Average Error: 6.1 → 3.2
Time: 2.4s
Precision: binary64
\[\frac{x \cdot y}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{\sqrt[3]{z}}\\ \frac{\frac{y}{\sqrt[3]{z}}}{t_1 \cdot t_1} \cdot \frac{\frac{x}{\sqrt[3]{z}}}{t_1} \end{array}\\ \mathbf{elif}\;t_0 \leq 4.6360253909051757 \cdot 10^{+285}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
t_0 := \frac{x \cdot y}{z}\\
\mathbf{if}\;t_0 \leq 0:\\
\;\;\;\;\begin{array}{l}
t_1 := \sqrt[3]{\sqrt[3]{z}}\\
\frac{\frac{y}{\sqrt[3]{z}}}{t_1 \cdot t_1} \cdot \frac{\frac{x}{\sqrt[3]{z}}}{t_1}
\end{array}\\

\mathbf{elif}\;t_0 \leq 4.6360253909051757 \cdot 10^{+285}:\\
\;\;\;\;t_0\\

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


\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 0.0)
     (let* ((t_1 (cbrt (cbrt z))))
       (* (/ (/ y (cbrt z)) (* t_1 t_1)) (/ (/ x (cbrt z)) t_1)))
     (if (<= t_0 4.6360253909051757e+285) t_0 (/ y (/ z x))))))
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 <= 0.0) {
		double t_1_1 = cbrt(cbrt(z));
		tmp = ((y / cbrt(z)) / (t_1_1 * t_1_1)) * ((x / cbrt(z)) / t_1_1);
	} else if (t_0 <= 4.6360253909051757e+285) {
		tmp = t_0;
	} else {
		tmp = y / (z / x);
	}
	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

Original6.1
Target6.1
Herbie3.2
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 (*.f64 x y) z) < 0.0

    1. Initial program 7.0

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

    3. Applied add-cube-cbrt_binary647.7

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

    4. Applied associate-/r*_binary647.7

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

    5. Simplified5.1

      \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\sqrt[3]{z}} \]
    6. Using strategy rm

    7. Applied *-un-lft-identity_binary645.1

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

    8. Applied times-frac_binary645.1

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

    9. Applied associate-*r*_binary644.4

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

    10. Simplified4.4

      \[\leadsto \frac{\color{blue}{\frac{y}{\sqrt[3]{z}}} \cdot \frac{x}{\sqrt[3]{z}}}{\sqrt[3]{z}} \]
    11. Using strategy rm

    12. Applied add-cube-cbrt_binary644.6

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

    13. Applied times-frac_binary644.7

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

    if 0.0 < (/.f64 (*.f64 x y) z) < 4.63602539090517566e285

    1. Initial program 0.5

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

    if 4.63602539090517566e285 < (/.f64 (*.f64 x y) z)

    1. Initial program 47.9

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

    3. Applied add-cube-cbrt_binary6448.2

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

    4. Applied associate-/r*_binary6448.2

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

    5. Simplified15.6

      \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}}{\sqrt[3]{z}} \]
    6. Using strategy rm

    7. Applied associate-/l*_binary644.9

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

    8. Simplified3.8

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

  4. Final simplification3.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;\frac{\frac{y}{\sqrt[3]{z}}}{\sqrt[3]{\sqrt[3]{z}} \cdot \sqrt[3]{\sqrt[3]{z}}} \cdot \frac{\frac{x}{\sqrt[3]{z}}}{\sqrt[3]{\sqrt[3]{z}}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 4.6360253909051757 \cdot 10^{+285}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))