?

Average Error: 0.6 → 0.5
Time: 12.2s
Precision: binary64
Cost: 14144

?

\[\left(0 \leq w \land w \leq 10\right) \land \left(0 \leq k \land k \leq 10000\right)\]
\[\frac{1}{16 \cdot {w}^{2}} \cdot \sqrt{\left(\left(\left(k + 1\right) \cdot \left(k + 2\right)\right) \cdot \left(k + 3\right)\right) \cdot \left(k + 4\right)} \]
↓
\[\frac{\frac{0.0625}{w}}{\frac{w}{\sqrt{k + 3}}} \cdot \sqrt{\left(k + 2\right) \cdot \left(\left(k + 4\right) \cdot \left(k + 1\right)\right)} \]
(FPCore (w k)
 :precision binary64
 (*
  (/ 1.0 (* 16.0 (pow w 2.0)))
  (sqrt (* (* (* (+ k 1.0) (+ k 2.0)) (+ k 3.0)) (+ k 4.0)))))
↓
(FPCore (w k)
 :precision binary64
 (*
  (/ (/ 0.0625 w) (/ w (sqrt (+ k 3.0))))
  (sqrt (* (+ k 2.0) (* (+ k 4.0) (+ k 1.0))))))
double code(double w, double k) {
	return (1.0 / (16.0 * pow(w, 2.0))) * sqrt(((((k + 1.0) * (k + 2.0)) * (k + 3.0)) * (k + 4.0)));
}
↓
double code(double w, double k) {
	return ((0.0625 / w) / (w / sqrt((k + 3.0)))) * sqrt(((k + 2.0) * ((k + 4.0) * (k + 1.0))));
}

real(8) function code(w, k)
    real(8), intent (in) :: w
    real(8), intent (in) :: k
    code = (1.0d0 / (16.0d0 * (w ** 2.0d0))) * sqrt(((((k + 1.0d0) * (k + 2.0d0)) * (k + 3.0d0)) * (k + 4.0d0)))
end function
↓
real(8) function code(w, k)
    real(8), intent (in) :: w
    real(8), intent (in) :: k
    code = ((0.0625d0 / w) / (w / sqrt((k + 3.0d0)))) * sqrt(((k + 2.0d0) * ((k + 4.0d0) * (k + 1.0d0))))
end function

public static double code(double w, double k) {
	return (1.0 / (16.0 * Math.pow(w, 2.0))) * Math.sqrt(((((k + 1.0) * (k + 2.0)) * (k + 3.0)) * (k + 4.0)));
}
↓
public static double code(double w, double k) {
	return ((0.0625 / w) / (w / Math.sqrt((k + 3.0)))) * Math.sqrt(((k + 2.0) * ((k + 4.0) * (k + 1.0))));
}

def code(w, k):
	return (1.0 / (16.0 * math.pow(w, 2.0))) * math.sqrt(((((k + 1.0) * (k + 2.0)) * (k + 3.0)) * (k + 4.0)))
↓
def code(w, k):
	return ((0.0625 / w) / (w / math.sqrt((k + 3.0)))) * math.sqrt(((k + 2.0) * ((k + 4.0) * (k + 1.0))))

function code(w, k)
	return Float64(Float64(1.0 / Float64(16.0 * (w ^ 2.0))) * sqrt(Float64(Float64(Float64(Float64(k + 1.0) * Float64(k + 2.0)) * Float64(k + 3.0)) * Float64(k + 4.0))))
end
↓
function code(w, k)
	return Float64(Float64(Float64(0.0625 / w) / Float64(w / sqrt(Float64(k + 3.0)))) * sqrt(Float64(Float64(k + 2.0) * Float64(Float64(k + 4.0) * Float64(k + 1.0)))))
end

function tmp = code(w, k)
	tmp = (1.0 / (16.0 * (w ^ 2.0))) * sqrt(((((k + 1.0) * (k + 2.0)) * (k + 3.0)) * (k + 4.0)));
end
↓
function tmp = code(w, k)
	tmp = ((0.0625 / w) / (w / sqrt((k + 3.0)))) * sqrt(((k + 2.0) * ((k + 4.0) * (k + 1.0))));
end

