Average Error: 7.7 → 0.7
Time: 6.3s
Precision: binary64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
↓
\[\begin{array}{l} t_1 := z \cdot t - x\\ \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t_1}}{x + 1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{t_1}, \frac{x - \frac{x}{t_1}}{x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := t \cdot \left(x + 1\right)\\ \left(\frac{x}{x + 1} + \frac{y}{t_2}\right) - \frac{x}{z \cdot t_2} \end{array}\\ \end{array} \]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
t_1 := z \cdot t - x\\
\mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t_1}}{x + 1} \leq \infty:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{t_1}, \frac{x - \frac{x}{t_1}}{x + 1}\right)\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_2 := t \cdot \left(x + 1\right)\\
\left(\frac{x}{x + 1} + \frac{y}{t_2}\right) - \frac{x}{z \cdot t_2}
\end{array}\\


\end{array}

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)))
   (if (<= (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0)) INFINITY)
     (fma (/ y (+ x 1.0)) (/ z t_1) (/ (- x (/ x t_1)) (+ x 1.0)))
     (let* ((t_2 (* t (+ x 1.0))))
       (- (+ (/ x (+ x 1.0)) (/ y t_2)) (/ x (* z t_2)))))))
double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double tmp;
	if (((x + (((y * z) - x) / t_1)) / (x + 1.0)) <= ((double) INFINITY)) {
		tmp = fma((y / (x + 1.0)), (z / t_1), ((x - (x / t_1)) / (x + 1.0)));
	} else {
		double t_2 = t * (x + 1.0);
		tmp = ((x / (x + 1.0)) + (y / t_2)) - (x / (z * t_2));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.7
Target0.4
Herbie0.7
\[\frac{x + \left(\frac{y}{t - \frac{x}{z}} - \frac{x}{t \cdot z - x}\right)}{x + 1} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < +inf.0

    1. Initial program 5.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

    2. Taylor expanded in y around 0 5.0

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

    3. Simplified0.7

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

    if +inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

    1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

    2. Applied fma-neg_binary6464.0

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

    3. Applied div-inv_binary6464.0

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

    4. Taylor expanded in t around inf 0.1

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

    5. Simplified0.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{z \cdot t - x}, \frac{x - \frac{x}{z \cdot t - x}}{x + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\right) - \frac{x}{z \cdot \left(t \cdot \left(x + 1\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))