Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 0.9

Time: 22.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r93621355 = x;
        double r93621356 = y;
        double r93621357 = r93621356 - r93621355;
        double r93621358 = z;
        double r93621359 = r93621357 * r93621358;
        double r93621360 = t;
        double r93621361 = r93621359 / r93621360;
        double r93621362 = r93621355 + r93621361;
        return r93621362;
}

↓

double f(double x, double y, double z, double t) {
        double r93621363 = x;
        double r93621364 = y;
        double r93621365 = r93621364 - r93621363;
        double r93621366 = cbrt(r93621365);
        double r93621367 = t;
        double r93621368 = cbrt(r93621367);
        double r93621369 = z;
        double r93621370 = cbrt(r93621369);
        double r93621371 = r93621368 / r93621370;
        double r93621372 = r93621366 / r93621371;
        double r93621373 = r93621372 * r93621372;
        double r93621374 = r93621373 * r93621372;
        double r93621375 = r93621363 + r93621374;
        return r93621375;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.3
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Initial program 6.5

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

3. Applied associate-/l* 2.1

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

5. Applied add-cube-cbrt 2.6

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

6. Applied add-cube-cbrt 2.8

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

7. Applied times-frac 2.8

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

8. Applied add-cube-cbrt 2.8

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

9. Applied times-frac 1.0

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

10. Simplified 0.9

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

11. Final simplification 0.9

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

Reproduce

herbie shell --seed 2019173 
(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)))