Average Error: 9.3 → 1.1
Time: 4.9s
Precision: binary64
\[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
↓
\[\mathsf{fma}\left(y, \frac{z}{z - a} - \frac{t}{z - a}, x\right) \]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\mathsf{fma}\left(y, \frac{z}{z - a} - \frac{t}{z - a}, x\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original9.3
Target1.1
Herbie1.1
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation

  1. Initial program 9.3

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

  2. Simplified1.1

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

  3. Applied clear-num_binary641.2

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

  4. Applied div-inv_binary641.2

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

  5. Applied add-cube-cbrt_binary641.2

    \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(z - a\right) \cdot \frac{1}{z - t}}, x\right) \]

  6. Applied times-frac_binary641.2

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{z - a} \cdot \frac{\sqrt[3]{1}}{\frac{1}{z - t}}}, x\right) \]

  7. Simplified1.2

    \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1}{z - a}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{z - t}}, x\right) \]

  8. Simplified1.2

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

  9. Taylor expanded in t around 0 1.1

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

  10. Final simplification1.1

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

Reproduce

herbie shell --seed 2022088 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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