Average Error: 11.0 → 0.5
Time: 18.0s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[x + \frac{1}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}} \cdot \left(t \cdot \frac{1}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{y - z}}}\right)\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
x + \frac{1}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}} \cdot \left(t \cdot \frac{1}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{y - z}}}\right)
double f(double x, double y, double z, double t, double a) {
        double r349911 = x;
        double r349912 = y;
        double r349913 = z;
        double r349914 = r349912 - r349913;
        double r349915 = t;
        double r349916 = r349914 * r349915;
        double r349917 = a;
        double r349918 = r349917 - r349913;
        double r349919 = r349916 / r349918;
        double r349920 = r349911 + r349919;
        return r349920;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r349921 = x;
        double r349922 = 1.0;
        double r349923 = a;
        double r349924 = z;
        double r349925 = r349923 - r349924;
        double r349926 = cbrt(r349925);
        double r349927 = r349926 * r349926;
        double r349928 = y;
        double r349929 = r349928 - r349924;
        double r349930 = cbrt(r349929);
        double r349931 = r349930 * r349930;
        double r349932 = r349927 / r349931;
        double r349933 = r349922 / r349932;
        double r349934 = t;
        double r349935 = r349926 / r349930;
        double r349936 = r349922 / r349935;
        double r349937 = r349934 * r349936;
        double r349938 = r349933 * r349937;
        double r349939 = r349921 + r349938;
        return r349939;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original11.0
Target0.6
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 11.0

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

  2. Simplified3.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)}\]
  3. Using strategy rm

  4. Applied div-inv3.2

    \[\leadsto \mathsf{fma}\left(y - z, \color{blue}{t \cdot \frac{1}{a - z}}, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef3.2

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

  7. Simplified11.0

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

  9. Applied clear-num11.0

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

  10. Simplified1.3

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

  12. Applied *-un-lft-identity1.3

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

  13. Applied add-cube-cbrt1.8

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

  14. Applied add-cube-cbrt1.7

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

  15. Applied times-frac1.7

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

  16. Applied times-frac0.5

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

  17. Applied add-cube-cbrt0.5

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

  18. Applied times-frac0.6

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

  19. Simplified0.6

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

  20. Simplified0.5

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

  21. Final simplification0.5

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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