Average Error: 6.6 → 1.2
Time: 10.1s
Precision: binary64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
↓
\[\begin{array}{l} t_1 := y \cdot z - z \cdot t\\ t_2 := \sqrt[3]{y - t}\\ \mathbf{if}\;t_1 \leq -1.7486352694326628 \cdot 10^{+296}:\\ \;\;\;\;\frac{x \cdot \frac{2}{y - t}}{z}\\ \mathbf{elif}\;t_1 \leq -7.916846244685067 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(t_2 \cdot t_2\right)} \cdot \frac{\frac{2}{t_2}}{\sqrt[3]{z}}\\ \end{array} \]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
t_1 := y \cdot z - z \cdot t\\
t_2 := \sqrt[3]{y - t}\\
\mathbf{if}\;t_1 \leq -1.7486352694326628 \cdot 10^{+296}:\\
\;\;\;\;\frac{x \cdot \frac{2}{y - t}}{z}\\

\mathbf{elif}\;t_1 \leq -7.916846244685067 \cdot 10^{-304}:\\
\;\;\;\;2 \cdot \frac{x}{z \cdot \left(y - t\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(t_2 \cdot t_2\right)} \cdot \frac{\frac{2}{t_2}}{\sqrt[3]{z}}\\


\end{array}

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) (* z t))) (t_2 (cbrt (- y t))))
   (if (<= t_1 -1.7486352694326628e+296)
     (/ (* x (/ 2.0 (- y t))) z)
     (if (<= t_1 -7.916846244685067e-304)
       (* 2.0 (/ x (* z (- y t))))
       (*
        (/ x (* (* (cbrt z) (cbrt z)) (* t_2 t_2)))
        (/ (/ 2.0 t_2) (cbrt z)))))))
double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - (z * t);
	double t_2 = cbrt((y - t));
	double tmp;
	if (t_1 <= -1.7486352694326628e+296) {
		tmp = (x * (2.0 / (y - t))) / z;
	} else if (t_1 <= -7.916846244685067e-304) {
		tmp = 2.0 * (x / (z * (y - t)));
	} else {
		tmp = (x / ((cbrt(z) * cbrt(z)) * (t_2 * t_2))) * ((2.0 / t_2) / cbrt(z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.6
Target2.0
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -1.74863526943266277e296

    1. Initial program 17.8

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

    2. Simplified16.4

      \[\leadsto \color{blue}{x \cdot \frac{\frac{2}{y - t}}{z}} \]

    3. Applied associate-*r/_binary640.1

      \[\leadsto \color{blue}{\frac{x \cdot \frac{2}{y - t}}{z}} \]

    if -1.74863526943266277e296 < (-.f64 (*.f64 y z) (*.f64 t z)) < -7.91684624468506695e-304

    1. Initial program 0.2

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

    2. Simplified0.3

      \[\leadsto \color{blue}{x \cdot \frac{\frac{2}{y - t}}{z}} \]

    3. Taylor expanded in x around 0 0.2

      \[\leadsto \color{blue}{2 \cdot \frac{x}{\left(y - t\right) \cdot z}} \]

    if -7.91684624468506695e-304 < (-.f64 (*.f64 y z) (*.f64 t z))

    1. Initial program 8.9

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]

    2. Simplified6.6

      \[\leadsto \color{blue}{x \cdot \frac{\frac{2}{y - t}}{z}} \]

    3. Applied add-cube-cbrt_binary647.3

      \[\leadsto x \cdot \frac{\frac{2}{y - t}}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} \]

    4. Applied add-cube-cbrt_binary647.4

      \[\leadsto x \cdot \frac{\frac{2}{\color{blue}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

    5. Applied *-un-lft-identity_binary647.4

      \[\leadsto x \cdot \frac{\frac{\color{blue}{1 \cdot 2}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

    6. Applied times-frac_binary647.4

      \[\leadsto x \cdot \frac{\color{blue}{\frac{1}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{2}{\sqrt[3]{y - t}}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

    7. Applied times-frac_binary647.4

      \[\leadsto x \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\frac{2}{\sqrt[3]{y - t}}}{\sqrt[3]{z}}\right)} \]

    8. Applied associate-*r*_binary641.9

      \[\leadsto \color{blue}{\left(x \cdot \frac{\frac{1}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{2}{\sqrt[3]{y - t}}}{\sqrt[3]{z}}} \]

    9. Simplified2.1

      \[\leadsto \color{blue}{\frac{x}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right)}} \cdot \frac{\frac{2}{\sqrt[3]{y - t}}}{\sqrt[3]{z}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - z \cdot t \leq -1.7486352694326628 \cdot 10^{+296}:\\ \;\;\;\;\frac{x \cdot \frac{2}{y - t}}{z}\\ \mathbf{elif}\;y \cdot z - z \cdot t \leq -7.916846244685067 \cdot 10^{-304}:\\ \;\;\;\;2 \cdot \frac{x}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right)} \cdot \frac{\frac{2}{\sqrt[3]{y - t}}}{\sqrt[3]{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))