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

Average Error: 24.8 → 11.7

Time: 6.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.0538724122977772 \cdot 10^{187} \lor \neg \left(z \le 2.80403393809698577 \cdot 10^{31}\right):\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r699825 = x;
        double r699826 = y;
        double r699827 = z;
        double r699828 = r699826 - r699827;
        double r699829 = t;
        double r699830 = r699829 - r699825;
        double r699831 = r699828 * r699830;
        double r699832 = a;
        double r699833 = r699832 - r699827;
        double r699834 = r699831 / r699833;
        double r699835 = r699825 + r699834;
        return r699835;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r699836 = z;
        double r699837 = -1.0538724122977772e+187;
        bool r699838 = r699836 <= r699837;
        double r699839 = 2.8040339380969858e+31;
        bool r699840 = r699836 <= r699839;
        double r699841 = !r699840;
        bool r699842 = r699838 || r699841;
        double r699843 = y;
        double r699844 = x;
        double r699845 = r699844 / r699836;
        double r699846 = t;
        double r699847 = r699846 / r699836;
        double r699848 = r699845 - r699847;
        double r699849 = r699843 * r699848;
        double r699850 = r699849 + r699846;
        double r699851 = r699843 - r699836;
        double r699852 = cbrt(r699851);
        double r699853 = r699852 * r699852;
        double r699854 = a;
        double r699855 = r699854 - r699836;
        double r699856 = cbrt(r699855);
        double r699857 = r699853 / r699856;
        double r699858 = r699852 / r699856;
        double r699859 = r699846 - r699844;
        double r699860 = r699859 / r699856;
        double r699861 = r699858 * r699860;
        double r699862 = r699857 * r699861;
        double r699863 = r699844 + r699862;
        double r699864 = r699842 ? r699850 : r699863;
        return r699864;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	12.0
Herbie	11.7

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

Derivation

1. Split input into 2 regimes

2. if z < -1.0538724122977772e+187 or 2.8040339380969858e+31 < z

   1. Initial program 43.4

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

   3. Applied add-cube-cbrt 43.7

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

   4. Applied times-frac 22.2

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

   6. Applied add-cube-cbrt 21.9

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

   7. Applied times-frac 21.9

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

   8. Applied associate-*l* 21.9

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

   9. Taylor expanded around inf 27.2

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

   10. Simplified 19.4

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

   if -1.0538724122977772e+187 < z < 2.8040339380969858e+31

   1. Initial program 14.2

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

   3. Applied add-cube-cbrt 14.7

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

   4. Applied times-frac 7.7

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

   6. Applied add-cube-cbrt 7.7

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

   7. Applied times-frac 7.7

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

   8. Applied associate-*l* 7.3

      \[\leadsto x + \color{blue}{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.0538724122977772 \cdot 10^{187} \lor \neg \left(z \le 2.80403393809698577 \cdot 10^{31}\right):\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))