Average Error: 1.8 → 0.2
Time: 22.6s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}\]
↓
\[\mathsf{fma}\left(\frac{z - y}{\left(1.0 + t\right) - z}, a, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}
↓
\mathsf{fma}\left(\frac{z - y}{\left(1.0 + t\right) - z}, a, x\right)
double f(double x, double y, double z, double t, double a) {
        double r29532301 = x;
        double r29532302 = y;
        double r29532303 = z;
        double r29532304 = r29532302 - r29532303;
        double r29532305 = t;
        double r29532306 = r29532305 - r29532303;
        double r29532307 = 1.0;
        double r29532308 = r29532306 + r29532307;
        double r29532309 = a;
        double r29532310 = r29532308 / r29532309;
        double r29532311 = r29532304 / r29532310;
        double r29532312 = r29532301 - r29532311;
        return r29532312;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r29532313 = z;
        double r29532314 = y;
        double r29532315 = r29532313 - r29532314;
        double r29532316 = 1.0;
        double r29532317 = t;
        double r29532318 = r29532316 + r29532317;
        double r29532319 = r29532318 - r29532313;
        double r29532320 = r29532315 / r29532319;
        double r29532321 = a;
        double r29532322 = x;
        double r29532323 = fma(r29532320, r29532321, r29532322);
        return r29532323;
}

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

Derivation

  1. Initial program 1.8

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

  2. Simplified0.2

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

  3. Final simplification0.2

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

Reproduce

herbie shell --seed 2019162 +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))))