Average Error: 2.1 → 0.2
Time: 9.7s
Precision: 64
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\]
↓
\[\mathsf{fma}\left(1, x, -\frac{a}{\sqrt{1}} \cdot \frac{\frac{y - z}{\left(t - z\right) + 1}}{\sqrt{1}}\right) + \mathsf{fma}\left(-\frac{a}{\sqrt{1}}, \frac{\frac{y - z}{\left(t - z\right) + 1}}{\sqrt{1}}, \frac{a}{\sqrt{1}} \cdot \frac{\frac{y - z}{\left(t - z\right) + 1}}{\sqrt{1}}\right)\]
x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}
↓
\mathsf{fma}\left(1, x, -\frac{a}{\sqrt{1}} \cdot \frac{\frac{y - z}{\left(t - z\right) + 1}}{\sqrt{1}}\right) + \mathsf{fma}\left(-\frac{a}{\sqrt{1}}, \frac{\frac{y - z}{\left(t - z\right) + 1}}{\sqrt{1}}, \frac{a}{\sqrt{1}} \cdot \frac{\frac{y - z}{\left(t - z\right) + 1}}{\sqrt{1}}\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) sqrt(1.0)))) * ((double) (((double) (((double) (y - z)) / ((double) (((double) (t - z)) + 1.0)))) / ((double) sqrt(1.0)))))))))) + ((double) fma(((double) -(((double) (a / ((double) sqrt(1.0)))))), ((double) (((double) (((double) (y - z)) / ((double) (((double) (t - z)) + 1.0)))) / ((double) sqrt(1.0)))), ((double) (((double) (a / ((double) sqrt(1.0)))) * ((double) (((double) (((double) (y - z)) / ((double) (((double) (t - z)) + 1.0)))) / ((double) sqrt(1.0))))))))));
}

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

Original2.1
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. Using strategy rm

  3. Applied div-inv2.1

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

  4. Applied associate-/r*0.3

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

  6. Applied add-sqr-sqrt0.3

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

  7. Applied associate-/l*0.3

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

  8. Applied associate-/r/0.2

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

  9. Applied *-un-lft-identity0.2

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

  10. Applied prod-diff0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2020113 +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))))