Average Error: 7.0 → 1.9
Time: 5.8s
Precision: binary64
\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]
↓
\[\left(x \cdot \frac{\sqrt[3]{\frac{2}{y - t}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}} \cdot \sqrt[3]{\frac{2}{\sqrt[3]{y - t}}}\right)}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z}}\]
\frac{x \cdot 2}{y \cdot z - t \cdot z}
↓
\left(x \cdot \frac{\sqrt[3]{\frac{2}{y - t}} \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}} \cdot \sqrt[3]{\frac{2}{\sqrt[3]{y - t}}}\right)}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{\frac{2}{y - t}}}{\sqrt[3]{z}}
(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))
↓
(FPCore (x y z t)
 :precision binary64
 (*
  (*
   x
   (/
    (*
     (cbrt (/ 2.0 (- y t)))
     (*
      (cbrt (/ 1.0 (* (cbrt (- y t)) (cbrt (- y t)))))
      (cbrt (/ 2.0 (cbrt (- y t))))))
    (* (cbrt z) (cbrt z))))
  (/ (cbrt (/ 2.0 (- y t))) (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) {
	return (x * ((cbrt(2.0 / (y - t)) * (cbrt(1.0 / (cbrt(y - t) * cbrt(y - t))) * cbrt(2.0 / cbrt(y - t)))) / (cbrt(z) * cbrt(z)))) * (cbrt(2.0 / (y - t)) / 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

Original7.0
Target2.2
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} < 1.0450278273301259 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

  1. Initial program 7.0

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

  2. Simplified5.7

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

  4. Applied add-cube-cbrt_binary646.4

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

  5. Applied add-cube-cbrt_binary646.5

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

  6. Applied times-frac_binary646.5

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

  7. Applied associate-*r*_binary641.9

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

  9. Applied add-cube-cbrt_binary641.9

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

  10. Applied *-un-lft-identity_binary641.9

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

  11. Applied times-frac_binary641.9

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

  12. Applied cbrt-prod_binary641.9

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

  13. Final simplification1.9

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

Reproduce

herbie shell --seed 2020219 
(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))) 1.0450278273301259e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

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