Average Error: 2.1 → 0.3
Time: 33.4s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \left(z - y\right) \cdot \frac{1}{1 + \left(t - z\right)}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \left(z - y\right) \cdot \frac{1}{1 + \left(t - z\right)}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r23765968 = x;
        double r23765969 = y;
        double r23765970 = z;
        double r23765971 = r23765969 - r23765970;
        double r23765972 = t;
        double r23765973 = r23765972 - r23765970;
        double r23765974 = 1.0;
        double r23765975 = r23765973 + r23765974;
        double r23765976 = a;
        double r23765977 = r23765975 / r23765976;
        double r23765978 = r23765971 / r23765977;
        double r23765979 = r23765968 - r23765978;
        return r23765979;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r23765980 = a;
        double r23765981 = z;
        double r23765982 = y;
        double r23765983 = r23765981 - r23765982;
        double r23765984 = 1.0;
        double r23765985 = 1.0;
        double r23765986 = t;
        double r23765987 = r23765986 - r23765981;
        double r23765988 = r23765985 + r23765987;
        double r23765989 = r23765984 / r23765988;
        double r23765990 = r23765983 * r23765989;
        double r23765991 = x;
        double r23765992 = fma(r23765980, r23765990, r23765991);
        return r23765992;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original2.1
Target0.3
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

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

  4. Applied div-inv0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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))))