Average Error: 1.2 → 0.4
Time: 7.1s
Precision: binary64
\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
↓
\[e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{0.05555555555555555}{y} \cdot \left(\frac{x}{z} \cdot \sqrt{t}\right)\right)\right)} + -1 \]
(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
 (+
  (exp
   (log1p
    (*
     0.3333333333333333
     (acos (* (/ 0.05555555555555555 y) (* (/ x z) (sqrt t)))))))
  -1.0))
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 exp(log1p((0.3333333333333333 * acos(((0.05555555555555555 / y) * ((x / z) * sqrt(t))))))) + -1.0;
}

public static double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * Math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * Math.sqrt(t)));
}
↓
public static double code(double x, double y, double z, double t) {
	return Math.exp(Math.log1p((0.3333333333333333 * Math.acos(((0.05555555555555555 / y) * ((x / z) * Math.sqrt(t))))))) + -1.0;
}

def code(x, y, z, t):
	return (1.0 / 3.0) * math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * math.sqrt(t)))
↓
def code(x, y, z, t):
	return math.exp(math.log1p((0.3333333333333333 * math.acos(((0.05555555555555555 / y) * ((x / z) * math.sqrt(t))))))) + -1.0

function code(x, y, z, t)
	return Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)) * sqrt(t))))
end
↓
function code(x, y, z, t)
	return Float64(exp(log1p(Float64(0.3333333333333333 * acos(Float64(Float64(0.05555555555555555 / y) * Float64(Float64(x / z) * sqrt(t))))))) + -1.0)
end

code[x_, y_, z_, t_] := N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(N[(3.0 * N[(x / N[(y * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(N[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(N[(0.05555555555555555 / y), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{0.05555555555555555}{y} \cdot \left(\frac{x}{z} \cdot \sqrt{t}\right)\right)\right)} + -1

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

Original1.2
Target1.3
Herbie0.4
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3} \]

Derivation

  1. Initial program 1.2

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

  2. Taylor expanded in x around 0 1.3

    \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(0.05555555555555555 \cdot \frac{x}{y \cdot z}\right)} \cdot \sqrt{t}\right) \]

  3. Simplified1.3

    \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{0.05555555555555555}{y} \cdot \frac{x}{z}\right)} \cdot \sqrt{t}\right) \]

  4. Applied egg-rr0.4

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{0.05555555555555555}{y} \cdot \left(\frac{x}{z} \cdot \sqrt{t}\right)\right)\right)} - 1} \]

  5. Final simplification0.4

    \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{0.05555555555555555}{y} \cdot \left(\frac{x}{z} \cdot \sqrt{t}\right)\right)\right)} + -1 \]

Reproduce

herbie shell --seed 2022165 
(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)))))