expax (section 3.5)

Average Error: 29.3 → 14.1

Time: 54.4s

Precision: binary64

Cost: 52810

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \leq -8.209112939621517 \cdot 10^{+154}:\\ \;\;\;\;-1 + \log \left(\sqrt[3]{e^{e^{a \cdot x}}} \cdot {\left(\sqrt[3]{e^{e^{a \cdot x}}}\right)}^{2}\right)\\ \mathbf{elif}\;a \leq 8.001029692762442 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;a \leq 1.7983423256931096 \cdot 10^{+63} \lor \neg \left(a \leq 2.8512023937374467 \cdot 10^{+91}\right):\\ \;\;\;\;-1 + \log \left(\sqrt[3]{e^{e^{a \cdot x}}} \cdot {\left(\sqrt[3]{e^{e^{a \cdot x}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

FPCore

(FPCore (a x) :precision binary64 (- (exp (* a x)) 1.0))

↓

(FPCore (a x)
 :precision binary64
 (if (<= a -8.209112939621517e+154)
   (+
    -1.0
    (log (* (cbrt (exp (exp (* a x)))) (pow (cbrt (exp (exp (* a x)))) 2.0))))
   (if (<= a 8.001029692762442e+30)
     (* (+ 1.0 (sqrt (exp (* a x)))) (* (* a x) 0.5))
     (if (or (<= a 1.7983423256931096e+63) (not (<= a 2.8512023937374467e+91)))
       (+
        -1.0
        (log
         (* (cbrt (exp (exp (* a x)))) (pow (cbrt (exp (exp (* a x)))) 2.0))))
       (* a x)))))

C

double code(double a, double x) {
	return exp(a * x) - 1.0;
}

↓

double code(double a, double x) {
	double tmp;
	if (a <= -8.209112939621517e+154) {
		tmp = -1.0 + log(cbrt(exp(exp(a * x))) * pow(cbrt(exp(exp(a * x))), 2.0));
	} else if (a <= 8.001029692762442e+30) {
		tmp = (1.0 + sqrt(exp(a * x))) * ((a * x) * 0.5);
	} else if ((a <= 1.7983423256931096e+63) || !(a <= 2.8512023937374467e+91)) {
		tmp = -1.0 + log(cbrt(exp(exp(a * x))) * pow(cbrt(exp(exp(a * x))), 2.0));
	} else {
		tmp = a * x;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.3
Target	0.2
Herbie	14.1

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

Alternatives

Alternative 1
Error	14.0
Cost	20289

\[\begin{array}{l} \mathbf{if}\;a \leq -5.004489129148046 \cdot 10^{+155}:\\ \;\;\;\;\frac{-1 + {\left(e^{a \cdot x}\right)}^{2}}{e^{a \cdot x} + 1}\\ \mathbf{elif}\;a \leq 9.746292023465319 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;a \leq 1.993024066462074 \cdot 10^{+63} \lor \neg \left(a \leq 5.3949203852007904 \cdot 10^{+91}\right):\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Alternative 2
Error	14.0
Cost	19905

\[\begin{array}{l} \mathbf{if}\;a \leq -1.4405171466963989 \cdot 10^{+156}:\\ \;\;\;\;\sqrt[3]{{\left(-1 + e^{a \cdot x}\right)}^{3}}\\ \mathbf{elif}\;a \leq 9.746292023465319 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;a \leq 1.993024066462074 \cdot 10^{+63} \lor \neg \left(a \leq 5.764384431142674 \cdot 10^{+92}\right):\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Alternative 3
Error	14.0
Cost	14146

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4228326418509253 \cdot 10^{+155}:\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{elif}\;a \leq 9.746292023465319 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;a \leq 2.7717510295379315 \cdot 10^{+63} \lor \neg \left(a \leq 3.015731236062579 \cdot 10^{+92}\right):\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Alternative 4
Error	14.0
Cost	7874

\[\begin{array}{l} \mathbf{if}\;a \leq -2.6082210744770845 \cdot 10^{+155}:\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{elif}\;a \leq 9.746292023465319 \cdot 10^{+30}:\\ \;\;\;\;\left(\left(a \cdot x\right) \cdot 0.5\right) \cdot \left(1 + e^{\left(a \cdot x\right) \cdot 0.5}\right)\\ \mathbf{elif}\;a \leq 2.966432770306896 \cdot 10^{+63} \lor \neg \left(a \leq 1.0544818775008219 \cdot 10^{+93}\right):\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Alternative 5
Error	14.1
Cost	7313

