Average Error: 29.0 → 3.4
Time: 3.0s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.2474497912437426 \cdot 10^{-13}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{2} + -1}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \cdot x \leq 8.874575160662803 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(a + \log \left({\left(e^{x}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right)}^{3}}\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -1.2474497912437426 \cdot 10^{-13}:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{2} + -1}{e^{a \cdot x} + 1}\\

\mathbf{elif}\;a \cdot x \leq 8.874575160662803 \cdot 10^{-107}:\\
\;\;\;\;x \cdot \left(a + \log \left({\left(e^{x}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right)}^{3}}\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1.2474497912437426e-13)
   (/ (+ (pow (exp (* a x)) 2.0) -1.0) (+ (exp (* a x)) 1.0))
   (if (<= (* a x) 8.874575160662803e-107)
     (*
      x
      (+
       a
       (log
        (pow (exp x) (* (* a a) (+ 0.5 (* a (* x 0.16666666666666666))))))))
     (cbrt
      (pow
       (* x (+ a (* x (* (* a a) (+ 0.5 (* a (* x 0.16666666666666666)))))))
       3.0)))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1.2474497912437426e-13) {
		tmp = (pow(exp(a * x), 2.0) + -1.0) / (exp(a * x) + 1.0);
	} else if ((a * x) <= 8.874575160662803e-107) {
		tmp = x * (a + log(pow(exp(x), ((a * a) * (0.5 + (a * (x * 0.16666666666666666)))))));
	} else {
		tmp = cbrt(pow((x * (a + (x * ((a * a) * (0.5 + (a * (x * 0.16666666666666666))))))), 3.0));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.0
Target0.2
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 a x) < -1.2474497912437426e-13

    1. Initial program 0.8

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

    3. Applied flip--_binary640.8

      \[\leadsto \color{blue}{\frac{e^{a \cdot x} \cdot e^{a \cdot x} - 1 \cdot 1}{e^{a \cdot x} + 1}}\]

    4. Simplified0.8

      \[\leadsto \frac{\color{blue}{{\left(e^{a \cdot x}\right)}^{2} + -1}}{e^{a \cdot x} + 1}\]

    if -1.2474497912437426e-13 < (*.f64 a x) < 8.8745751606628034e-107

    1. Initial program 41.8

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

    2. Taylor expanded around 0 9.0

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666666 \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    3. Simplified4.8

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp_binary645.7

      \[\leadsto x \cdot \left(a + \color{blue}{\log \left(e^{x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)}\right)}\right)\]

    6. Simplified2.6

      \[\leadsto x \cdot \left(a + \log \color{blue}{\left({\left(e^{x}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right)}\right)\]

    if 8.8745751606628034e-107 < (*.f64 a x)

    1. Initial program 53.8

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

    2. Taylor expanded around 0 35.8

      \[\leadsto \color{blue}{0.5 \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(0.16666666666666666 \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

    3. Simplified20.9

      \[\leadsto \color{blue}{x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube_binary6421.5

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\right) \cdot \left(x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right) + x \cdot \left(0.16666666666666666 \cdot {a}^{3}\right)\right)\right)\right)}}\]

    6. Simplified15.4

      \[\leadsto \sqrt[3]{\color{blue}{{\left(x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)\right)\right)}^{3}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.2474497912437426 \cdot 10^{-13}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{2} + -1}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \cdot x \leq 8.874575160662803 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \left(a + \log \left({\left(e^{x}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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))