Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.3 → 1.5

Time: 22.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -2.398417684101355072336787554272342478967 \cdot 10^{-89}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r357718 = x;
        double r357719 = y;
        double r357720 = r357719 - r357718;
        double r357721 = z;
        double r357722 = r357720 * r357721;
        double r357723 = t;
        double r357724 = r357722 / r357723;
        double r357725 = r357718 + r357724;
        return r357725;
}

↓

double f(double x, double y, double z, double t) {
        double r357726 = x;
        double r357727 = y;
        double r357728 = r357727 - r357726;
        double r357729 = z;
        double r357730 = r357728 * r357729;
        double r357731 = t;
        double r357732 = r357730 / r357731;
        double r357733 = r357726 + r357732;
        double r357734 = -inf.0;
        bool r357735 = r357733 <= r357734;
        double r357736 = r357728 / r357731;
        double r357737 = fma(r357736, r357729, r357726);
        double r357738 = -2.398417684101355e-89;
        bool r357739 = r357733 <= r357738;
        double r357740 = r357731 / r357729;
        double r357741 = r357728 / r357740;
        double r357742 = r357726 + r357741;
        double r357743 = r357739 ? r357733 : r357742;
        double r357744 = r357735 ? r357737 : r357743;
        return r357744;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.3
Target	2.2
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 0.2

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

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < -2.398417684101355e-89

   1. Initial program 0.2

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

   3. Applied *-un-lft-identity 0.2

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

   4. Applied times-frac 2.2

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

   5. Simplified 2.2

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

   7. Applied associate-*r/ 0.2

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

   if -2.398417684101355e-89 < (+ x (/ (* (- y x) z) t))

   1. Initial program 5.7

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

   3. Applied associate-/l* 2.4

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -2.398417684101355072336787554272342478967 \cdot 10^{-89}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :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)))