Average Error: 1.4 → 1.7
Time: 3.4min
Precision: binary64
Cost: 2176
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)\right)}\right) \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)\right)}\right) \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}
(FPCore (x y z t)
 :precision binary64
 (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))
↓
(FPCore (x y z t)
 :precision binary64
 (*
  (*
   0.3333333333333333
   (sqrt
    (acos
     (* (* (cbrt x) (cbrt x)) (* (cbrt x) (/ (/ (sqrt t) z) (* y 18.0)))))))
  (sqrt (acos (* x (/ (/ (sqrt t) z) (* y 18.0)))))))
double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * acos(((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t));
}
↓
double code(double x, double y, double z, double t) {
	return (0.3333333333333333 * sqrt(acos((cbrt(x) * cbrt(x)) * (cbrt(x) * ((sqrt(t) / z) / (y * 18.0)))))) * sqrt(acos(x * ((sqrt(t) / z) / (y * 18.0))));
}

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

Original1.4
Target1.3
Herbie1.7
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

  1. Initial program 1.4

    \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

  2. Simplified1.7

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt_binary64_195372.2

    \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)} \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\right)}\]

  5. Applied associate-*r*_binary64_194551.7

    \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\right) \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt_binary64_195501.7

    \[\leadsto \left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\right) \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\]

  8. Applied associate-*l*_binary64_194561.7

    \[\leadsto \left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)\right)}}\right) \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\]

  9. Simplified1.7

    \[\leadsto \left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt{t}}{z}}{y \cdot 18} \cdot \sqrt[3]{x}\right)}\right)}\right) \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\]

  10. Final simplification1.7

    \[\leadsto \left(0.3333333333333333 \cdot \sqrt{\cos^{-1} \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)\right)}\right) \cdot \sqrt{\cos^{-1} \left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y \cdot 18}\right)}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27.0) (* y z)) (/ (sqrt t) (/ 2.0 3.0)))) 3.0)

  (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))