Average Error: 5.2 → 1.2
Time: 2.2s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq -1.5845934776653614 \cdot 10^{-284}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 1.0283373703112465 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;t_0 \leq 1.8018552614952002 \cdot 10^{+180}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
t_0 := \frac{x \cdot y}{z}\\
\mathbf{if}\;t_0 \leq -\infty:\\
\;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\

\mathbf{elif}\;t_0 \leq -1.5845934776653614 \cdot 10^{-284}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;t_0 \leq 1.0283373703112465 \cdot 10^{-308}:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;t_0 \leq 1.8018552614952002 \cdot 10^{+180}:\\
\;\;\;\;\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))
     (/ 1.0 (* (/ 1.0 x) (/ z y)))
     (if (<= t_0 -1.5845934776653614e-284)
       t_0
       (if (<= t_0 1.0283373703112465e-308)
         (* x (/ y z))
         (if (<= t_0 1.8018552614952002e+180)
           (* (* 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 = 1.0 / ((1.0 / x) * (z / y));
	} else if (t_0 <= -1.5845934776653614e-284) {
		tmp = t_0;
	} else if (t_0 <= 1.0283373703112465e-308) {
		tmp = x * (y / z);
	} else if (t_0 <= 1.8018552614952002e+180) {
		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

Original5.2
Target5.0
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 5 regimes

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

    1. Initial program 12.0

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

    2. Applied clear-num_binary6412.0

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

    3. Applied *-un-lft-identity_binary6412.0

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

    4. Applied times-frac_binary640.1

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

    if -inf.0 < (/.f64 (*.f64 x y) z) < -1.5845934776653614e-284

    1. Initial program 0.5

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

    if -1.5845934776653614e-284 < (/.f64 (*.f64 x y) z) < 1.0283373703112465e-308

    1. Initial program 11.3

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

    2. Applied *-un-lft-identity_binary6411.3

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

    3. Applied times-frac_binary640.9

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

    4. Simplified0.9

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

    if 1.0283373703112465e-308 < (/.f64 (*.f64 x y) z) < 1.80185526149520024e180

    1. Initial program 0.6

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

    2. Applied div-inv_binary640.7

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

    if 1.80185526149520024e180 < (/.f64 (*.f64 x y) z)

    1. Initial program 8.2

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

    2. Applied associate-/l*_binary644.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}} \]
  3. Recombined 5 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{1}{x} \cdot \frac{z}{y}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq -1.5845934776653614 \cdot 10^{-284}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 1.0283373703112465 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 1.8018552614952002 \cdot 10^{+180}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))