Average Error: 10.9 → 0.5
Time: 3.5s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{y - z}}} + x\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t}{\frac{\sqrt[3]{a - z}}{\sqrt[3]{y - z}}} + x
double f(double x, double y, double z, double t, double a) {
        double r578187 = x;
        double r578188 = y;
        double r578189 = z;
        double r578190 = r578188 - r578189;
        double r578191 = t;
        double r578192 = r578190 * r578191;
        double r578193 = a;
        double r578194 = r578193 - r578189;
        double r578195 = r578192 / r578194;
        double r578196 = r578187 + r578195;
        return r578196;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r578197 = y;
        double r578198 = z;
        double r578199 = r578197 - r578198;
        double r578200 = cbrt(r578199);
        double r578201 = r578200 * r578200;
        double r578202 = a;
        double r578203 = r578202 - r578198;
        double r578204 = cbrt(r578203);
        double r578205 = r578204 * r578204;
        double r578206 = r578201 / r578205;
        double r578207 = t;
        double r578208 = r578204 / r578200;
        double r578209 = r578207 / r578208;
        double r578210 = r578206 * r578209;
        double r578211 = x;
        double r578212 = r578210 + r578211;
        return r578212;
}

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

Original10.9
Target0.6
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;t \lt -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t \lt 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Initial program 10.9

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

  2. Simplified1.4

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

  4. Applied add-cube-cbrt1.9

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

  5. Applied associate-/r*1.9

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

  7. Applied fma-udef1.9

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

  8. Simplified1.3

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

  10. Applied add-cube-cbrt1.8

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

  11. Applied add-cube-cbrt1.7

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

  12. Applied times-frac1.7

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

  13. Applied *-un-lft-identity1.7

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

  14. Applied times-frac0.5

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2020024 +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))))