Average Error: 2.0 → 0.3
Time: 19.9s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \frac{z - y}{\left(t - z\right) + 1}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \frac{z - y}{\left(t - z\right) + 1}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r419611 = x;
        double r419612 = y;
        double r419613 = z;
        double r419614 = r419612 - r419613;
        double r419615 = t;
        double r419616 = r419615 - r419613;
        double r419617 = 1.0;
        double r419618 = r419616 + r419617;
        double r419619 = a;
        double r419620 = r419618 / r419619;
        double r419621 = r419614 / r419620;
        double r419622 = r419611 - r419621;
        return r419622;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r419623 = a;
        double r419624 = z;
        double r419625 = y;
        double r419626 = r419624 - r419625;
        double r419627 = t;
        double r419628 = r419627 - r419624;
        double r419629 = 1.0;
        double r419630 = r419628 + r419629;
        double r419631 = r419626 / r419630;
        double r419632 = x;
        double r419633 = fma(r419623, r419631, r419632);
        return r419633;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original2.0
Target0.3
Herbie0.3
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 2.0

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

  2. Simplified0.3

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

  3. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 +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))))