Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 1.0

Time: 3.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 2.334828876963019126638500573266416450503 \cdot 10^{262}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r491656 = x;
        double r491657 = y;
        double r491658 = r491657 - r491656;
        double r491659 = z;
        double r491660 = r491658 * r491659;
        double r491661 = t;
        double r491662 = r491660 / r491661;
        double r491663 = r491656 + r491662;
        return r491663;
}

↓

double f(double x, double y, double z, double t) {
        double r491664 = x;
        double r491665 = y;
        double r491666 = r491665 - r491664;
        double r491667 = z;
        double r491668 = r491666 * r491667;
        double r491669 = t;
        double r491670 = r491668 / r491669;
        double r491671 = r491664 + r491670;
        double r491672 = -inf.0;
        bool r491673 = r491671 <= r491672;
        double r491674 = r491669 / r491667;
        double r491675 = r491666 / r491674;
        double r491676 = r491664 + r491675;
        double r491677 = 2.334828876963019e+262;
        bool r491678 = r491671 <= r491677;
        double r491679 = r491667 / r491669;
        double r491680 = r491666 * r491679;
        double r491681 = r491664 + r491680;
        double r491682 = r491678 ? r491671 : r491681;
        double r491683 = r491673 ? r491676 : r491682;
        return r491683;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.2
Herbie	1.0

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

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. Using strategy rm

   3. Applied associate-/l* 0.2

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

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < 2.334828876963019e+262

   1. Initial program 0.7

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

   3. Applied *-un-lft-identity 0.7

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

   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/ 0.7

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

   if 2.334828876963019e+262 < (+ x (/ (* (- y x) z) t))

   1. Initial program 32.4

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

   3. Applied *-un-lft-identity 32.4

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

   4. Applied times-frac 4.0

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

   5. Simplified 4.0

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \le 2.334828876963019126638500573266416450503 \cdot 10^{262}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))