Average Error: 6.7 → 0.7
Time: 4.3s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot z}{t} \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ t_1 \leq -\infty \lor \neg \left(t_1 \leq 1.4451404226086927 \cdot 10^{+306}\right) \end{array}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) - \frac{x \cdot z}{t}\\ \end{array} \]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\
t_1 \leq -\infty \lor \neg \left(t_1 \leq 1.4451404226086927 \cdot 10^{+306}\right)
\end{array}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\

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


\end{array}

(FPCore (x y z t) :precision binary64 (+ x (/ (* (- y x) z) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (if (let* ((t_1 (+ x (/ (* (- y x) z) t))))
       (or (<= t_1 (- INFINITY)) (not (<= t_1 1.4451404226086927e+306))))
   (fma (- y x) (/ z t) x)
   (- (+ x (/ (* y z) t)) (/ (* x z) 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 t_1 = x + (((y - x) * z) / t);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1.4451404226086927e+306)) {
		tmp = fma((y - x), (z / t), x);
	} else {
		tmp = (x + ((y * z) / t)) - ((x * z) / t);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.7
Target2.0
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -inf.0 or 1.4451404226086927e306 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

    1. Initial program 63.0

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

    2. Simplified0.4

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

    if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 1.4451404226086927e306

    1. Initial program 0.7

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

    2. Simplified2.4

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

    3. Taylor expanded in y around 0 0.7

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -\infty \lor \neg \left(x + \frac{\left(y - x\right) \cdot z}{t} \leq 1.4451404226086927 \cdot 10^{+306}\right):\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{y \cdot z}{t}\right) - \frac{x \cdot z}{t}\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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