Average Error: 1.8 → 1.7
Time: 21.0s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\mathsf{fma}\left(z - t, \frac{x}{y}, t\right)\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\mathsf{fma}\left(z - t, \frac{x}{y}, t\right)
double f(double x, double y, double z, double t) {
        double r333493 = x;
        double r333494 = y;
        double r333495 = r333493 / r333494;
        double r333496 = z;
        double r333497 = t;
        double r333498 = r333496 - r333497;
        double r333499 = r333495 * r333498;
        double r333500 = r333499 + r333497;
        return r333500;
}
↓
double f(double x, double y, double z, double t) {
        double r333501 = z;
        double r333502 = t;
        double r333503 = r333501 - r333502;
        double r333504 = x;
        double r333505 = y;
        double r333506 = r333504 / r333505;
        double r333507 = fma(r333503, r333506, r333502);
        return r333507;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original1.8
Target2.0
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 1.8

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

  3. Applied associate-*l/6.6

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

  4. Simplified6.6

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

  6. Applied *-un-lft-identity6.6

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

  7. Applied times-frac1.8

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

  8. Applied fma-def1.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{1}, \frac{x}{y}, t\right)}\]

  9. Final simplification1.7

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.7594565545626922e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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