Average Error: 29.7 → 0.3
Time: 8.0s
Precision: binary64
Cost: 26753
\[e^{a \cdot x} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.0010265468494552537:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}\right)\\ \end{array}\]
e^{a \cdot x} - 1
↓
\begin{array}{l}
\mathbf{if}\;a \cdot x \leq -0.0010265468494552537:\\
\;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\

\mathbf{else}:\\
\;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}\right)\\

\end{array}
(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))
↓
(FPCore (a x)
 :precision binary64
 (if (<= (* a x) -0.0010265468494552537)
   (* (+ 1.0 (sqrt (exp (* a x)))) (+ (sqrt (exp (* a x))) -1.0))
   (*
    (+ 1.0 (sqrt (exp (* a x))))
    (+
     (* (* a x) (+ 0.5 (* (* a x) 0.125)))
     (* 0.020833333333333332 (pow (* a x) 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) <= -0.0010265468494552537) {
		tmp = (1.0 + sqrt(exp(a * x))) * (sqrt(exp(a * x)) + -1.0);
	} else {
		tmp = (1.0 + sqrt(exp(a * x))) * (((a * x) * (0.5 + ((a * x) * 0.125))) + (0.020833333333333332 * pow((a * x), 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

Original29.7
Target0.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| < 0.1:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Alternatives

Alternative 1
Error29.8
Cost71872
\[\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}} + -1} \cdot \left(\sqrt[3]{\sqrt{e^{a \cdot x}} + -1} \cdot \sqrt[3]{\sqrt{e^{a \cdot x}} + -1}\right)\right)\]
Alternative 2
Error29.8
Cost52288
\[\left(\sqrt{e^{a \cdot x}} + -1\right) \cdot \left(\sqrt{1 + \sqrt{e^{a \cdot x}}} \cdot \sqrt{1 + \sqrt{e^{a \cdot x}}}\right)\]
Alternative 3
Error38.0
Cost41152
\[\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}} \cdot \sqrt{\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}}\right)\]
Alternative 4
Error29.7
Cost39488
\[\sqrt[3]{e^{a \cdot x} + -1} \cdot \left(\sqrt[3]{e^{a \cdot x} + -1} \cdot \sqrt[3]{e^{a \cdot x} + -1}\right)\]
Alternative 5
Error33.5
Cost34048
\[\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + \left(0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3} + 0.0026041666666666665 \cdot \left({a}^{4} \cdot {x}^{4}\right)\right)\right)\]
Alternative 6
Error29.8
Cost26752
\[\frac{-1 + {\left(e^{a \cdot x}\right)}^{3}}{1 + e^{a \cdot x} \cdot \left(1 + e^{a \cdot x}\right)}\]
Alternative 7
Error29.8
Cost26304
\[\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\]
Alternative 8
Error51.1
Cost20864
\[\left(\sqrt{e^{a \cdot x}} + -1\right) \cdot \left(2 + \left(\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}\right)\right)\]
Alternative 9
Error21.2
Cost20736
\[\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}\right)\]
Alternative 10
Error29.8
Cost19520
\[\log \left(e^{e^{a \cdot x} + -1}\right)\]
Alternative 11
Error21.6
Cost13888
\[\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right)\right)\]
Alternative 12
Error36.2
Cost13056
\[-1 + {\left(e^{a}\right)}^{x}\]
Alternative 13
Error29.7
Cost6720
\[e^{a \cdot x} + -1\]
Alternative 14
Error21.3
Cost192
\[a \cdot x\]
Alternative 15
Error62.0
Cost64
\[1\]
Alternative 16
Error51.7
Cost64
\[0\]
Alternative 17
Error41.3
Cost64
\[-1\]

Error

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 a x) < -0.0010265468494552537

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt_binary64_27570.0

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

    4. Applied difference-of-sqr-1_binary64_27050.0

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

    5. Simplified0.0

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

    6. Simplified0.0

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

    7. Simplified0.0

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

    if -0.0010265468494552537 < (*.f64 a x)

    1. Initial program 44.7

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

    3. Applied add-sqr-sqrt_binary64_275744.7

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

    4. Applied difference-of-sqr-1_binary64_270544.7

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

    5. Simplified44.7

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

    6. Simplified44.7

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

    7. Taylor expanded around 0 14.6

      \[\leadsto \left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \color{blue}{\left(0.5 \cdot \left(a \cdot x\right) + \left(0.020833333333333332 \cdot \left({a}^{3} \cdot {x}^{3}\right) + 0.125 \cdot \left({a}^{2} \cdot {x}^{2}\right)\right)\right)}\]

    8. Simplified0.4

      \[\leadsto \left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \color{blue}{\left(\left(a \cdot x\right) \cdot \left(0.5 + 0.125 \cdot \left(a \cdot x\right)\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}\right)}\]

    9. Simplified0.4

      \[\leadsto \color{blue}{\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \leq -0.0010265468494552537:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\sqrt{e^{a \cdot x}} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(0.5 + \left(a \cdot x\right) \cdot 0.125\right) + 0.020833333333333332 \cdot {\left(a \cdot x\right)}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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))