Average Error: 2.0 → 0.2
Time: 5.5s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(1, x, -\left(-a\right) \cdot \frac{y - z}{-\left(\left(t - z\right) + 1\right)}\right) + \frac{y - z}{-\left(\left(t - z\right) + 1\right)} \cdot \left(a + \left(-a\right)\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(1, x, -\left(-a\right) \cdot \frac{y - z}{-\left(\left(t - z\right) + 1\right)}\right) + \frac{y - z}{-\left(\left(t - z\right) + 1\right)} \cdot \left(a + \left(-a\right)\right)
double code(double x, double y, double z, double t, double a) {
	return ((double) (x - ((double) (((double) (y - z)) / ((double) (((double) (((double) (t - z)) + 1.0)) / a))))));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) fma(1.0, x, ((double) -(((double) (((double) -(a)) * ((double) (((double) (y - z)) / ((double) -(((double) (((double) (t - z)) + 1.0)))))))))))) + ((double) (((double) (((double) (y - z)) / ((double) -(((double) (((double) (t - z)) + 1.0)))))) * ((double) (a + ((double) -(a))))))));
}

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. Using strategy rm

  3. Applied frac-2neg2.0

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

  4. Applied associate-/r/0.2

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

  5. Applied add-sqr-sqrt32.0

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

  6. Applied prod-diff32.0

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

  7. Simplified0.2

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020114 +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))))