Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.2 → 1.1

Time: 15.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 2.4829011027507563 \cdot 10^{+286}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r11784546 = x;
        double r11784547 = y;
        double r11784548 = r11784547 - r11784546;
        double r11784549 = z;
        double r11784550 = r11784548 * r11784549;
        double r11784551 = t;
        double r11784552 = r11784550 / r11784551;
        double r11784553 = r11784546 + r11784552;
        return r11784553;
}

↓

double f(double x, double y, double z, double t) {
        double r11784554 = x;
        double r11784555 = y;
        double r11784556 = r11784555 - r11784554;
        double r11784557 = z;
        double r11784558 = r11784556 * r11784557;
        double r11784559 = t;
        double r11784560 = r11784558 / r11784559;
        double r11784561 = r11784554 + r11784560;
        double r11784562 = -inf.0;
        bool r11784563 = r11784561 <= r11784562;
        double r11784564 = r11784557 / r11784559;
        double r11784565 = r11784556 * r11784564;
        double r11784566 = r11784554 + r11784565;
        double r11784567 = 2.4829011027507563e+286;
        bool r11784568 = r11784561 <= r11784567;
        double r11784569 = r11784556 / r11784559;
        double r11784570 = fma(r11784569, r11784557, r11784554);
        double r11784571 = r11784568 ? r11784561 : r11784570;
        double r11784572 = r11784563 ? r11784566 : r11784571;
        return r11784572;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.2
Target	2.0
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) z) t)) < -inf.0

   1. Initial program 60.3

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

   3. Applied *-un-lft-identity 60.3

      \[\leadsto x + \frac{\left(y - x\right) \cdot z}{\color{blue}{1 \cdot t}}\]

   4. Applied times-frac 0.2

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

   5. Simplified 0.2

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

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < 2.4829011027507563e+286

   1. Initial program 0.8

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

   if 2.4829011027507563e+286 < (+ x (/ (* (- y x) z) t))

   1. Initial program 42.6

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

   2. Simplified 6.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 2.4829011027507563 \cdot 10^{+286}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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)))