Average Error: 29.8 → 0.8
Time: 3.0s
Precision: binary64
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.434635393425338 \cdot 10^{-08}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{e^{a \cdot x}} + 1} \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}} + 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} + 1}\right)\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -1.434635393425338 \cdot 10^{-08}:\\
\;\;\;\;\left(\sqrt[3]{\sqrt{e^{a \cdot x}} + 1} \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}} + 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} + 1}\right)\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -1.434635393425338e-08)
   (*
    (*
     (cbrt (+ (sqrt (exp (* a x))) 1.0))
     (*
      (cbrt (+ (sqrt (exp (* a x))) 1.0))
      (cbrt (+ (sqrt (exp (* a x))) 1.0))))
    (+ (sqrt (exp (* a x))) -1.0))
   (* (+ (sqrt (exp (* a x))) 1.0) (* (* a x) 0.5))))
double code(double a, double x) {
	return exp(a * x) - 1.0;
}
↓
double code(double a, double x) {
	double tmp;
	if ((a * x) <= -1.434635393425338e-08) {
		tmp = (cbrt(sqrt(exp(a * x)) + 1.0) * (cbrt(sqrt(exp(a * x)) + 1.0) * cbrt(sqrt(exp(a * x)) + 1.0))) * (sqrt(exp(a * x)) + -1.0);
	} else {
		tmp = (sqrt(exp(a * x)) + 1.0) * ((a * x) * 0.5);
	}
	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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -1.4346353934253381e-8

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt_binary640.3

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

    4. Applied difference-of-sqr-1_binary640.3

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

    5. Simplified0.3

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

    6. Simplified0.3

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

    8. Applied add-cube-cbrt_binary640.3

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

    9. Simplified0.3

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

    10. Simplified0.3

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

    if -1.4346353934253381e-8 < (*.f64 a x)

    1. Initial program 44.8

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

    3. Applied add-sqr-sqrt_binary6444.9

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

    4. Applied difference-of-sqr-1_binary6444.9

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

    5. Simplified44.9

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

    6. Simplified44.9

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

    7. Taylor expanded around 0 1.1

      \[\leadsto \left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -1.434635393425338 \cdot 10^{-08}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{e^{a \cdot x}} + 1} \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}} + 1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} + 1}\right)\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{e^{a \cdot x}} + 1\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(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))