Average Error: 11.2 → 1.2
Time: 19.4s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z} \]
↓
\[\left(t \cdot \frac{y}{a - z} - t \cdot \frac{z}{a - z}\right) + x \]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\left(t \cdot \frac{y}{a - z} - t \cdot \frac{z}{a - z}\right) + x

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

Original11.2
Target0.6
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

  1. Initial program 11.2

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

  2. Simplified2.7

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

  3. Applied fma-udef_binary642.7

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

  4. Simplified1.2

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

  5. Applied sub-neg_binary641.2

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

  6. Applied distribute-lft-in_binary641.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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