applied flip3--_binary6429.9

    \[\leadsto \color{blue}{\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\]

  4. Simplified29.9

    \[\leadsto \frac{\color{blue}{-1 + {\left(e^{a \cdot x}\right)}^{3}}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]

  5. Simplified29.9

    \[\leadsto \frac{-1 + {\left(e^{a \cdot x}\right)}^{3}}{\color{blue}{1 + e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right)}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt_binary6429.9

    \[\leadsto \frac{-1 + {\left(e^{a \cdot x}\right)}^{3}}{1 + \color{blue}{\left(\left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right) \cdot \sqrt[3]{e^{a \cdot x}}\right)} \cdot \left(e^{a \cdot x} + 1\right)}\]

  8. Final simplification29.9

    \[\leadsto \frac{-1 + {\left(e^{a \cdot x}\right)}^{3}}{1 + \left(\sqrt[3]{e^{a \cdot x}} \cdot \left(\sqrt[3]{e^{a \cdot x}} \cdot \sqrt[3]{e^{a \cdot x}}\right)\right) \cdot \left(e^{a \cdot x} + 1\right)}\]

Reproduce

herbie shell --seed 2020285 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :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))