Average Error: 10.0 → 0.2
Time: 5.0s
Precision: binary64
\[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ t_0 \leq -4.083961478419301 \cdot 10^{-70} \lor \neg \left(t_0 \leq 1.5305641720450072 \cdot 10^{+32}\right) \end{array}:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\ \end{array} \]
\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_0 := \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\
t_0 \leq -4.083961478419301 \cdot 10^{-70} \lor \neg \left(t_0 \leq 1.5305641720450072 \cdot 10^{+32}\right)
\end{array}:\\
\;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (let* ((t_0 (/ (* x (+ (- y z) 1.0)) z)))
       (or (<= t_0 -4.083961478419301e-70)
           (not (<= t_0 1.5305641720450072e+32))))
   (- (* (+ y 1.0) (/ x z)) x)
   (- (/ (fma y x x) z) x)))
double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}
↓
double code(double x, double y, double z) {
	double t_0 = (x * ((y - z) + 1.0)) / z;
	double tmp;
	if ((t_0 <= -4.083961478419301e-70) || !(t_0 <= 1.5305641720450072e+32)) {
		tmp = ((y + 1.0) * (x / z)) - x;
	} else {
		tmp = (fma(y, x, x) / z) - x;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original10.0
Target0.5
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + y\right) \cdot \frac{x}{z} - x\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < -4.08396147841930076e-70 or 1.5305641720450072e32 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z)

    1. Initial program 15.6

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

    2. Simplified15.6

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

    3. Taylor expanded in y around 0 5.5

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

    4. Simplified0.2

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

    5. Applied add-cube-cbrt_binary640.9

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)}} - x \]

    6. Taylor expanded in y around 0 5.5

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

    7. Simplified0.2

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

    if -4.08396147841930076e-70 < (/.f64 (*.f64 x (+.f64 (-.f64 y z) 1)) z) < 1.5305641720450072e32

    1. Initial program 0.1

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

    2. Simplified0.1

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

    3. Taylor expanded in y around 0 0.1

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

    4. Simplified3.7

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

    5. Applied add-cube-cbrt_binary644.6

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right) - \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} \]

    6. Applied add-cube-cbrt_binary645.0

      \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)}} - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x} \]

    7. Applied prod-diff_binary645.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)}, \sqrt[3]{\mathsf{fma}\left(\frac{x}{z}, y, \frac{x}{z}\right)}, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)} \]

    8. Simplified0.1

      \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\right)} + \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \]

    9. Simplified0.1

      \[\leadsto \left(\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\right) + \color{blue}{0} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq -4.083961478419301 \cdot 10^{-70} \lor \neg \left(\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \leq 1.5305641720450072 \cdot 10^{+32}\right):\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, x\right)}{z} - x\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

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