Average Error: 2.0 → 0.2
Time: 47.6s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(\frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}, a, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(\frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}, a, x\right)
double f(double x, double y, double z, double t, double a) {
        double r24931316 = x;
        double r24931317 = y;
        double r24931318 = z;
        double r24931319 = r24931317 - r24931318;
        double r24931320 = t;
        double r24931321 = r24931320 - r24931318;
        double r24931322 = 1.0;
        double r24931323 = r24931321 + r24931322;
        double r24931324 = a;
        double r24931325 = r24931323 / r24931324;
        double r24931326 = r24931319 / r24931325;
        double r24931327 = r24931316 - r24931326;
        return r24931327;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r24931328 = z;
        double r24931329 = t;
        double r24931330 = r24931329 - r24931328;
        double r24931331 = 1.0;
        double r24931332 = r24931330 + r24931331;
        double r24931333 = r24931328 / r24931332;
        double r24931334 = y;
        double r24931335 = r24931334 / r24931332;
        double r24931336 = r24931333 - r24931335;
        double r24931337 = a;
        double r24931338 = x;
        double r24931339 = fma(r24931336, r24931337, r24931338);
        return r24931339;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original2.0
Target0.2
Herbie0.2
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 2.0

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

  2. Simplified0.2

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

  4. Applied div-sub0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019200 +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))))