Average Error: 1.2 → 1.2
Time: 7.8s
Precision: binary64
\[x + y \cdot \frac{z - t}{a - t} \]
↓
\[\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right) \]
x + y \cdot \frac{z - t}{a - t}
↓
\mathsf{fma}\left(y, \frac{z - t}{a - t}, x\right)

(FPCore (x y z t a) :precision binary64 (+ x (* y (/ (- z t) (- a t)))))
↓
(FPCore (x y z t a) :precision binary64 (fma y (/ (- z t) (- a t)) x))
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 fma(y, ((z - t) / (a - t)), x);
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original1.2
Target0.5
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

  1. Initial program 1.2

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

  2. Simplified1.2

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

  3. Final simplification1.2

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

Reproduce

herbie shell --seed 2022068 
(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)))))