Average Error: 9.4 → 1.3
Time: 6.7s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{a - t} \]
↓
\[\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{t}{a - t}, x\right) \]
x + \frac{y \cdot \left(z - t\right)}{a - t}
↓
\mathsf{fma}\left(y, \frac{z}{a - t} - \frac{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 (- a t)) (/ 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 / (a - t)) - (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

Original9.4
Target1.2
Herbie1.3
\[x + \frac{y}{\frac{a - t}{z - t}} \]

Derivation

  1. Initial program 9.4

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

  2. Simplified1.3

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

  3. Taylor expanded in z around 0 1.3

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

  4. Applied sub-neg_binary641.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- a t) (- z t))))

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