Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.6 → 2.2

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.01649379191452465 \cdot 10^{-92}:\\ \;\;\;\;\left(\left(y - x\right) \cdot \frac{1}{t}\right) \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r477473 = x;
        double r477474 = y;
        double r477475 = r477474 - r477473;
        double r477476 = z;
        double r477477 = r477475 * r477476;
        double r477478 = t;
        double r477479 = r477477 / r477478;
        double r477480 = r477473 + r477479;
        return r477480;
}

↓

double f(double x, double y, double z, double t) {
        double r477481 = z;
        double r477482 = -4.0164937919145246e-92;
        bool r477483 = r477481 <= r477482;
        double r477484 = y;
        double r477485 = x;
        double r477486 = r477484 - r477485;
        double r477487 = 1.0;
        double r477488 = t;
        double r477489 = r477487 / r477488;
        double r477490 = r477486 * r477489;
        double r477491 = r477490 * r477481;
        double r477492 = r477491 + r477485;
        double r477493 = r477481 / r477488;
        double r477494 = r477486 * r477493;
        double r477495 = r477494 + r477485;
        double r477496 = r477483 ? r477492 : r477495;
        return r477496;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	2.2
Herbie	2.2

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

Derivation

1. Split input into 2 regimes

2. if z < -4.0164937919145246e-92

   1. Initial program 10.3

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

   2. Simplified 2.9

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

   4. Applied fma-udef 2.9

      \[\leadsto \color{blue}{\frac{y - x}{t} \cdot z + x}\]
   5. Using strategy rm

   6. Applied div-inv 2.9

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

   if -4.0164937919145246e-92 < z

   1. Initial program 5.2

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

   2. Simplified 8.0

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

   4. Applied fma-udef 8.0

      \[\leadsto \color{blue}{\frac{y - x}{t} \cdot z + x}\]
   5. Using strategy rm

   6. Applied div-inv 8.1

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

   7. Applied associate-*l* 2.0

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

   8. Simplified 2.0

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.01649379191452465 \cdot 10^{-92}:\\ \;\;\;\;\left(\left(y - x\right) \cdot \frac{1}{t}\right) \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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)))