\[\begin{array}{l} \mathbf{if}\;a \leq -6.302208157531161 \cdot 10^{+155} \lor \neg \left(a \leq 9.528134232127459 \cdot 10^{+30} \lor \neg \left(a \leq 4.5886738336610246 \cdot 10^{+63}\right) \land a \leq 2.044028689187582 \cdot 10^{+93}\right):\\ \;\;\;\;-1 + e^{a \cdot x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Alternative 6
Error	14.6
Cost	973

\[\begin{array}{l} \mathbf{if}\;a \leq -1.0394536637398784 \cdot 10^{+168}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 9.746292023465319 \cdot 10^{+30} \lor \neg \left(a \leq 1.895683196077592 \cdot 10^{+63}\right) \land a \leq 2.0781189471071655 \cdot 10^{+91}:\\ \;\;\;\;a \cdot x\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 7
Error	34.1
Cost	706

\[\begin{array}{l} \mathbf{if}\;a \leq -1.312928803269667 \cdot 10^{-100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;a \leq 2.9859537436308303 \cdot 10^{-90}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 8
Error	51.7
Cost	64

\[0\]

Derivation

1. Split input into 3 regimes

2. if a < -8.2091129396215167e154 or 8.0010296927624417e30 < a < 1.79834232569310959e63 or 2.8512023937374467e91 < a

   1. Initial program 15.6

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

   3. Applied add-log-exp_binary64_1822 15.9

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

   5. Applied add-log-exp_binary64_1822 15.9

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

   7. Applied add-cube-cbrt_binary64_1818 15.9

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

   8. Applied log-prod_binary64_1869 15.9

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

   9. Applied exp-sum_binary64_1829 15.9

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

   10. Simplified 15.9

       \[\leadsto \log \left(\color{blue}{{\left(\sqrt[3]{e^{e^{a \cdot x}}}\right)}^{2}} \cdot e^{\log \left(\sqrt[3]{e^{e^{a \cdot x}}}\right)}\right) - 1\]

   11. Simplified 15.9

       \[\leadsto \log \left({\left(\sqrt[3]{e^{e^{a \cdot x}}}\right)}^{2} \cdot \color{blue}{\sqrt[3]{e^{e^{a \cdot x}}}}\right) - 1\]

   12. Simplified 15.9

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

   if -8.2091129396215167e154 < a < 8.0010296927624417e30

   1. Initial program 34.1

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

   3. Applied add-sqr-sqrt_binary64_1805 34.2

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

   4. Applied difference-of-sqr-1_binary64_1753 34.2

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

   5. Simplified 34.2

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

   6. Simplified 34.2

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

   7. Taylor expanded around 0 13.0

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

   8. Simplified 13.0

      \[\leadsto \left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \color{blue}{\left(\left(a \cdot x\right) \cdot 0.5\right)}\]
   9. Using strategy rm

   10. Applied pow1_binary64_1844 13.0

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

   11. Simplified 13.0

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

   if 1.79834232569310959e63 < a < 2.8512023937374467e91

   1. Initial program 28.3

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

   2. Taylor expanded around 0 59.6

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

   4. Applied associate--l+_binary64_1720 32.2

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

   5. Simplified 32.2

      \[\leadsto a \cdot x + \color{blue}{0}\]

   6. Simplified 32.2

      \[\leadsto \color{blue}{x \cdot a}\]
3. Recombined 3 regimes into one program.

4. Final simplification 14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.209112939621517 \cdot 10^{+154}:\\ \;\;\;\;-1 + \log \left(\sqrt[3]{e^{e^{a \cdot x}}} \cdot {\left(\sqrt[3]{e^{e^{a \cdot x}}}\right)}^{2}\right)\\ \mathbf{elif}\;a \leq 8.001029692762442 \cdot 10^{+30}:\\ \;\;\;\;\left(1 + \sqrt{e^{a \cdot x}}\right) \cdot \left(\left(a \cdot x\right) \cdot 0.5\right)\\ \mathbf{elif}\;a \leq 1.7983423256931096 \cdot 10^{+63} \lor \neg \left(a \leq 2.8512023937374467 \cdot 10^{+91}\right):\\ \;\;\;\;-1 + \log \left(\sqrt[3]{e^{e^{a \cdot x}}} \cdot {\left(\sqrt[3]{e^{e^{a \cdot x}}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2021065 
(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))