Average Error: 1.9 → 0.3
Time: 4.1s
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 code(double x, double y, double z, double t, double a) {
	return (x - ((y - z) / (((t - z) + 1.0) / a)));
}
↓
double code(double x, double y, double z, double t, double a) {
	return ((a * ((z - y) / ((t - z) + 1.0))) + x);
}

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

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. Simplified1.7

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

  4. Applied fma-udef1.7

    \[\leadsto \color{blue}{\frac{a}{\left(t - z\right) + 1} \cdot \left(z - y\right) + x}\]
  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*0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020102 +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))))