Average Error: 10.8 → 0.5
Time: 18.3s
Precision: 64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot t\right) + x\]
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 \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot t\right) + x
double f(double x, double y, double z, double t, double a) {
        double r392817 = x;
        double r392818 = y;
        double r392819 = z;
        double r392820 = r392818 - r392819;
        double r392821 = t;
        double r392822 = r392820 * r392821;
        double r392823 = a;
        double r392824 = r392823 - r392819;
        double r392825 = r392822 / r392824;
        double r392826 = r392817 + r392825;
        return r392826;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r392827 = y;
        double r392828 = z;
        double r392829 = r392827 - r392828;
        double r392830 = cbrt(r392829);
        double r392831 = r392830 * r392830;
        double r392832 = a;
        double r392833 = r392832 - r392828;
        double r392834 = cbrt(r392833);
        double r392835 = r392834 * r392834;
        double r392836 = r392831 / r392835;
        double r392837 = r392830 / r392834;
        double r392838 = t;
        double r392839 = r392837 * r392838;
        double r392840 = r392836 * r392839;
        double r392841 = x;
        double r392842 = r392840 + r392841;
        return r392842;
}

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

Original10.8
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 10.8

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

  2. Simplified1.2

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

  4. Applied fma-udef1.2

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

  6. Applied add-cube-cbrt1.8

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

  7. Applied add-cube-cbrt1.6

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

  8. Applied times-frac1.6

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

  9. Applied associate-*l*0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019303 +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))))