Average Error: 2.1 → 0.2
Time: 18.5s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \frac{1}{\frac{\left(t - z\right) + 1}{z - y}}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \frac{1}{\frac{\left(t - z\right) + 1}{z - y}}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r515872 = x;
        double r515873 = y;
        double r515874 = z;
        double r515875 = r515873 - r515874;
        double r515876 = t;
        double r515877 = r515876 - r515874;
        double r515878 = 1.0;
        double r515879 = r515877 + r515878;
        double r515880 = a;
        double r515881 = r515879 / r515880;
        double r515882 = r515875 / r515881;
        double r515883 = r515872 - r515882;
        return r515883;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r515884 = a;
        double r515885 = 1.0;
        double r515886 = t;
        double r515887 = z;
        double r515888 = r515886 - r515887;
        double r515889 = 1.0;
        double r515890 = r515888 + r515889;
        double r515891 = y;
        double r515892 = r515887 - r515891;
        double r515893 = r515890 / r515892;
        double r515894 = r515885 / r515893;
        double r515895 = x;
        double r515896 = fma(r515884, r515894, r515895);
        return r515896;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

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

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 clear-num0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 +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))))