Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.1 → 1.0

Time: 3.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -6.7778335529442914 \cdot 10^{300}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 1.7124072696210518 \cdot 10^{169}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r406377 = x;
        double r406378 = y;
        double r406379 = r406378 - r406377;
        double r406380 = z;
        double r406381 = r406379 * r406380;
        double r406382 = t;
        double r406383 = r406381 / r406382;
        double r406384 = r406377 + r406383;
        return r406384;
}

↓

double f(double x, double y, double z, double t) {
        double r406385 = x;
        double r406386 = y;
        double r406387 = r406386 - r406385;
        double r406388 = z;
        double r406389 = r406387 * r406388;
        double r406390 = t;
        double r406391 = r406389 / r406390;
        double r406392 = r406385 + r406391;
        double r406393 = -6.777833552944291e+300;
        bool r406394 = r406392 <= r406393;
        double r406395 = r406390 / r406388;
        double r406396 = r406387 / r406395;
        double r406397 = r406385 + r406396;
        double r406398 = 1.7124072696210518e+169;
        bool r406399 = r406392 <= r406398;
        double r406400 = r406388 / r406390;
        double r406401 = r406387 * r406400;
        double r406402 = r406385 + r406401;
        double r406403 = r406399 ? r406392 : r406402;
        double r406404 = r406394 ? r406397 : r406403;
        return r406404;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.1
Target	1.8
Herbie	1.0

\[\begin{array}{l} \mathbf{if}\;x \lt -9.0255111955330046 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.2750321637007147 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) z) t)) < -6.777833552944291e+300

   1. Initial program 55.1

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

   3. Applied associate-/l* 1.1

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

   if -6.777833552944291e+300 < (+ x (/ (* (- y x) z) t)) < 1.7124072696210518e+169

   1. Initial program 0.8

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

   if 1.7124072696210518e+169 < (+ x (/ (* (- y x) z) t))

   1. Initial program 14.4

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

   3. Applied *-un-lft-identity 14.4

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

   4. Applied times-frac 2.1

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

   5. Simplified 2.1

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \le -6.7778335529442914 \cdot 10^{300}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 1.7124072696210518 \cdot 10^{169}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))