Average Error: 1.6 → 0.9
Time: 4.3s
Precision: 64
\[x + \left(y - x\right) \cdot \frac{z}{t}\]
↓
\[\left(\left(\left(y - x\right) \cdot \frac{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}} + x\]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\left(\left(\left(y - x\right) \cdot \frac{\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}} + x
double f(double x, double y, double z, double t) {
        double r520188 = x;
        double r520189 = y;
        double r520190 = r520189 - r520188;
        double r520191 = z;
        double r520192 = t;
        double r520193 = r520191 / r520192;
        double r520194 = r520190 * r520193;
        double r520195 = r520188 + r520194;
        return r520195;
}
↓
double f(double x, double y, double z, double t) {
        double r520196 = y;
        double r520197 = x;
        double r520198 = r520196 - r520197;
        double r520199 = z;
        double r520200 = cbrt(r520199);
        double r520201 = r520200 * r520200;
        double r520202 = cbrt(r520201);
        double r520203 = cbrt(r520200);
        double r520204 = r520202 * r520203;
        double r520205 = t;
        double r520206 = cbrt(r520205);
        double r520207 = r520204 / r520206;
        double r520208 = r520198 * r520207;
        double r520209 = r520200 / r520206;
        double r520210 = r520208 * r520209;
        double r520211 = r520210 * r520209;
        double r520212 = r520211 + r520197;
        return r520212;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.6
Target1.7
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} \lt -1013646692435.887:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} \lt -0.0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 1.6

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

  2. Simplified1.6

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

  4. Applied fma-udef1.6

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

  6. Applied add-cube-cbrt2.1

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

  7. Applied add-cube-cbrt2.3

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

  8. Applied times-frac2.3

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

  9. Applied associate-*r*1.0

    \[\leadsto \color{blue}{\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}}} + x\]
  10. Using strategy rm

  11. Applied add-cube-cbrt1.0

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

  12. Applied cbrt-prod1.0

    \[\leadsto \left(\left(y - x\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}\right)} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}} + x\]
  13. Using strategy rm

  14. Applied times-frac1.0

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

  15. Applied associate-*r*0.9

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

  16. Final simplification0.9

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.887) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) -0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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