Average Error: 2.0 → 0.3
Time: 21.2s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}\]
↓
\[\mathsf{fma}\left(\frac{z}{\left(1.0 + t\right) - z} - \frac{y}{\left(1.0 + t\right) - z}, a, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}
↓
\mathsf{fma}\left(\frac{z}{\left(1.0 + t\right) - z} - \frac{y}{\left(1.0 + t\right) - z}, a, x\right)
double f(double x, double y, double z, double t, double a) {
        double r29248046 = x;
        double r29248047 = y;
        double r29248048 = z;
        double r29248049 = r29248047 - r29248048;
        double r29248050 = t;
        double r29248051 = r29248050 - r29248048;
        double r29248052 = 1.0;
        double r29248053 = r29248051 + r29248052;
        double r29248054 = a;
        double r29248055 = r29248053 / r29248054;
        double r29248056 = r29248049 / r29248055;
        double r29248057 = r29248046 - r29248056;
        return r29248057;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r29248058 = z;
        double r29248059 = 1.0;
        double r29248060 = t;
        double r29248061 = r29248059 + r29248060;
        double r29248062 = r29248061 - r29248058;
        double r29248063 = r29248058 / r29248062;
        double r29248064 = y;
        double r29248065 = r29248064 / r29248062;
        double r29248066 = r29248063 - r29248065;
        double r29248067 = a;
        double r29248068 = x;
        double r29248069 = fma(r29248066, r29248067, r29248068);
        return r29248069;
}

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
\[x - \frac{y - z}{\left(t - z\right) + 1.0} \cdot a\]

Derivation

  1. Initial program 2.0

    \[x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{\left(1.0 + t\right) - z}, a, x\right)}\]
  3. Using strategy rm

  4. Applied div-sub0.3

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z}{\left(1.0 + t\right) - z} - \frac{y}{\left(1.0 + t\right) - z}}, a, x\right)\]

  5. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\frac{z}{\left(1.0 + t\right) - z} - \frac{y}{\left(1.0 + t\right) - z}, a, x\right)\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1.0)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))