Average Error: 6.5 → 1.0
Time: 2.9s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{z} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.604516891076534 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -1.3010356389337067 \cdot 10^{-72}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \leq 2.833726387088633 \cdot 10^{-265}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.6279132317167615 \cdot 10^{+190}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \end{array} \]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \leq -2.604516891076534 \cdot 10^{+217}:\\
\;\;\;\;y \cdot \frac{x}{z}\\

\mathbf{elif}\;x \cdot y \leq -1.3010356389337067 \cdot 10^{-72}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\

\mathbf{elif}\;x \cdot y \leq 2.833726387088633 \cdot 10^{-265}:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;x \cdot y \leq 1.6279132317167615 \cdot 10^{+190}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\


\end{array}

(FPCore (x y z) :precision binary64 (/ (* x y) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -2.604516891076534e+217)
   (* y (/ x z))
   (if (<= (* x y) -1.3010356389337067e-72)
     (* (* x y) (/ 1.0 z))
     (if (<= (* x y) 2.833726387088633e-265)
       (* x (/ y z))
       (if (<= (* x y) 1.6279132317167615e+190)
         (/ (* x y) z)
         (/ 1.0 (/ (/ z x) y)))))))
double code(double x, double y, double z) {
	return (x * y) / z;
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -2.604516891076534e+217) {
		tmp = y * (x / z);
	} else if ((x * y) <= -1.3010356389337067e-72) {
		tmp = (x * y) * (1.0 / z);
	} else if ((x * y) <= 2.833726387088633e-265) {
		tmp = x * (y / z);
	} else if ((x * y) <= 1.6279132317167615e+190) {
		tmp = (x * y) / z;
	} else {
		tmp = 1.0 / ((z / x) / 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

Original6.5
Target6.2
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 5 regimes

  2. if (*.f64 x y) < -2.60451689107653394e217

    1. Initial program 28.4

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

    2. Applied clear-num_binary6428.5

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

    3. Applied associate-/r*_binary640.7

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

    4. Applied div-inv_binary640.7

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

    5. Applied add-cube-cbrt_binary640.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{z}{x} \cdot \frac{1}{y}} \]

    6. Applied times-frac_binary640.8

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{z}{x}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}}} \]

    7. Simplified0.6

      \[\leadsto \color{blue}{\frac{x}{z}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{y}} \]

    8. Simplified0.6

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

    if -2.60451689107653394e217 < (*.f64 x y) < -1.30103563893370669e-72

    1. Initial program 0.2

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

    2. Applied clear-num_binary640.5

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

    3. Applied associate-/r*_binary6411.1

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

    4. Applied *-un-lft-identity_binary6411.1

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

    5. Applied div-inv_binary6411.2

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

    6. Applied times-frac_binary640.6

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

    7. Applied add-cube-cbrt_binary640.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{z}{1} \cdot \frac{\frac{1}{x}}{y}} \]

    8. Applied times-frac_binary640.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{z}{1}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{1}{x}}{y}}} \]

    9. Simplified0.5

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{1}{x}}{y}} \]

    10. Simplified0.4

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

    if -1.30103563893370669e-72 < (*.f64 x y) < 2.8337263870886332e-265

    1. Initial program 9.5

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

    2. Applied *-un-lft-identity_binary649.5

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

    3. Applied times-frac_binary642.1

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

    if 2.8337263870886332e-265 < (*.f64 x y) < 1.6279132317167615e190

    1. Initial program 0.2

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

    if 1.6279132317167615e190 < (*.f64 x y)

    1. Initial program 22.9

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

    2. Applied clear-num_binary6423.0

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

    3. Applied associate-/r*_binary641.5

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{z}{x}}{y}}} \]
  3. Recombined 5 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2.604516891076534 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq -1.3010356389337067 \cdot 10^{-72}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{elif}\;x \cdot y \leq 2.833726387088633 \cdot 10^{-265}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq 1.6279132317167615 \cdot 10^{+190}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{y}}\\ \end{array} \]

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))