Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 0.9

Time: 25.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \le 8.46409780581755386792226817444994499769 \cdot 10^{302}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r306805 = x;
        double r306806 = y;
        double r306807 = r306806 - r306805;
        double r306808 = z;
        double r306809 = r306807 * r306808;
        double r306810 = t;
        double r306811 = r306809 / r306810;
        double r306812 = r306805 + r306811;
        return r306812;
}

↓

double f(double x, double y, double z, double t) {
        double r306813 = x;
        double r306814 = y;
        double r306815 = r306814 - r306813;
        double r306816 = z;
        double r306817 = r306815 * r306816;
        double r306818 = t;
        double r306819 = r306817 / r306818;
        double r306820 = r306813 + r306819;
        double r306821 = -inf.0;
        bool r306822 = r306820 <= r306821;
        double r306823 = 8.464097805817554e+302;
        bool r306824 = r306820 <= r306823;
        double r306825 = !r306824;
        bool r306826 = r306822 || r306825;
        double r306827 = r306815 / r306818;
        double r306828 = fma(r306827, r306816, r306813);
        double r306829 = r306826 ? r306828 : r306820;
        return r306829;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.0
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) z) t)) < -inf.0 or 8.464097805817554e+302 < (+ x (/ (* (- y x) z) t))

   1. Initial program 59.6

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

   2. Simplified 2.0

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

   if -inf.0 < (+ x (/ (* (- y x) z) t)) < 8.464097805817554e+302

   1. Initial program 0.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} = -\infty \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \le 8.46409780581755386792226817444994499769 \cdot 10^{302}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{t}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))