Average Error: 2.1 → 0.2
Time: 18.6s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \frac{1}{\frac{\left(t - z\right) + 1}{z}} - \frac{y}{\left(t - z\right) + 1}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \frac{1}{\frac{\left(t - z\right) + 1}{z}} - \frac{y}{\left(t - z\right) + 1}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r450112 = x;
        double r450113 = y;
        double r450114 = z;
        double r450115 = r450113 - r450114;
        double r450116 = t;
        double r450117 = r450116 - r450114;
        double r450118 = 1.0;
        double r450119 = r450117 + r450118;
        double r450120 = a;
        double r450121 = r450119 / r450120;
        double r450122 = r450115 / r450121;
        double r450123 = r450112 - r450122;
        return r450123;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r450124 = a;
        double r450125 = 1.0;
        double r450126 = t;
        double r450127 = z;
        double r450128 = r450126 - r450127;
        double r450129 = 1.0;
        double r450130 = r450128 + r450129;
        double r450131 = r450130 / r450127;
        double r450132 = r450125 / r450131;
        double r450133 = y;
        double r450134 = r450133 / r450130;
        double r450135 = r450132 - r450134;
        double r450136 = x;
        double r450137 = fma(r450124, r450135, r450136);
        return r450137;
}

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

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

  4. Applied div-sub0.2

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

  6. Applied clear-num0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1) a))))