Average Error: 1.9 → 0.3
Time: 17.1s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(a, \left(z - y\right) \cdot \frac{1}{\left(t - z\right) + 1}, x\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(a, \left(z - y\right) \cdot \frac{1}{\left(t - z\right) + 1}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r796673 = x;
        double r796674 = y;
        double r796675 = z;
        double r796676 = r796674 - r796675;
        double r796677 = t;
        double r796678 = r796677 - r796675;
        double r796679 = 1.0;
        double r796680 = r796678 + r796679;
        double r796681 = a;
        double r796682 = r796680 / r796681;
        double r796683 = r796676 / r796682;
        double r796684 = r796673 - r796683;
        return r796684;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r796685 = a;
        double r796686 = z;
        double r796687 = y;
        double r796688 = r796686 - r796687;
        double r796689 = 1.0;
        double r796690 = t;
        double r796691 = r796690 - r796686;
        double r796692 = 1.0;
        double r796693 = r796691 + r796692;
        double r796694 = r796689 / r796693;
        double r796695 = r796688 * r796694;
        double r796696 = x;
        double r796697 = fma(r796685, r796695, r796696);
        return r796697;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.9
Target0.3
Herbie0.3
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 1.9

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

  2. Simplified0.3

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

  4. Applied div-inv0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2020042 +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))))