Average Error: 1.8 → 1.7
Time: 18.5s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\frac{x}{y} \cdot z + \mathsf{fma}\left(\frac{x}{y}, -t, t\right)\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\frac{x}{y} \cdot z + \mathsf{fma}\left(\frac{x}{y}, -t, t\right)
double f(double x, double y, double z, double t) {
        double r273118 = x;
        double r273119 = y;
        double r273120 = r273118 / r273119;
        double r273121 = z;
        double r273122 = t;
        double r273123 = r273121 - r273122;
        double r273124 = r273120 * r273123;
        double r273125 = r273124 + r273122;
        return r273125;
}
↓
double f(double x, double y, double z, double t) {
        double r273126 = x;
        double r273127 = y;
        double r273128 = r273126 / r273127;
        double r273129 = z;
        double r273130 = r273128 * r273129;
        double r273131 = t;
        double r273132 = -r273131;
        double r273133 = fma(r273128, r273132, r273131);
        double r273134 = r273130 + r273133;
        return r273134;
}

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 sub-neg1.8

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

  4. Applied distribute-lft-in1.8

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

  5. Applied associate-+l+1.8

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

  6. Simplified1.7

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

  7. Final simplification1.7

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

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))