Average Error: 1.9 → 0.2
Time: 16.9s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \frac{1}{\frac{\left(t - z\right) + 1}{z - y}}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \frac{1}{\frac{\left(t - z\right) + 1}{z - y}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r459547 = x;
        double r459548 = y;
        double r459549 = z;
        double r459550 = r459548 - r459549;
        double r459551 = t;
        double r459552 = r459551 - r459549;
        double r459553 = 1.0;
        double r459554 = r459552 + r459553;
        double r459555 = a;
        double r459556 = r459554 / r459555;
        double r459557 = r459550 / r459556;
        double r459558 = r459547 - r459557;
        return r459558;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r459559 = a;
        double r459560 = 1.0;
        double r459561 = t;
        double r459562 = z;
        double r459563 = r459561 - r459562;
        double r459564 = 1.0;
        double r459565 = r459563 + r459564;
        double r459566 = y;
        double r459567 = r459562 - r459566;
        double r459568 = r459565 / r459567;
        double r459569 = r459560 / r459568;
        double r459570 = x;
        double r459571 = fma(r459559, r459569, r459570);
        return r459571;
}

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 clear-num0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019212 +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))))