Average Error: 29.8 → 2.9
Time: 3.0s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -9.859809998267145 \cdot 10^{-08}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{1 + e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{elif}\;a \cdot x \leq -2.206613079296664 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 2.3830218440335053 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \left(a + \log \left({\left(e^{x}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -9.859809998267145 \cdot 10^{-08}:\\
\;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{1 + e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right)}\\

\mathbf{elif}\;a \cdot x \leq -2.206613079296664 \cdot 10^{-73}:\\
\;\;\;\;x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\\

\mathbf{elif}\;a \cdot x \leq 2.3830218440335053 \cdot 10^{-64}:\\
\;\;\;\;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}:\\
\;\;\;\;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)\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -9.859809998267145e-08)
   (/
    (- (pow (exp (* a x)) 3.0) 1.0)
    (+ 1.0 (* (exp (* a x)) (+ (exp (* a x)) 1.0))))
   (if (<= (* a x) -2.206613079296664e-73)
     (* x (+ a (* x (* 0.5 (* a a)))))
     (if (<= (* a x) 2.3830218440335053e-64)
       (*
        x
        (+
         a
         (log
          (pow (exp x) (* (* a a) (+ 0.5 (* a (* x 0.16666666666666666))))))))
       (*
        x
        (+ a (* x (* (* a a) (+ 0.5 (* a (* x 0.16666666666666666)))))))))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -9.859809998267145e-08) {
		tmp = (pow(exp(a * x), 3.0) - 1.0) / (1.0 + (exp(a * x) * (exp(a * x) + 1.0)));
	} else if ((a * x) <= -2.206613079296664e-73) {
		tmp = x * (a + (x * (0.5 * (a * a))));
	} else if ((a * x) <= 2.3830218440335053e-64) {
		tmp = x * (a + log(pow(exp(x), ((a * a) * (0.5 + (a * (x * 0.16666666666666666)))))));
	} else {
		tmp = x * (a + (x * ((a * a) * (0.5 + (a * (x * 0.16666666666666666))))));
	}
	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

Original29.8
Target0.2
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if (*.f64 a x) < -9.85980999826714522e-8

    1. Initial program 0.2

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

    3. Applied flip3--_binary64_14460.2

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

    4. Simplified0.2

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

    5. Simplified0.2

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

    if -9.85980999826714522e-8 < (*.f64 a x) < -2.20661307929666384e-73

    1. Initial program 57.9

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

    2. Taylor expanded around 0 33.2

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

    3. Simplified17.6

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

    4. Taylor expanded around 0 21.8

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

    5. Simplified11.7

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

    if -2.20661307929666384e-73 < (*.f64 a x) < 2.38302184403350533e-64

    1. Initial program 42.2

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

    2. Taylor expanded around 0 7.8

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

    3. Simplified3.9

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

    5. Applied add-log-exp_binary64_14814.1

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

    6. Simplified1.8

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

    if 2.38302184403350533e-64 < (*.f64 a x)

    1. Initial program 48.4

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

    2. Taylor expanded around 0 40.6

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

    3. Simplified23.0

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

    4. Taylor expanded around 0 23.0

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

    5. Simplified16.4

      \[\leadsto x \cdot \left(a + x \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(0.5 + \left(x \cdot 0.16666666666666666\right) \cdot a\right)\right)}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -9.859809998267145 \cdot 10^{-08}:\\ \;\;\;\;\frac{{\left(e^{a \cdot x}\right)}^{3} - 1}{1 + e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right)}\\ \mathbf{elif}\;a \cdot x \leq -2.206613079296664 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;a \cdot x \leq 2.3830218440335053 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \left(a + \log \left({\left(e^{x}\right)}^{\left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(a + x \cdot \left(\left(a \cdot a\right) \cdot \left(0.5 + a \cdot \left(x \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(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))