Average Error: 7.7 → 0.2
Time: 3.4s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}} \]
↓
\[\begin{array}{l} t_0 := 1 - \frac{y}{z}\\ t_1 := \frac{x + y}{t_0}\\ \mathbf{if}\;t_1 \leq -8.573458850050232 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\ \mathbf{elif}\;t_1 \leq 2.403585899629984 \cdot 10^{-298}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{t_0}\\ \end{array} \]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
t_0 := 1 - \frac{y}{z}\\
t_1 := \frac{x + y}{t_0}\\
\mathbf{if}\;t_1 \leq -8.573458850050232 \cdot 10^{-275}:\\
\;\;\;\;\frac{x}{t_0} + \frac{y}{t_0}\\

\mathbf{elif}\;t_1 \leq 2.403585899629984 \cdot 10^{-298}:\\
\;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x + y\right) \cdot \frac{1}{t_0}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y z))) (t_1 (/ (+ x y) t_0)))
   (if (<= t_1 -8.573458850050232e-275)
     (+ (/ x t_0) (/ y t_0))
     (if (<= t_1 2.403585899629984e-298)
       (- (fma z (/ x y) z))
       (* (+ x y) (/ 1.0 t_0))))))
double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}
↓
double code(double x, double y, double z) {
	double t_0 = 1.0 - (y / z);
	double t_1 = (x + y) / t_0;
	double tmp;
	if (t_1 <= -8.573458850050232e-275) {
		tmp = (x / t_0) + (y / t_0);
	} else if (t_1 <= 2.403585899629984e-298) {
		tmp = -fma(z, (x / y), z);
	} else {
		tmp = (x + y) * (1.0 / t_0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original7.7
Target4.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < -8.573458850050232e-275

    1. Initial program 0.1

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

    2. Taylor expanded in x around 0 0.1

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

    if -8.573458850050232e-275 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z))) < 2.4035858996299839e-298

    1. Initial program 56.7

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

    2. Taylor expanded in x around 0 56.7

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

    3. Taylor expanded in z around 0 1.1

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

    4. Simplified1.1

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

    if 2.4035858996299839e-298 < (/.f64 (+.f64 x y) (-.f64 1 (/.f64 y z)))

    1. Initial program 0.1

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

    2. Taylor expanded in x around 0 0.1

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

    3. Applied div-inv_binary640.1

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

    4. Applied div-inv_binary640.1

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

    5. Applied distribute-rgt-out_binary640.2

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

    6. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \leq -8.573458850050232 \cdot 10^{-275}:\\ \;\;\;\;\frac{x}{1 - \frac{y}{z}} + \frac{y}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{x + y}{1 - \frac{y}{z}} \leq 2.403585899629984 \cdot 10^{-298}:\\ \;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1}{1 - \frac{y}{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

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