Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 1.5

Time: 18.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 3.115020440267368122135799538881200874305 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right) - \frac{x}{\frac{t}{z}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 3.52827741147431349904142835074504593956 \cdot 10^{239}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right) - \frac{x}{\frac{t}{z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r25117252 = x;
        double r25117253 = y;
        double r25117254 = r25117253 - r25117252;
        double r25117255 = z;
        double r25117256 = r25117254 * r25117255;
        double r25117257 = t;
        double r25117258 = r25117256 / r25117257;
        double r25117259 = r25117252 + r25117258;
        return r25117259;
}

↓

double f(double x, double y, double z, double t) {
        double r25117260 = x;
        double r25117261 = y;
        double r25117262 = r25117261 - r25117260;
        double r25117263 = z;
        double r25117264 = r25117262 * r25117263;
        double r25117265 = t;
        double r25117266 = r25117264 / r25117265;
        double r25117267 = r25117260 + r25117266;
        double r25117268 = 3.115020440267368e-212;
        bool r25117269 = r25117267 <= r25117268;
        double r25117270 = r25117263 / r25117265;
        double r25117271 = fma(r25117270, r25117261, r25117260);
        double r25117272 = r25117265 / r25117263;
        double r25117273 = r25117260 / r25117272;
        double r25117274 = r25117271 - r25117273;
        double r25117275 = 3.5282774114743135e+239;
        bool r25117276 = r25117267 <= r25117275;
        double r25117277 = r25117276 ? r25117267 : r25117274;
        double r25117278 = r25117269 ? r25117274 : r25117277;
        return r25117278;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.1
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 2 regimes

2. if (+ x (/ (* (- y x) z) t)) < 3.115020440267368e-212 or 3.5282774114743135e+239 < (+ x (/ (* (- y x) z) t))

   1. Initial program 10.0

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

   2. Simplified 7.5

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

   3. Taylor expanded around 0 10.0

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

   4. Simplified 6.6

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

   5. Taylor expanded around 0 10.0

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

   6. Simplified 2.1

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

   if 3.115020440267368e-212 < (+ x (/ (* (- y x) z) t)) < 3.5282774114743135e+239

   1. Initial program 0.5

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 3.115020440267368122135799538881200874305 \cdot 10^{-212}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right) - \frac{x}{\frac{t}{z}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 3.52827741147431349904142835074504593956 \cdot 10^{239}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right) - \frac{x}{\frac{t}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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)))