Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.1 → 1.4

Time: 11.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -5.989972514868663359283165761582355934773 \cdot 10^{283} \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \le -7.684478048840749020220214110509354970101 \cdot 10^{-81}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{x}{\frac{t}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r393256 = x;
        double r393257 = y;
        double r393258 = r393257 - r393256;
        double r393259 = z;
        double r393260 = r393258 * r393259;
        double r393261 = t;
        double r393262 = r393260 / r393261;
        double r393263 = r393256 + r393262;
        return r393263;
}

↓

double f(double x, double y, double z, double t) {
        double r393264 = x;
        double r393265 = y;
        double r393266 = r393265 - r393264;
        double r393267 = z;
        double r393268 = r393266 * r393267;
        double r393269 = t;
        double r393270 = r393268 / r393269;
        double r393271 = r393264 + r393270;
        double r393272 = -5.989972514868663e+283;
        bool r393273 = r393271 <= r393272;
        double r393274 = -7.684478048840749e-81;
        bool r393275 = r393271 <= r393274;
        double r393276 = !r393275;
        bool r393277 = r393273 || r393276;
        double r393278 = r393267 / r393269;
        double r393279 = r393269 / r393267;
        double r393280 = r393264 / r393279;
        double r393281 = r393264 - r393280;
        double r393282 = fma(r393278, r393265, r393281);
        double r393283 = r393277 ? r393282 : r393271;
        return r393283;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.1
Target	1.9
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) z) t)) < -5.989972514868663e+283 or -7.684478048840749e-81 < (+ x (/ (* (- y x) z) t))

   1. Initial program 9.2

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

   2. Simplified 5.6

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

   3. Taylor expanded around 0 9.2

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

   4. Simplified 7.1

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

   6. Applied associate-/l* 2.1

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

   if -5.989972514868663e+283 < (+ x (/ (* (- y x) z) t)) < -7.684478048840749e-81

   1. Initial program 0.2

      \[x + \frac{\left(y - x\right) \cdot z}{t}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -5.989972514868663359283165761582355934773 \cdot 10^{283} \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \le -7.684478048840749020220214110509354970101 \cdot 10^{-81}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{x}{\frac{t}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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)))