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

Average Error: 24.6 → 9.9

Time: 32.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.413972289654817505443274974299844032514 \cdot 10^{128}:\\ \;\;\;\;\left(t + \frac{x}{\frac{z}{y}}\right) - \frac{t}{\frac{z}{y}}\\ \mathbf{elif}\;z \le 1.119836987767907781053345179829644266102 \cdot 10^{148}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z}}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}}{\frac{\frac{1}{t - x}}{\frac{\sqrt[3]{y - z}}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{\sqrt[3]{y - z}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(t + \frac{x}{\frac{z}{y}}\right) - \frac{t}{\frac{z}{y}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r34378830 = x;
        double r34378831 = y;
        double r34378832 = z;
        double r34378833 = r34378831 - r34378832;
        double r34378834 = t;
        double r34378835 = r34378834 - r34378830;
        double r34378836 = r34378833 * r34378835;
        double r34378837 = a;
        double r34378838 = r34378837 - r34378832;
        double r34378839 = r34378836 / r34378838;
        double r34378840 = r34378830 + r34378839;
        return r34378840;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r34378841 = z;
        double r34378842 = -1.4139722896548175e+128;
        bool r34378843 = r34378841 <= r34378842;
        double r34378844 = t;
        double r34378845 = x;
        double r34378846 = y;
        double r34378847 = r34378841 / r34378846;
        double r34378848 = r34378845 / r34378847;
        double r34378849 = r34378844 + r34378848;
        double r34378850 = r34378844 / r34378847;
        double r34378851 = r34378849 - r34378850;
        double r34378852 = 1.1198369877679078e+148;
        bool r34378853 = r34378841 <= r34378852;
        double r34378854 = r34378846 - r34378841;
        double r34378855 = cbrt(r34378854);
        double r34378856 = a;
        double r34378857 = r34378856 - r34378841;
        double r34378858 = cbrt(r34378857);
        double r34378859 = r34378858 * r34378858;
        double r34378860 = r34378855 * r34378855;
        double r34378861 = cbrt(r34378860);
        double r34378862 = r34378859 / r34378861;
        double r34378863 = r34378855 / r34378862;
        double r34378864 = 1.0;
        double r34378865 = r34378844 - r34378845;
        double r34378866 = r34378864 / r34378865;
        double r34378867 = cbrt(r34378855);
        double r34378868 = r34378858 / r34378867;
        double r34378869 = r34378855 / r34378868;
        double r34378870 = r34378866 / r34378869;
        double r34378871 = r34378863 / r34378870;
        double r34378872 = r34378845 + r34378871;
        double r34378873 = r34378853 ? r34378872 : r34378851;
        double r34378874 = r34378843 ? r34378851 : r34378873;
        return r34378874;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.6
Target	12.0
Herbie	9.9

\[\begin{array}{l} \mathbf{if}\;z \lt -1.253613105609503593846459977496550767343 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -1.4139722896548175e+128 or 1.1198369877679078e+148 < z

   1. Initial program 46.1

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

   3. Applied associate-/l* 26.8

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

   5. Applied div-inv 26.9

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

   6. Applied associate-/r* 21.5

      \[\leadsto x + \color{blue}{\frac{\frac{y - z}{a - z}}{\frac{1}{t - x}}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 22.2

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

   9. Applied associate-/l* 22.2

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

   10. Taylor expanded around inf 25.2

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

   11. Simplified 15.7

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

   if -1.4139722896548175e+128 < z < 1.1198369877679078e+148

   1. Initial program 14.6

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

   3. Applied associate-/l* 9.2

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

   5. Applied div-inv 9.3

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

   6. Applied associate-/r* 7.3

      \[\leadsto x + \color{blue}{\frac{\frac{y - z}{a - z}}{\frac{1}{t - x}}}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 7.8

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

   9. Applied associate-/l* 7.8

      \[\leadsto x + \frac{\color{blue}{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\frac{a - z}{\sqrt[3]{y - z}}}}}{\frac{1}{t - x}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 7.8

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

   12. Applied cbrt-prod 7.9

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

   13. Applied add-cube-cbrt 7.9

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

   14. Applied times-frac 7.9

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

   15. Applied times-frac 7.9

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

   16. Applied associate-/l* 7.1

       \[\leadsto x + \color{blue}{\frac{\frac{\sqrt[3]{y - z}}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}}{\frac{\frac{1}{t - x}}{\frac{\sqrt[3]{y - z}}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{\sqrt[3]{y - z}}}}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.413972289654817505443274974299844032514 \cdot 10^{128}:\\ \;\;\;\;\left(t + \frac{x}{\frac{z}{y}}\right) - \frac{t}{\frac{z}{y}}\\ \mathbf{elif}\;z \le 1.119836987767907781053345179829644266102 \cdot 10^{148}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z}}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}}{\frac{\frac{1}{t - x}}{\frac{\sqrt[3]{y - z}}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{\sqrt[3]{y - z}}}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(t + \frac{x}{\frac{z}{y}}\right) - \frac{t}{\frac{z}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"

  :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))))