Average Error: 11.3 → 1.0
Time: 4.6s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\frac{\sqrt[3]{t}}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{t}}} \cdot \frac{\sqrt[3]{t}}{\frac{\sqrt[3]{a - z}}{y - z}} + x\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\frac{\sqrt[3]{t}}{\frac{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}{\sqrt[3]{t}}} \cdot \frac{\sqrt[3]{t}}{\frac{\sqrt[3]{a - z}}{y - z}} + x
double f(double x, double y, double z, double t, double a) {
        double r567522 = x;
        double r567523 = y;
        double r567524 = z;
        double r567525 = r567523 - r567524;
        double r567526 = t;
        double r567527 = r567525 * r567526;
        double r567528 = a;
        double r567529 = r567528 - r567524;
        double r567530 = r567527 / r567529;
        double r567531 = r567522 + r567530;
        return r567531;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r567532 = t;
        double r567533 = cbrt(r567532);
        double r567534 = a;
        double r567535 = z;
        double r567536 = r567534 - r567535;
        double r567537 = cbrt(r567536);
        double r567538 = r567537 * r567537;
        double r567539 = r567538 / r567533;
        double r567540 = r567533 / r567539;
        double r567541 = y;
        double r567542 = r567541 - r567535;
        double r567543 = r567537 / r567542;
        double r567544 = r567533 / r567543;
        double r567545 = r567540 * r567544;
        double r567546 = x;
        double r567547 = r567545 + r567546;
        return r567547;
}

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

Original11.3
Target0.5
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;t \lt -1.068297449017406694366747246993994850729 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.911094988758637497591020599238553861375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 11.3

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

  2. Simplified1.3

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

  4. Applied add-cube-cbrt1.8

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

  5. Applied associate-/r*1.8

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

  7. Applied fma-udef1.8

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

  8. Simplified1.2

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

  10. Applied *-un-lft-identity1.2

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

  11. Applied add-cube-cbrt1.7

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

  12. Applied times-frac1.7

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

  13. Applied add-cube-cbrt1.9

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

  14. Applied times-frac1.0

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

  15. Simplified1.0

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

  16. Final simplification1.0

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :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))))