Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.6 → 2.8

Time: 13.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -7.3731441332722536 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r433982 = x;
        double r433983 = y;
        double r433984 = r433983 - r433982;
        double r433985 = z;
        double r433986 = r433984 * r433985;
        double r433987 = t;
        double r433988 = r433986 / r433987;
        double r433989 = r433982 + r433988;
        return r433989;
}

↓

double f(double x, double y, double z, double t) {
        double r433990 = z;
        double r433991 = -7.373144133272254e-181;
        bool r433992 = r433990 <= r433991;
        double r433993 = y;
        double r433994 = x;
        double r433995 = r433993 - r433994;
        double r433996 = t;
        double r433997 = r433995 / r433996;
        double r433998 = fma(r433997, r433990, r433994);
        double r433999 = r433996 / r433990;
        double r434000 = r433995 / r433999;
        double r434001 = r433994 + r434000;
        double r434002 = r433992 ? r433998 : r434001;
        return r434002;
}

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	2.8

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

Derivation

1. Split input into 2 regimes

2. if z < -7.373144133272254e-181

   1. Initial program 8.2

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

   2. Simplified 4.3

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

   if -7.373144133272254e-181 < z

   1. Initial program 5.8

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

   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. Simplified 2.0

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

   7. Applied +-commutative 2.0

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.3731441332722536 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

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)))