expax (section 3.5)

Average Error: 29.0 → 8.7

Time: 4.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -1.97221861134718174 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}} \cdot \sqrt[3]{\log \left(e^{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}\right)}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}} \cdot \frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\\ \mathbf{elif}\;a \cdot x \le 6.8409562196091337 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}} \cdot \sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}} \cdot \frac{\sqrt[3]{\left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}} - {\left(\sqrt{1}\right)}^{3}\right)}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\\ \end{array}\]

C

double f(double a, double x) {
        double r112151 = a;
        double r112152 = x;
        double r112153 = r112151 * r112152;
        double r112154 = exp(r112153);
        double r112155 = 1.0;
        double r112156 = r112154 - r112155;
        return r112156;
}

↓

double f(double a, double x) {
        double r112157 = a;
        double r112158 = x;
        double r112159 = r112157 * r112158;
        double r112160 = -1.9722186113471817e-05;
        bool r112161 = r112159 <= r112160;
        double r112162 = 3.0;
        double r112163 = r112159 * r112162;
        double r112164 = exp(r112163);
        double r112165 = 1.0;
        double r112166 = pow(r112165, r112162);
        double r112167 = r112164 - r112166;
        double r112168 = cbrt(r112167);
        double r112169 = exp(r112167);
        double r112170 = log(r112169);
        double r112171 = cbrt(r112170);
        double r112172 = r112168 * r112171;
        double r112173 = exp(r112159);
        double r112174 = r112173 + r112165;
        double r112175 = r112173 * r112174;
        double r112176 = r112165 * r112165;
        double r112177 = r112175 + r112176;
        double r112178 = cbrt(r112177);
        double r112179 = r112178 * r112178;
        double r112180 = r112172 / r112179;
        double r112181 = r112168 / r112178;
        double r112182 = r112180 * r112181;
        double r112183 = 6.840956219609134e-13;
        bool r112184 = r112159 <= r112183;
        double r112185 = 0.5;
        double r112186 = 2.0;
        double r112187 = pow(r112157, r112186);
        double r112188 = r112185 * r112187;
        double r112189 = r112188 * r112158;
        double r112190 = r112157 + r112189;
        double r112191 = r112158 * r112190;
        double r112192 = 0.16666666666666666;
        double r112193 = pow(r112157, r112162);
        double r112194 = pow(r112158, r112162);
        double r112195 = r112193 * r112194;
        double r112196 = r112192 * r112195;
        double r112197 = r112191 + r112196;
        double r112198 = r112168 * r112168;
        double r112199 = r112198 / r112179;
        double r112200 = sqrt(r112164);
        double r112201 = sqrt(r112165);
        double r112202 = pow(r112201, r112162);
        double r112203 = r112200 + r112202;
        double r112204 = r112200 - r112202;
        double r112205 = r112203 * r112204;
        double r112206 = cbrt(r112205);
        double r112207 = r112206 / r112178;
        double r112208 = r112199 * r112207;
        double r112209 = r112184 ? r112197 : r112208;
        double r112210 = r112161 ? r112182 : r112209;
        return r112210;
}

Error

Bits error versus a

Bits error versus x

Target

Original	29.0
Target	0.2
Herbie	8.7

\[\begin{array}{l} \mathbf{if}\;\left|a \cdot x\right| \lt 0.10000000000000001:\\ \;\;\;\;\left(a \cdot x\right) \cdot \left(1 + \left(\frac{a \cdot x}{2} + \frac{{\left(a \cdot x\right)}^{2}}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{a \cdot x} - 1\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* a x) < -1.9722186113471817e-05

   1. Initial program 0.1

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

   3. Applied flip3-- 0.1

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

   4. Simplified 0.1

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

   6. Applied pow-exp 0.1

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

   8. Applied add-cube-cbrt 0.1

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

   9. Applied add-cube-cbrt 0.1

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

   10. Applied times-frac 0.1

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

   12. Applied add-log-exp 0.1

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

   13. Applied add-log-exp 0.1

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

   14. Applied diff-log 0.1

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

   15. Simplified 0.1

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

   if -1.9722186113471817e-05 < (* a x) < 6.840956219609134e-13

   1. Initial program 44.8

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

   2. Taylor expanded around 0 13.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(\frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right) + a \cdot x\right)}\]

   3. Simplified 13.1

      \[\leadsto \color{blue}{x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)}\]

   if 6.840956219609134e-13 < (* a x)

   1. Initial program 14.9

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

   3. Applied flip3-- 15.8

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

   4. Simplified 15.8

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

   6. Applied pow-exp 14.9

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

   8. Applied add-cube-cbrt 15.0

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

   9. Applied add-cube-cbrt 15.0

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

   10. Applied times-frac 15.0

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

   12. Applied add-sqr-sqrt 15.0

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

   13. Applied unpow-prod-down 15.0

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

   14. Applied add-sqr-sqrt 15.1

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

   15. Applied difference-of-squares 15.2

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

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -1.97221861134718174 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}} \cdot \sqrt[3]{\log \left(e^{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}\right)}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}} \cdot \frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\\ \mathbf{elif}\;a \cdot x \le 6.8409562196091337 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(a + \left(\frac{1}{2} \cdot {a}^{2}\right) \cdot x\right) + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}} \cdot \sqrt[3]{e^{\left(a \cdot x\right) \cdot 3} - {1}^{3}}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1} \cdot \sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}} \cdot \frac{\sqrt[3]{\left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}} + {\left(\sqrt{1}\right)}^{3}\right) \cdot \left(\sqrt{e^{\left(a \cdot x\right) \cdot 3}} - {\left(\sqrt{1}\right)}^{3}\right)}}{\sqrt[3]{e^{a \cdot x} \cdot \left(e^{a \cdot x} + 1\right) + 1 \cdot 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a x)
  :name "expax (section 3.5)"
  :precision binary64
  :herbie-expected 14

  :herbie-target
  (if (< (fabs (* a x)) 0.1) (* (* a x) (+ 1 (+ (/ (* a x) 2) (/ (pow (* a x) 2) 6)))) (- (exp (* a x)) 1))

  (- (exp (* a x)) 1))