Average Error: 6.8 → 1.8
Time: 5.1s
Precision: binary64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{y - t}\\ \left(x \cdot \frac{\frac{1}{t_1 \cdot t_1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{2}{t_1}}{\sqrt[3]{z}} \end{array} \]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
t_1 := \sqrt[3]{y - t}\\
\left(x \cdot \frac{\frac{1}{t_1 \cdot t_1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\frac{2}{t_1}}{\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 (cbrt (- y t))))
   (*
    (* x (/ (/ 1.0 (* t_1 t_1)) (* (cbrt z) (cbrt z))))
    (/ (/ 2.0 t_1) (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 = cbrt(y - t);
	return (x * ((1.0 / (t_1 * t_1)) / (cbrt(z) * cbrt(z)))) * ((2.0 / t_1) / cbrt(z));
}

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

Original6.8
Target2.0
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array} \]

Derivation

  1. Initial program 6.8

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

  2. Simplified5.4

    \[\leadsto \color{blue}{x \cdot \frac{\frac{2}{y - t}}{z}} \]
  3. Using strategy rm

  4. Applied add-cube-cbrt_binary646.1

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

  5. Applied add-cube-cbrt_binary646.2

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

  6. Applied *-un-lft-identity_binary646.2

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

  7. Applied times-frac_binary646.2

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

  8. Applied times-frac_binary646.2

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

  9. Applied associate-*r*_binary641.8

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

  10. Final simplification1.8

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

Reproduce

herbie shell --seed 2021204 
(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))))