Average Error: 5.8 → 1.5
Time: 2.8s
Precision: binary64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.0861216218609165 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -3.9699086841145227 \cdot 10^{-296}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt[3]{z}}\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \leq -5.0861216218609165 \cdot 10^{+116}:\\
\;\;\;\;x \cdot \frac{y}{z}\\

\mathbf{elif}\;x \cdot y \leq -3.9699086841145227 \cdot 10^{-296}:\\
\;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt[3]{z}}\\

\end{array}
(FPCore (x y z) :precision binary64 (/ (* x y) z))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -5.0861216218609165e+116)
   (* x (/ y z))
   (if (<= (* x y) -3.9699086841145227e-296)
     (* (* x y) (/ 1.0 z))
     (*
      (* x (* (cbrt y) (/ (cbrt y) (* (cbrt z) (cbrt z)))))
      (/ (* (cbrt (cbrt y)) (* (cbrt (cbrt y)) (cbrt (cbrt y)))) (cbrt z))))))
double code(double x, double y, double z) {
	return (((double) (x * y)) / z);
}
↓
double code(double x, double y, double z) {
	double tmp;
	if ((((double) (x * y)) <= -5.0861216218609165e+116)) {
		tmp = ((double) (x * (y / z)));
	} else {
		double tmp_1;
		if ((((double) (x * y)) <= -3.9699086841145227e-296)) {
			tmp_1 = ((double) (((double) (x * y)) * (1.0 / z)));
		} else {
			tmp_1 = ((double) (((double) (x * ((double) (((double) cbrt(y)) * (((double) cbrt(y)) / ((double) (((double) cbrt(z)) * ((double) cbrt(z))))))))) * (((double) (((double) cbrt(((double) cbrt(y)))) * ((double) (((double) cbrt(((double) cbrt(y)))) * ((double) cbrt(((double) cbrt(y)))))))) / ((double) cbrt(z)))));
		}
		tmp = tmp_1;
	}
	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

Original5.8
Target6.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z < -4.262230790519429 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z < 1.7042130660650472 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -5.0861216218609165e116

    1. Initial program Error: 15.6 bits

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

    2. SimplifiedError: 3.1 bits

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

    if -5.0861216218609165e116 < (* x y) < -3.96990868411452272e-296

    1. Initial program Error: 0.2 bits

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

    2. SimplifiedError: 8.9 bits

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

    4. Applied div-invError: 9.0 bits

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

    5. Applied associate-*r*Error: 0.3 bits

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

    if -3.96990868411452272e-296 < (* x y)

    1. Initial program Error: 7.1 bits

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

    2. SimplifiedError: 5.6 bits

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

    4. Applied add-cube-cbrtError: 6.3 bits

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

    5. Applied add-cube-cbrtError: 6.5 bits

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

    6. Applied times-fracError: 6.5 bits

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

    7. Applied associate-*r*Error: 1.7 bits

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

    8. SimplifiedError: 1.7 bits

      \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right)} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrtError: 1.9 bits

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

  4. Final simplificationError: 1.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.0861216218609165 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;x \cdot y \leq -3.9699086841145227 \cdot 10^{-296}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}{\sqrt[3]{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))