Average Error: 1.9 → 0.3
Time: 17.2s
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 r770159 = x;
        double r770160 = y;
        double r770161 = z;
        double r770162 = r770160 - r770161;
        double r770163 = t;
        double r770164 = r770163 - r770161;
        double r770165 = 1.0;
        double r770166 = r770164 + r770165;
        double r770167 = a;
        double r770168 = r770166 / r770167;
        double r770169 = r770162 / r770168;
        double r770170 = r770159 - r770169;
        return r770170;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r770171 = a;
        double r770172 = z;
        double r770173 = y;
        double r770174 = r770172 - r770173;
        double r770175 = 1.0;
        double r770176 = t;
        double r770177 = r770176 - r770172;
        double r770178 = 1.0;
        double r770179 = r770177 + r770178;
        double r770180 = r770175 / r770179;
        double r770181 = r770174 * r770180;
        double r770182 = x;
        double r770183 = fma(r770171, r770181, r770182);
        return r770183;
}

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))))