Average Error: 10.7 → 1.2
Time: 2.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{z - a}\]
↓
\[\frac{y}{\frac{z - a}{z - t}} + x\]
x + \frac{y \cdot \left(z - t\right)}{z - a}
↓
\frac{y}{\frac{z - a}{z - t}} + 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 ((y / ((z - a) / (z - t))) + 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

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Target

Original10.7
Target1.2
Herbie1.2
\[x + \frac{y}{\frac{z - a}{z - t}}\]

Derivation

  1. Initial program 10.7

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

  2. Simplified3.2

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

  4. Applied fma-udef3.2

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

  6. Applied div-inv3.2

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

  7. Applied associate-*l*1.4

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

  8. Simplified1.3

    \[\leadsto y \cdot \color{blue}{\frac{z - t}{z - a}} + x\]
  9. Using strategy rm

  10. Applied clear-num1.4

    \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{z - a}{z - t}}} + x\]

  11. Applied un-div-inv1.2

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

  12. Final simplification1.2

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(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))))