Average Error: 4.8 → 2.9
Time: 2.9s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq -7.11017140165214 \cdot 10^{-291}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
t_0 := \frac{x \cdot y}{z}\\
\mathbf{if}\;t_0 \leq -\infty:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;t_0 \leq -7.11017140165214 \cdot 10^{-291}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\

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


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x y) z))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) z)))
   (if (<= t_0 (- INFINITY))
     (* x (/ y z))
     (if (<= t_0 -7.11017140165214e-291)
       (* (* x y) (/ 1.0 z))
       (/ x (/ z y))))))
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 (t_0 <= -((double) INFINITY)) {
		tmp = x * (y / z);
	} else if (t_0 <= -7.11017140165214e-291) {
		tmp = (x * y) * (1.0 / z);
	} else {
		tmp = x / (z / y);
	}
	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

Original4.8
Target5.2
Herbie2.9
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 x y) z) < -inf.0

    1. Initial program 11.7

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

    2. Applied *-un-lft-identity_binary6411.7

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

    3. Applied times-frac_binary640.1

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

    if -inf.0 < (/.f64 (*.f64 x y) z) < -7.1101714016521394e-291

    1. Initial program 0.4

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

    2. Applied div-inv_binary640.5

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

    if -7.1101714016521394e-291 < (/.f64 (*.f64 x y) z)

    1. Initial program 5.8

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

    2. Applied associate-/l*_binary644.5

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

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -\infty:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -7.11017140165214 \cdot 10^{-291}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))