Average Error: 7.2 → 0.7
Time: 7.6s
Precision: binary64
\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
↓
\[\begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x - \frac{x}{t_1}}{x + 1}\\ \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t_1}}{x + 1} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x + 1}, \frac{z}{t_1}, t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\ \end{array} \]
\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}
↓
\begin{array}{l}
t_1 := z \cdot t - x\\
t_2 := \frac{x - \frac{x}{t_1}}{x + 1}\\
\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}, t_2\right)\\

\mathbf{else}:\\
\;\;\;\;t_2 + \frac{y}{\mathsf{fma}\left(x, t, t\right)}\\


\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)) (t_2 (/ (- x (/ x t_1)) (+ x 1.0))))
   (if (<= (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0)) INFINITY)
     (fma (/ y (+ x 1.0)) (/ z t_1) t_2)
     (+ t_2 (/ y (fma x t t))))))
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 t_2 = (x - (x / t_1)) / (x + 1.0);
	double tmp;
	if (((x + (((y * z) - x) / t_1)) / (x + 1.0)) <= ((double) INFINITY)) {
		tmp = fma((y / (x + 1.0)), (z / t_1), t_2);
	} else {
		tmp = t_2 + (y / fma(x, t, t));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.2
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 4.7

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

    2. Taylor expanded in y around 0 4.7

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

    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. Taylor expanded in y around 0 64.0

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

    3. Simplified30.3

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

    4. Applied fma-udef_binary6430.3

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

    5. Simplified64.0

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

    6. Taylor expanded in z around inf 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2022068 
(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)))