Average Error: 2.0 → 0.2
Time: 3.8s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[a \cdot \frac{z - y}{\left(t - z\right) + 1} + x\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
a \cdot \frac{z - y}{\left(t - z\right) + 1} + x
double f(double x, double y, double z, double t, double a) {
        double r567976 = x;
        double r567977 = y;
        double r567978 = z;
        double r567979 = r567977 - r567978;
        double r567980 = t;
        double r567981 = r567980 - r567978;
        double r567982 = 1.0;
        double r567983 = r567981 + r567982;
        double r567984 = a;
        double r567985 = r567983 / r567984;
        double r567986 = r567979 / r567985;
        double r567987 = r567976 - r567986;
        return r567987;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r567988 = a;
        double r567989 = z;
        double r567990 = y;
        double r567991 = r567989 - r567990;
        double r567992 = t;
        double r567993 = r567992 - r567989;
        double r567994 = 1.0;
        double r567995 = r567993 + r567994;
        double r567996 = r567991 / r567995;
        double r567997 = r567988 * r567996;
        double r567998 = x;
        double r567999 = r567997 + r567998;
        return r567999;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.0
Target0.2
Herbie0.2
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a\]

Derivation

  1. Initial program 2.0

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

  2. Simplified1.8

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

  4. Applied fma-udef1.8

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

  6. Applied div-inv1.8

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

  7. Applied associate-*l*0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020001 +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))))