Average Error: 1.3 → 0.2
Time: 14.2s
Precision: binary64
Cost: 26432
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
↓
\[e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\frac{z \cdot \left(18 \cdot y\right)}{x}}\right)\right)} + -1 \]
(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 (/ (sqrt t) (/ (* z (* 18.0 y)) x))))))
  -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((sqrt(t) / ((z * (18.0 * y)) / x)))))) + -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((Math.sqrt(t) / ((z * (18.0 * y)) / x)))))) + -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((math.sqrt(t) / ((z * (18.0 * y)) / x)))))) + -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(sqrt(t) / Float64(Float64(z * Float64(18.0 * y)) / x)))))) + -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[Sqrt[t], $MachinePrecision] / N[(N[(z * N[(18.0 * y), $MachinePrecision]), $MachinePrecision] / x), $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{\sqrt{t}}{\frac{z \cdot \left(18 \cdot y\right)}{x}}\right)\right)} + -1

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 1.3

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

  2. Simplified1.3

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{\frac{\frac{x}{18}}{y}}{z}\right)} \]
    Proof
    (*.f64 1/3 (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (/.f64 x 18) y) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (Rewrite<= metadata-eval (/.f64 1 3)) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (/.f64 x 18) y) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (/.f64 x (Rewrite<= metadata-eval (*.f64 2 9))) y) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (/.f64 x (*.f64 2 (Rewrite<= metadata-eval (/.f64 27 3)))) y) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 x (/.f64 27 3)) 2)) y) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 3) 27)) 2) y) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 3 x)) 27) 2) y) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 (/.f64 (*.f64 3 x) 27) (*.f64 2 y))) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (Rewrite=> associate-/l/_binary64 (/.f64 (*.f64 3 x) (*.f64 (*.f64 2 y) 27))) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 x 3)) (*.f64 (*.f64 2 y) 27)) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (Rewrite=> times-frac_binary64 (*.f64 (/.f64 x (*.f64 2 y)) (/.f64 3 27))) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 x y) 2)) (/.f64 3 27)) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 3 27) (/.f64 (/.f64 x y) 2))) z)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (/.f64 3 27) z) (/.f64 (/.f64 x y) 2)))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (/.f64 3 27) (/.f64 x y)) (*.f64 z 2)))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 3 x) (*.f64 27 y))) (*.f64 z 2))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (/.f64 (*.f64 3 x) (Rewrite<= *-commutative_binary64 (*.f64 y 27))) (*.f64 z 2))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (*.f64 (sqrt.f64 t) (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 3 (/.f64 x (*.f64 y 27)))) (*.f64 z 2))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (/.f64 1 3) (acos.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 3 (/.f64 x (*.f64 y 27))) (*.f64 z 2)) (sqrt.f64 t))))): 0 points increase in error, 0 points decrease in error

  3. Applied egg-rr0.2

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\frac{z}{x} \cdot \left(18 \cdot y\right)}\right)\right)} - 1} \]

  4. Applied egg-rr0.2

    \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\color{blue}{\frac{z \cdot \left(18 \cdot y\right)}{x}}}\right)\right)} - 1 \]

  5. Final simplification0.2

    \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\frac{z \cdot \left(18 \cdot y\right)}{x}}\right)\right)} + -1 \]

Alternatives

Alternative 1
Error1.2
Cost13504
\[0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \left(x \cdot \frac{\frac{0.05555555555555555}{z}}{y}\right)\right) \]

Error

Reproduce

herbie shell --seed 2022291 
(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)))))