Average Error: 2.2 → 1.0
Time: 3.4s
Precision: binary64
\[x + \left(y - x\right) \cdot \frac{z}{t} \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1.5422195534991153 \cdot 10^{-205} \lor \neg \left(\frac{z}{t} \leq 0\right) \land \frac{z}{t} \leq 8.323090403460942 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{z \cdot y}{t}\right) - \frac{z \cdot x}{t}\\ \end{array} \]
x + \left(y - x\right) \cdot \frac{z}{t}
↓
\begin{array}{l}
\mathbf{if}\;\frac{z}{t} \leq -1.5422195534991153 \cdot 10^{-205} \lor \neg \left(\frac{z}{t} \leq 0\right) \land \frac{z}{t} \leq 8.323090403460942 \cdot 10^{+171}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\

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


\end{array}

(FPCore (x y z t) :precision binary64 (+ x (* (- y x) (/ z t))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ z t) -1.5422195534991153e-205)
         (and (not (<= (/ z t) 0.0)) (<= (/ z t) 8.323090403460942e+171)))
   (fma (- y x) (/ z t) x)
   (- (+ x (/ (* z y) t)) (/ (* z x) t))))
double code(double x, double y, double z, double t) {
	return x + ((y - x) * (z / t));
}
↓
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -1.5422195534991153e-205) || (!((z / t) <= 0.0) && ((z / t) <= 8.323090403460942e+171))) {
		tmp = fma((y - x), (z / t), x);
	} else {
		tmp = (x + ((z * y) / t)) - ((z * x) / t);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original2.2
Target2.3
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} < -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 z t) < -1.5422195534991153e-205 or 0.0 < (/.f64 z t) < 8.32309040346094174e171

    1. Initial program 1.2

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

    2. Simplified1.2

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

    if -1.5422195534991153e-205 < (/.f64 z t) < 0.0 or 8.32309040346094174e171 < (/.f64 z t)

    1. Initial program 4.8

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

    2. Simplified4.8

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

    3. Taylor expanded in y around 0 0.6

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

    4. Applied sub-neg_binary640.6

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

    5. Simplified0.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1.5422195534991153 \cdot 10^{-205} \lor \neg \left(\frac{z}{t} \leq 0\right) \land \frac{z}{t} \leq 8.323090403460942 \cdot 10^{+171}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{z \cdot y}{t}\right) - \frac{z \cdot x}{t}\\ \end{array} \]

Reproduce

herbie shell --seed 2021307 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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