Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 0.8

Time: 22.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{t}{y - x}}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 1.812525037796768018373255724443839618269 \cdot 10^{293}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r263975 = x;
        double r263976 = y;
        double r263977 = r263976 - r263975;
        double r263978 = z;
        double r263979 = r263977 * r263978;
        double r263980 = t;
        double r263981 = r263979 / r263980;
        double r263982 = r263975 + r263981;
        return r263982;
}

↓

double f(double x, double y, double z, double t) {
        double r263983 = x;
        double r263984 = y;
        double r263985 = r263984 - r263983;
        double r263986 = z;
        double r263987 = r263985 * r263986;
        double r263988 = t;
        double r263989 = r263987 / r263988;
        double r263990 = r263983 + r263989;
        double r263991 = -inf.0;
        bool r263992 = r263990 <= r263991;
        double r263993 = 1.0;
        double r263994 = r263988 / r263985;
        double r263995 = r263993 / r263994;
        double r263996 = fma(r263995, r263986, r263983);
        double r263997 = 1.812525037796768e+293;
        bool r263998 = r263990 <= r263997;
        double r263999 = r263986 / r263988;
        double r264000 = r263985 * r263999;
        double r264001 = r263983 + r264000;
        double r264002 = r263998 ? r263990 : r264001;
        double r264003 = r263992 ? r263996 : r264002;
        return r264003;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	1.9
Herbie	0.8

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

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

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

   4. Applied clear-num 0.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{t}{y - x}}}, z, x\right)\]

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < 1.812525037796768e+293

   1. Initial program 0.8

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

   if 1.812525037796768e+293 < (+ x (/ (* (- y x) z) t))

   1. Initial program 50.1

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

   3. Applied *-un-lft-identity 50.1

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

   4. Applied times-frac 2.2

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

   5. Simplified 2.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{t}{y - x}}, z, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 1.812525037796768018373255724443839618269 \cdot 10^{293}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.0255111955330046e-135) (- x (* (/ z t) (- x y))) (if (< x 4.2750321637007147e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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