Average Error: 2.1 → 0.3
Time: 20.3s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \frac{1}{\frac{t}{z - y} - \frac{z - 1}{z - y}}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \frac{1}{\frac{t}{z - y} - \frac{z - 1}{z - y}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r338460 = x;
        double r338461 = y;
        double r338462 = z;
        double r338463 = r338461 - r338462;
        double r338464 = t;
        double r338465 = r338464 - r338462;
        double r338466 = 1.0;
        double r338467 = r338465 + r338466;
        double r338468 = a;
        double r338469 = r338467 / r338468;
        double r338470 = r338463 / r338469;
        double r338471 = r338460 - r338470;
        return r338471;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r338472 = a;
        double r338473 = 1.0;
        double r338474 = t;
        double r338475 = z;
        double r338476 = y;
        double r338477 = r338475 - r338476;
        double r338478 = r338474 / r338477;
        double r338479 = 1.0;
        double r338480 = r338475 - r338479;
        double r338481 = r338480 / r338477;
        double r338482 = r338478 - r338481;
        double r338483 = r338473 / r338482;
        double r338484 = x;
        double r338485 = fma(r338472, r338483, r338484);
        return r338485;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original2.1
Target0.2
Herbie0.3
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 2.1

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

  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 clear-num0.3

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

  6. Applied associate-+l-0.3

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

  7. Applied div-sub0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019347 +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))))