Average Error: 12.3 → 2.2
Time: 2.3s
Precision: binary64
\[\frac{x \cdot \left(y - z\right)}{y} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.569308653238193 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;x \leq -4.706085207486899 \cdot 10^{-289}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array} \]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.569308653238193 \cdot 10^{-19}:\\
\;\;\;\;x \cdot \frac{y - z}{y}\\

\mathbf{elif}\;x \leq -4.706085207486899 \cdot 10^{-289}:\\
\;\;\;\;x - \frac{x \cdot z}{y}\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.569308653238193e-19)
   (* x (/ (- y z) y))
   (if (<= x -4.706085207486899e-289)
     (- x (/ (* x z) y))
     (/ x (/ y (- y z))))))
double code(double x, double y, double z) {
	return (x * (y - z)) / y;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.569308653238193e-19) {
		tmp = x * ((y - z) / y);
	} else if (x <= -4.706085207486899e-289) {
		tmp = x - ((x * z) / y);
	} else {
		tmp = x / (y / (y - z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.3
Target3.1
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.56930865323819311e-19

    1. Initial program 21.3

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

    2. Applied *-un-lft-identity_binary6421.3

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

    3. Applied times-frac_binary640.1

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

    4. Simplified0.1

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

    if -1.56930865323819311e-19 < x < -4.70608520748689864e-289

    1. Initial program 4.3

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

    2. Taylor expanded in y around 0 2.1

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

    if -4.70608520748689864e-289 < x

    1. Initial program 12.4

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

    2. Applied associate-/l*_binary643.3

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.569308653238193 \cdot 10^{-19}:\\ \;\;\;\;x \cdot \frac{y - z}{y}\\ \mathbf{elif}\;x \leq -4.706085207486899 \cdot 10^{-289}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

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