Average Error: 2.1 → 0.2
Time: 22.1s
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 r391570 = x;
        double r391571 = y;
        double r391572 = z;
        double r391573 = r391571 - r391572;
        double r391574 = t;
        double r391575 = r391574 - r391572;
        double r391576 = 1.0;
        double r391577 = r391575 + r391576;
        double r391578 = a;
        double r391579 = r391577 / r391578;
        double r391580 = r391573 / r391579;
        double r391581 = r391570 - r391580;
        return r391581;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r391582 = a;
        double r391583 = z;
        double r391584 = t;
        double r391585 = r391584 - r391583;
        double r391586 = 1.0;
        double r391587 = r391585 + r391586;
        double r391588 = r391583 / r391587;
        double r391589 = y;
        double r391590 = r391589 / r391587;
        double r391591 = r391588 - r391590;
        double r391592 = x;
        double r391593 = fma(r391582, r391591, r391592);
        return r391593;
}

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. 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 2019322 +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))))