Average Error: 6.4 → 1.3
Time: 2.8s
Precision: binary64
\[[x, y] = \mathsf{sort}([x, y]) \\]
\[\frac{x \cdot y}{z} \]
↓
\[\begin{array}{l} t_0 := \frac{x \cdot y}{z}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\ \mathbf{elif}\;t_0 \leq 3.951917573061312 \cdot 10^{+304}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
t_0 := \frac{x \cdot y}{z}\\
\mathbf{if}\;t_0 \leq 0:\\
\;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\

\mathbf{elif}\;t_0 \leq 3.951917573061312 \cdot 10^{+304}:\\
\;\;\;\;t_0\\

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


\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)
     (*
      (/ (* (cbrt y) (cbrt y)) (* (cbrt z) (cbrt z)))
      (/ x (/ (cbrt z) (cbrt y))))
     (if (<= t_0 3.951917573061312e+304) t_0 (* x (/ y z))))))
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) {
		tmp = ((cbrt(y) * cbrt(y)) / (cbrt(z) * cbrt(z))) * (x / (cbrt(z) / cbrt(y)));
	} else if (t_0 <= 3.951917573061312e+304) {
		tmp = t_0;
	} else {
		tmp = x * (y / 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

Original6.4
Target6.3
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 7.3

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

    2. Applied associate-/l*_binary644.9

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

    3. Applied add-cube-cbrt_binary645.6

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

    4. Applied add-cube-cbrt_binary645.8

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

    5. Applied times-frac_binary645.8

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

    6. Applied *-un-lft-identity_binary645.8

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

    7. Applied times-frac_binary641.8

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

    8. Simplified1.8

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

    if -0.0 < (/.f64 (*.f64 x y) z) < 3.9519175730613121e304

    1. Initial program 0.5

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

    if 3.9519175730613121e304 < (/.f64 (*.f64 x y) z)

    1. Initial program 62.2

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

    2. Applied *-un-lft-identity_binary6462.2

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

    3. Applied times-frac_binary640.8

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

    4. Simplified0.8

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{z} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}\\ \mathbf{elif}\;\frac{x \cdot y}{z} \leq 3.951917573061312 \cdot 10^{+304}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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))