Average Error: 1.9 → 0.2
Time: 21.5s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r307611 = x;
        double r307612 = y;
        double r307613 = z;
        double r307614 = r307612 - r307613;
        double r307615 = t;
        double r307616 = r307615 - r307613;
        double r307617 = 1.0;
        double r307618 = r307616 + r307617;
        double r307619 = a;
        double r307620 = r307618 / r307619;
        double r307621 = r307614 / r307620;
        double r307622 = r307611 - r307621;
        return r307622;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r307623 = a;
        double r307624 = z;
        double r307625 = t;
        double r307626 = r307625 - r307624;
        double r307627 = 1.0;
        double r307628 = r307626 + r307627;
        double r307629 = r307624 / r307628;
        double r307630 = y;
        double r307631 = r307630 / r307628;
        double r307632 = r307629 - r307631;
        double r307633 = x;
        double r307634 = fma(r307623, r307632, r307633);
        return r307634;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.9
Target0.2
Herbie0.2
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 1.9

    \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{z - y}{\left(t - z\right) + 1}, x\right)}\]
  3. Using strategy rm

  4. Applied div-sub0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1) a))))