Average Error: 1.4 → 1.3
Time: 7.9s
Precision: binary64
\[x + y \cdot \frac{z - t}{a - t} \]
↓
\[x + \frac{y}{\frac{a - t}{z - t}} \]
x + y \cdot \frac{z - t}{a - t}
↓
x + \frac{y}{\frac{a - t}{z - t}}

(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- a t)))))
↓
(FPCore (x y z t a) :precision binary64 (+ x (/ y (/ (- a t) (- z t)))))
double code(double x, double y, double z, double t, double a) {
	return x + (y * ((z - t) / (a - t)));
}
↓
double code(double x, double y, double z, double t, double a) {
	return x + (y / ((a - t) / (z - t)));
}

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

Original1.4
Target0.4
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

  1. Initial program 1.4

    \[x + y \cdot \frac{z - t}{a - t} \]

  2. Simplified1.4

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

  3. Taylor expanded in y around 0 10.7

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

  4. Simplified3.1

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

  5. Applied egg-rr1.3

    \[\leadsto x + \color{blue}{\frac{y}{\frac{a - t}{z - t}}} \]

  6. Final simplification1.3

    \[\leadsto x + \frac{y}{\frac{a - t}{z - t}} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

  (+ x (* y (/ (- z t) (- a t)))))