Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.5 → 12.9

Time: 17.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.2582793557416086 \cdot 10^{232}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \le 2.17941341566278614 \cdot 10^{139}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{elif}\;z \le 3.37471355853467873 \cdot 10^{210}:\\ \;\;\;\;x + \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r782130 = x;
        double r782131 = y;
        double r782132 = z;
        double r782133 = r782131 - r782132;
        double r782134 = t;
        double r782135 = r782134 - r782130;
        double r782136 = r782133 * r782135;
        double r782137 = a;
        double r782138 = r782137 - r782132;
        double r782139 = r782136 / r782138;
        double r782140 = r782130 + r782139;
        return r782140;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r782141 = z;
        double r782142 = -3.2582793557416086e+232;
        bool r782143 = r782141 <= r782142;
        double r782144 = t;
        double r782145 = 2.1794134156627861e+139;
        bool r782146 = r782141 <= r782145;
        double r782147 = x;
        double r782148 = y;
        double r782149 = r782148 - r782141;
        double r782150 = a;
        double r782151 = r782150 - r782141;
        double r782152 = r782144 - r782147;
        double r782153 = r782151 / r782152;
        double r782154 = r782149 / r782153;
        double r782155 = r782147 + r782154;
        double r782156 = 3.374713558534679e+210;
        bool r782157 = r782141 <= r782156;
        double r782158 = cbrt(r782151);
        double r782159 = r782158 * r782158;
        double r782160 = r782149 / r782159;
        double r782161 = cbrt(r782160);
        double r782162 = r782161 * r782161;
        double r782163 = r782152 / r782158;
        double r782164 = r782161 * r782163;
        double r782165 = r782162 * r782164;
        double r782166 = r782147 + r782165;
        double r782167 = r782147 * r782148;
        double r782168 = r782167 / r782141;
        double r782169 = r782168 + r782144;
        double r782170 = r782144 * r782148;
        double r782171 = r782170 / r782141;
        double r782172 = r782169 - r782171;
        double r782173 = r782157 ? r782166 : r782172;
        double r782174 = r782146 ? r782155 : r782173;
        double r782175 = r782143 ? r782144 : r782174;
        return r782175;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.5
Target	11.7
Herbie	12.9

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if z < -3.2582793557416086e+232

   1. Initial program 52.1

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 52.4

      \[\leadsto x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}}\]

   4. Applied times-frac 26.3

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 26.3

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}}}\]

   7. Applied cbrt-prod 26.4

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\color{blue}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}}\]

   8. Applied add-cube-cbrt 26.5

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{\color{blue}{\left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right) \cdot \sqrt[3]{t - x}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}\]

   9. Applied times-frac 26.5

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \color{blue}{\left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z}}}\right)}\]

   10. Applied associate-*r* 26.0

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

   12. Applied *-un-lft-identity 26.0

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

   13. Applied times-frac 26.1

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

   14. Applied associate-*l* 28.9

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

   15. Taylor expanded around 0 21.7

       \[\leadsto \color{blue}{t}\]

   if -3.2582793557416086e+232 < z < 2.1794134156627861e+139

   1. Initial program 17.5

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
   2. Using strategy rm

   3. Applied associate-/l* 10.4

      \[\leadsto x + \color{blue}{\frac{y - z}{\frac{a - z}{t - x}}}\]

   if 2.1794134156627861e+139 < z < 3.374713558534679e+210

   1. Initial program 41.9

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 42.2

      \[\leadsto x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{\color{blue}{\left(\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}\right) \cdot \sqrt[3]{a - z}}}\]

   4. Applied times-frac 18.2

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 18.4

      \[\leadsto x + \color{blue}{\left(\left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right)} \cdot \frac{t - x}{\sqrt[3]{a - z}}\]

   7. Applied associate-*l* 18.4

      \[\leadsto x + \color{blue}{\left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)}\]

   if 3.374713558534679e+210 < z

   1. Initial program 51.4

      \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\]

   2. Taylor expanded around inf 24.3

      \[\leadsto \color{blue}{\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.2582793557416086 \cdot 10^{232}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \le 2.17941341566278614 \cdot 10^{139}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{elif}\;z \le 3.37471355853467873 \cdot 10^{210}:\\ \;\;\;\;x + \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}\right) \cdot \left(\sqrt[3]{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))