Average Error: 6.2 → 2.1
Time: 3.2s
Precision: binary64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq 1.3289674181225 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y}}}\\ \mathbf{elif}\;x \cdot y \leq 5.36460452411368 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \leq 1.3289674181225 \cdot 10^{-318}:\\
\;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y}}}\\

\mathbf{elif}\;x \cdot y \leq 5.36460452411368 \cdot 10^{+147}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x y) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) 1.3289674181225e-318)
   (*
    (* (* (cbrt x) (cbrt x)) (* (cbrt y) (cbrt y)))
    (/ (cbrt x) (/ z (cbrt y))))
   (if (<= (* x y) 5.36460452411368e+147) (/ (* x y) z) (* y (/ x z)))))
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) <= 1.3289674181225e-318) {
		tmp = ((cbrt(x) * cbrt(x)) * (cbrt(y) * cbrt(y))) * (cbrt(x) / (z / cbrt(y)));
	} else if ((x * y) <= 5.36460452411368e+147) {
		tmp = (x * y) / z;
	} else {
		tmp = y * (x / 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

Original6.2
Target6.2
Herbie2.1
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x y) < 1.3289674e-318

    1. Initial program 7.6

      \[\frac{x \cdot y}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*_binary64_129815.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary64_130716.1

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

    6. Applied *-un-lft-identity_binary64_130366.1

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

    7. Applied times-frac_binary64_130426.1

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

    8. Applied add-cube-cbrt_binary64_130716.3

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

    9. Applied times-frac_binary64_130422.9

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

    10. Simplified2.9

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

    if 1.3289674e-318 < (*.f64 x y) < 5.36460452411368e147

    1. Initial program 0.3

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

    if 5.36460452411368e147 < (*.f64 x y)

    1. Initial program 19.4

      \[\frac{x \cdot y}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*_binary64_129812.5

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
    4. Using strategy rm

    5. Applied associate-/r/_binary64_129823.4

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 1.3289674181225 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) \cdot \frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y}}}\\ \mathbf{elif}\;x \cdot y \leq 5.36460452411368 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(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))