Average Error: 1.8 → 0.2
Time: 14.3s
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 r800510 = x;
        double r800511 = y;
        double r800512 = z;
        double r800513 = r800511 - r800512;
        double r800514 = t;
        double r800515 = r800514 - r800512;
        double r800516 = 1.0;
        double r800517 = r800515 + r800516;
        double r800518 = a;
        double r800519 = r800517 / r800518;
        double r800520 = r800513 / r800519;
        double r800521 = r800510 - r800520;
        return r800521;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r800522 = a;
        double r800523 = z;
        double r800524 = t;
        double r800525 = r800524 - r800523;
        double r800526 = 1.0;
        double r800527 = r800525 + r800526;
        double r800528 = r800523 / r800527;
        double r800529 = y;
        double r800530 = r800529 / r800527;
        double r800531 = r800528 - r800530;
        double r800532 = x;
        double r800533 = fma(r800522, r800531, r800532);
        return r800533;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.8
Target0.2
Herbie0.2
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 1.8

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

  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 2019351 +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))))