Average Error: 6.3 → 2.0
Time: 2.0s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{x \cdot y}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;x \cdot y \leq -1.9971221197923187 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 7.340460290394229 \cdot 10^{-198}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2.5921356819505716 \cdot 10^{+209}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
t_0 := \frac{x \cdot y}{z}\\
\mathbf{if}\;x \cdot y \leq -1.9971221197923187 \cdot 10^{-165}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x \cdot y \leq 7.340460290394229 \cdot 10^{-198}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;x \cdot y \leq 2.5921356819505716 \cdot 10^{+209}:\\
\;\;\;\;t_0\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x y) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= (* x y) -1.9971221197923187e-165)
     t_0
     (if (<= (* x y) 7.340460290394229e-198)
       (/ x (/ z y))
       (if (<= (* x y) 2.5921356819505716e+209) t_0 (* x (/ y z)))))))
double code(double x, double y, double z) {
	return (x * y) / z;
}
↓
double code(double x, double y, double z) {
	double t_0 = (x * y) / z;
	double tmp;
	if ((x * y) <= -1.9971221197923187e-165) {
		tmp = t_0;
	} else if ((x * y) <= 7.340460290394229e-198) {
		tmp = x / (z / y);
	} else if ((x * y) <= 2.5921356819505716e+209) {
		tmp = t_0;
	} else {
		tmp = x * (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

Original6.3
Target6.3
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x y) < -1.9971221197923187e-165 or 7.3404602903942289e-198 < (*.f64 x y) < 2.5921356819505716e209

    1. Initial program 2.8

      \[\frac{x \cdot y}{z} \]

    if -1.9971221197923187e-165 < (*.f64 x y) < 7.3404602903942289e-198

    1. Initial program 9.8

      \[\frac{x \cdot y}{z} \]

    2. Applied associate-/l*_binary640.6

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

    if 2.5921356819505716e209 < (*.f64 x y)

    1. Initial program 28.5

      \[\frac{x \cdot y}{z} \]

    2. Applied *-un-lft-identity_binary6428.5

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

    3. Applied times-frac_binary640.6

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

    4. Simplified0.6

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.9971221197923187 \cdot 10^{-165}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \leq 7.340460290394229 \cdot 10^{-198}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \leq 2.5921356819505716 \cdot 10^{+209}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))