Average Error: 6.5 → 0.9
Time: 13.5s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[x + \frac{\frac{y - x}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
x + \frac{\frac{y - x}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\frac{\sqrt[3]{t}}{\sqrt[3]{z}}}
double f(double x, double y, double z, double t) {
        double r332684 = x;
        double r332685 = y;
        double r332686 = r332685 - r332684;
        double r332687 = z;
        double r332688 = r332686 * r332687;
        double r332689 = t;
        double r332690 = r332688 / r332689;
        double r332691 = r332684 + r332690;
        return r332691;
}
↓
double f(double x, double y, double z, double t) {
        double r332692 = x;
        double r332693 = y;
        double r332694 = r332693 - r332692;
        double r332695 = t;
        double r332696 = cbrt(r332695);
        double r332697 = r332696 * r332696;
        double r332698 = z;
        double r332699 = cbrt(r332698);
        double r332700 = r332699 * r332699;
        double r332701 = r332697 / r332700;
        double r332702 = r332694 / r332701;
        double r332703 = r332696 / r332699;
        double r332704 = r332702 / r332703;
        double r332705 = r332692 + r332704;
        return r332705;
}

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

Original6.5
Target2.0
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  3. Applied associate-/l*1.8

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

  5. Applied add-cube-cbrt2.3

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

  6. Applied add-cube-cbrt2.5

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

  7. Applied times-frac2.5

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

  8. Applied associate-/r*0.9

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

  9. Final simplification0.9

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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