Average Error: 1.9 → 0.2
Time: 21.2s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \frac{z}{\left(t - z\right) + 1} - \frac{y}{\left(t - z\right) + 1}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r437111 = x;
        double r437112 = y;
        double r437113 = z;
        double r437114 = r437112 - r437113;
        double r437115 = t;
        double r437116 = r437115 - r437113;
        double r437117 = 1.0;
        double r437118 = r437116 + r437117;
        double r437119 = a;
        double r437120 = r437118 / r437119;
        double r437121 = r437114 / r437120;
        double r437122 = r437111 - r437121;
        return r437122;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r437123 = a;
        double r437124 = z;
        double r437125 = t;
        double r437126 = r437125 - r437124;
        double r437127 = 1.0;
        double r437128 = r437126 + r437127;
        double r437129 = r437124 / r437128;
        double r437130 = y;
        double r437131 = r437130 / r437128;
        double r437132 = r437129 - r437131;
        double r437133 = x;
        double r437134 = fma(r437123, r437132, r437133);
        return r437134;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.9
Target0.2
Herbie0.2
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 1.9

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 +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))))