Average Error: 1.4 → 0.3
Time: 10.6s
Precision: binary64
Cost: 20032
\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
↓
\[\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\sqrt{t} \cdot \frac{\frac{x}{z}}{18 \cdot y}\right), 1\right) + -1 \]
(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
 (+
  (fma 0.3333333333333333 (acos (* (sqrt t) (/ (/ x z) (* 18.0 y)))) 1.0)
  -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 fma(0.3333333333333333, acos((sqrt(t) * ((x / z) / (18.0 * y)))), 1.0) + -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(fma(0.3333333333333333, acos(Float64(sqrt(t) * Float64(Float64(x / z) / Float64(18.0 * y)))), 1.0) + -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[(0.3333333333333333 * N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / N[(18.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $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)
↓
\mathsf{fma}\left(0.3333333333333333, \cos^{-1} \left(\sqrt{t} \cdot \frac{\frac{x}{z}}{18 \cdot y}\right), 1\right) + -1

Error

Target

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

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. Simplified1.4

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{\frac{\frac{x}{18}}{y}}{z}\right)} \]
    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)))): 1 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, 1 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-rr1.2

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

  4. Applied egg-rr0.3

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

  5. Final simplification0.3

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

Alternatives

Alternative 1
Error1.2
Cost13504
\[0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\left(18 \cdot y\right) \cdot \frac{z}{x}}\right) \]

Error

Reproduce

herbie shell --seed 2022295 
(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)))))