Average Error: 10.6 → 1.9
Time: 10.6s
Precision: binary64
\[x + \frac{\left(y - z\right) \cdot t}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -2.0367736854431388 \cdot 10^{+80} \lor \neg \left(z \leq 2.8021645666856244 \cdot 10^{-103}\right):\\ \;\;\;\;x + t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t \cdot y - z \cdot t\right) \cdot \frac{1}{a - z}\\ \end{array}\]
x + \frac{\left(y - z\right) \cdot t}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \leq -2.0367736854431388 \cdot 10^{+80} \lor \neg \left(z \leq 2.8021645666856244 \cdot 10^{-103}\right):\\
\;\;\;\;x + t \cdot \frac{y - z}{a - z}\\

\mathbf{else}:\\
\;\;\;\;x + \left(t \cdot y - z \cdot t\right) \cdot \frac{1}{a - z}\\

\end{array}
(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y z) t) (- a z))))
↓
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.0367736854431388e+80) (not (<= z 2.8021645666856244e-103)))
   (+ x (* t (/ (- y z) (- a z))))
   (+ x (* (- (* t y) (* z t)) (/ 1.0 (- a z))))))
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) {
	double tmp;
	if ((z <= -2.0367736854431388e+80) || !(z <= 2.8021645666856244e-103)) {
		tmp = x + (t * ((y - z) / (a - z)));
	} else {
		tmp = x + (((t * y) - (z * t)) * (1.0 / (a - z)));
	}
	return tmp;
}

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

Original10.6
Target0.6
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.03677368544313876e80 or 2.8021645666856244e-103 < z

    1. Initial program 16.3

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

    2. Taylor expanded around 0 16.3

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

    3. Simplified0.3

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

    if -2.03677368544313876e80 < z < 2.8021645666856244e-103

    1. Initial program 3.9

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

    3. Applied *-un-lft-identity_binary64_178103.9

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

    4. Applied times-frac_binary64_178162.8

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

    5. Simplified2.8

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

    7. Applied div-inv_binary64_178072.8

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

    8. Applied associate-*r*_binary64_177503.9

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

    9. Simplified3.9

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

    11. Applied sub-neg_binary64_178033.9

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

    12. Applied distribute-rgt-in_binary64_177603.9

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

    13. Simplified3.9

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

    14. Simplified3.9

      \[\leadsto x + \left(t \cdot y + \color{blue}{\left(-t \cdot z\right)}\right) \cdot \frac{1}{a - z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.0367736854431388 \cdot 10^{+80} \lor \neg \left(z \leq 2.8021645666856244 \cdot 10^{-103}\right):\\ \;\;\;\;x + t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t \cdot y - z \cdot t\right) \cdot \frac{1}{a - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(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))))