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

Average Error: 24.7 → 12.4

Time: 26.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.647463183269898 \cdot 10^{-65}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\\ \mathbf{elif}\;a \le 500051093016.9032:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{\frac{t - x}{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{a - z}}}\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}} \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r474897 = x;
        double r474898 = y;
        double r474899 = z;
        double r474900 = r474898 - r474899;
        double r474901 = t;
        double r474902 = r474901 - r474897;
        double r474903 = r474900 * r474902;
        double r474904 = a;
        double r474905 = r474904 - r474899;
        double r474906 = r474903 / r474905;
        double r474907 = r474897 + r474906;
        return r474907;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r474908 = a;
        double r474909 = -6.647463183269898e-65;
        bool r474910 = r474908 <= r474909;
        double r474911 = x;
        double r474912 = y;
        double r474913 = z;
        double r474914 = r474912 - r474913;
        double r474915 = r474908 - r474913;
        double r474916 = cbrt(r474915);
        double r474917 = r474916 * r474916;
        double r474918 = r474914 / r474917;
        double r474919 = t;
        double r474920 = r474919 - r474911;
        double r474921 = r474920 / r474916;
        double r474922 = r474918 * r474921;
        double r474923 = r474911 + r474922;
        double r474924 = 500051093016.9032;
        bool r474925 = r474908 <= r474924;
        double r474926 = r474911 / r474913;
        double r474927 = r474919 / r474913;
        double r474928 = r474926 - r474927;
        double r474929 = r474912 * r474928;
        double r474930 = r474919 + r474929;
        double r474931 = cbrt(r474921);
        double r474932 = r474931 * r474931;
        double r474933 = r474918 * r474932;
        double r474934 = 1.0;
        double r474935 = cbrt(r474916);
        double r474936 = r474935 * r474935;
        double r474937 = r474934 / r474936;
        double r474938 = cbrt(r474937);
        double r474939 = r474920 / r474935;
        double r474940 = cbrt(r474939);
        double r474941 = r474938 * r474940;
        double r474942 = r474933 * r474941;
        double r474943 = r474911 + r474942;
        double r474944 = r474925 ? r474930 : r474943;
        double r474945 = r474910 ? r474923 : r474944;
        return r474945;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.7
Target	12.1
Herbie	12.4

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

Derivation

1. Split input into 3 regimes

2. if a < -6.647463183269898e-65

   1. Initial program 22.7

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

   3. Applied add-cube-cbrt 23.0

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

   4. Applied times-frac 8.6

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

   if -6.647463183269898e-65 < a < 500051093016.9032

   1. Initial program 27.9

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

   3. Applied add-cube-cbrt 28.5

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

   4. Applied times-frac 20.5

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

   6. Applied add-cube-cbrt 20.7

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

   7. Applied add-cube-cbrt 20.8

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

   8. Applied times-frac 20.8

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

   9. Applied associate-*r* 19.5

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

   10. Taylor expanded around inf 21.7

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

   11. Simplified 18.5

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

   if 500051093016.9032 < a

   1. Initial program 22.1

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

   3. Applied add-cube-cbrt 22.4

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

   4. Applied times-frac 6.8

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

   6. Applied add-cube-cbrt 7.0

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

   7. Applied associate-*r* 7.0

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

   9. Applied add-cube-cbrt 7.0

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

   10. Applied *-un-lft-identity 7.0

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

   11. Applied times-frac 7.0

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

   12. Applied cbrt-prod 7.0

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.647463183269898 \cdot 10^{-65}:\\ \;\;\;\;x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\\ \mathbf{elif}\;a \le 500051093016.9032:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\sqrt[3]{\frac{t - x}{\sqrt[3]{a - z}}} \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{a - z}}}\right)\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}} \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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))))