Average Error: 6.4 → 1.3
Time: 5.4s
Precision: binary64
\[x + \frac{\left(y - x\right) \cdot z}{t} \]
↓
\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq 1.7047806052311535 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;t_1 \leq 4.139611946340411 \cdot 10^{+303}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\begin{array}{l}
t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\
\mathbf{if}\;t_1 \leq 1.7047806052311535 \cdot 10^{-214}:\\
\;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\

\mathbf{elif}\;t_1 \leq 4.139611946340411 \cdot 10^{+303}:\\
\;\;\;\;t_1\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\


\end{array}

(FPCore (x y z t) :precision binary64 (+ x (/ (* (- y x) z) t)))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) z) t))))
   (if (<= t_1 1.7047806052311535e-214)
     (fma (- y x) (/ z t) x)
     (if (<= t_1 4.139611946340411e+303) t_1 (+ x (/ (- y x) (/ t z)))))))
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 <= 1.7047806052311535e-214) {
		tmp = fma((y - x), (z / t), x);
	} else if (t_1 <= 4.139611946340411e+303) {
		tmp = t_1;
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.4
Target2.1
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 1.70478060523115349e-214

    1. Initial program 6.4

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

    2. Simplified2.0

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

    if 1.70478060523115349e-214 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 4.1396119463404107e303

    1. Initial program 0.4

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

    if 4.1396119463404107e303 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

    1. Initial program 60.3

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

    2. Applied associate-/l*_binary640.5

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq 1.7047806052311535 \cdot 10^{-214}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq 4.139611946340411 \cdot 10^{+303}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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)))