Average Error: 2.0 → 0.3
Time: 20.5s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}\]
↓
\[\mathsf{fma}\left(\left(z - y\right) \cdot \frac{1}{\left(t + 1.0\right) - z}, a, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1.0}{a}}
↓
\mathsf{fma}\left(\left(z - y\right) \cdot \frac{1}{\left(t + 1.0\right) - z}, a, x\right)
double f(double x, double y, double z, double t, double a) {
        double r22852887 = x;
        double r22852888 = y;
        double r22852889 = z;
        double r22852890 = r22852888 - r22852889;
        double r22852891 = t;
        double r22852892 = r22852891 - r22852889;
        double r22852893 = 1.0;
        double r22852894 = r22852892 + r22852893;
        double r22852895 = a;
        double r22852896 = r22852894 / r22852895;
        double r22852897 = r22852890 / r22852896;
        double r22852898 = r22852887 - r22852897;
        return r22852898;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r22852899 = z;
        double r22852900 = y;
        double r22852901 = r22852899 - r22852900;
        double r22852902 = 1.0;
        double r22852903 = t;
        double r22852904 = 1.0;
        double r22852905 = r22852903 + r22852904;
        double r22852906 = r22852905 - r22852899;
        double r22852907 = r22852902 / r22852906;
        double r22852908 = r22852901 * r22852907;
        double r22852909 = a;
        double r22852910 = x;
        double r22852911 = fma(r22852908, r22852909, r22852910);
        return r22852911;
}

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
Herbie0.3
\[x - \frac{y - z}{\left(t - z\right) + 1.0} \cdot a\]

Derivation

  1. Initial program 2.0

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

  2. Simplified0.2

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

  4. Applied div-inv0.3

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z - y\right) \cdot \frac{1}{\left(1.0 + t\right) - z}}, a, x\right)\]

  5. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\left(z - y\right) \cdot \frac{1}{\left(t + 1.0\right) - z}, a, x\right)\]

Reproduce

herbie shell --seed 2019158 +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))))