Average Error: 10.6 → 1.0
Time: 20.7s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\left(\left(y - z\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} + x\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\left(\left(y - z\right) \cdot \frac{\sqrt[3]{t}}{\sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} + x
double f(double x, double y, double z, double t, double a) {
        double r25219235 = x;
        double r25219236 = y;
        double r25219237 = z;
        double r25219238 = r25219236 - r25219237;
        double r25219239 = t;
        double r25219240 = r25219238 * r25219239;
        double r25219241 = a;
        double r25219242 = r25219241 - r25219237;
        double r25219243 = r25219240 / r25219242;
        double r25219244 = r25219235 + r25219243;
        return r25219244;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r25219245 = y;
        double r25219246 = z;
        double r25219247 = r25219245 - r25219246;
        double r25219248 = t;
        double r25219249 = cbrt(r25219248);
        double r25219250 = a;
        double r25219251 = r25219250 - r25219246;
        double r25219252 = cbrt(r25219251);
        double r25219253 = r25219249 / r25219252;
        double r25219254 = r25219247 * r25219253;
        double r25219255 = r25219249 * r25219249;
        double r25219256 = r25219252 * r25219252;
        double r25219257 = r25219255 / r25219256;
        double r25219258 = r25219254 * r25219257;
        double r25219259 = x;
        double r25219260 = r25219258 + r25219259;
        return r25219260;
}

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

Original10.6
Target0.6
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 10.6

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

  2. Simplified1.4

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

  4. Applied clear-num1.4

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

  6. Applied fma-udef1.4

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

  7. Simplified2.9

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

  9. Applied add-cube-cbrt3.3

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

  10. Applied add-cube-cbrt3.4

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

  11. Applied times-frac3.4

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

  12. Applied associate-*l*1.0

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

  13. Final simplification1.0

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

Reproduce

herbie shell --seed 2019170 +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))))