Average Error: 6.6 → 2.2
Time: 9.5s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{x}{\frac{t}{z}}\right)\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\mathsf{fma}\left(\frac{z}{t}, y, x - \frac{x}{\frac{t}{z}}\right)
double f(double x, double y, double z, double t) {
        double r371104 = x;
        double r371105 = y;
        double r371106 = r371105 - r371104;
        double r371107 = z;
        double r371108 = r371106 * r371107;
        double r371109 = t;
        double r371110 = r371108 / r371109;
        double r371111 = r371104 + r371110;
        return r371111;
}
↓
double f(double x, double y, double z, double t) {
        double r371112 = z;
        double r371113 = t;
        double r371114 = r371112 / r371113;
        double r371115 = y;
        double r371116 = x;
        double r371117 = r371113 / r371112;
        double r371118 = r371116 / r371117;
        double r371119 = r371116 - r371118;
        double r371120 = fma(r371114, r371115, r371119);
        return r371120;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.6
Target2.1
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  2. Simplified6.1

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

  3. Taylor expanded around 0 6.6

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

  4. Simplified5.8

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

  6. Applied associate-/l*2.2

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

  7. Final simplification2.2

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.0255111955330046e-135) (- x (* (/ z t) (- x y))) (if (< x 4.2750321637007147e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

  (+ x (/ (* (- y x) z) t)))