code[w_, k_] := N[(N[(1.0 / N[(16.0 * N[Power[w, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(k + 1.0), $MachinePrecision] * N[(k + 2.0), $MachinePrecision]), $MachinePrecision] * N[(k + 3.0), $MachinePrecision]), $MachinePrecision] * N[(k + 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[w_, k_] := N[(N[(N[(0.0625 / w), $MachinePrecision] / N[(w / N[Sqrt[N[(k + 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(k + 2.0), $MachinePrecision] * N[(N[(k + 4.0), $MachinePrecision] * N[(k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\frac{1}{16 \cdot {w}^{2}} \cdot \sqrt{\left(\left(\left(k + 1\right) \cdot \left(k + 2\right)\right) \cdot \left(k + 3\right)\right) \cdot \left(k + 4\right)}
↓
\frac{\frac{0.0625}{w}}{\frac{w}{\sqrt{k + 3}}} \cdot \sqrt{\left(k + 2\right) \cdot \left(\left(k + 4\right) \cdot \left(k + 1\right)\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.6

    \[\frac{1}{16 \cdot {w}^{2}} \cdot \sqrt{\left(\left(\left(k + 1\right) \cdot \left(k + 2\right)\right) \cdot \left(k + 3\right)\right) \cdot \left(k + 4\right)} \]

  2. Simplified0.6

    \[\leadsto \color{blue}{\frac{0.0625}{w \cdot w} \cdot \sqrt{\left(k + 3\right) \cdot \left(\left(\left(1 + k\right) \cdot \left(2 + k\right)\right) \cdot \left(k + 4\right)\right)}} \]
    Proof

    [Start]0.6

    \[ \frac{1}{16 \cdot {w}^{2}} \cdot \sqrt{\left(\left(\left(k + 1\right) \cdot \left(k + 2\right)\right) \cdot \left(k + 3\right)\right) \cdot \left(k + 4\right)} \]

    associate-/r* [=>]0.6

    \[ \color{blue}{\frac{\frac{1}{16}}{{w}^{2}}} \cdot \sqrt{\left(\left(\left(k + 1\right) \cdot \left(k + 2\right)\right) \cdot \left(k + 3\right)\right) \cdot \left(k + 4\right)} \]

    metadata-eval [=>]0.6

    \[ \frac{\color{blue}{0.0625}}{{w}^{2}} \cdot \sqrt{\left(\left(\left(k + 1\right) \cdot \left(k + 2\right)\right) \cdot \left(k + 3\right)\right) \cdot \left(k + 4\right)} \]

    unpow2 [=>]0.6

    \[ \frac{0.0625}{\color{blue}{w \cdot w}} \cdot \sqrt{\left(\left(\left(k + 1\right) \cdot \left(k + 2\right)\right) \cdot \left(k + 3\right)\right) \cdot \left(k + 4\right)} \]

    *-commutative [=>]0.6

    \[ \frac{0.0625}{w \cdot w} \cdot \sqrt{\color{blue}{\left(\left(k + 3\right) \cdot \left(\left(k + 1\right) \cdot \left(k + 2\right)\right)\right)} \cdot \left(k + 4\right)} \]

    associate-*l* [=>]0.6

    \[ \frac{0.0625}{w \cdot w} \cdot \sqrt{\color{blue}{\left(k + 3\right) \cdot \left(\left(\left(k + 1\right) \cdot \left(k + 2\right)\right) \cdot \left(k + 4\right)\right)}} \]

    +-commutative [=>]0.6

    \[ \frac{0.0625}{w \cdot w} \cdot \sqrt{\left(k + 3\right) \cdot \left(\left(\color{blue}{\left(1 + k\right)} \cdot \left(k + 2\right)\right) \cdot \left(k + 4\right)\right)} \]

    +-commutative [=>]0.6

    \[ \frac{0.0625}{w \cdot w} \cdot \sqrt{\left(k + 3\right) \cdot \left(\left(\left(1 + k\right) \cdot \color{blue}{\left(2 + k\right)}\right) \cdot \left(k + 4\right)\right)} \]

  3. Applied egg-rr0.6

    \[\leadsto \color{blue}{\frac{\frac{0.0625}{w} \cdot \sqrt{\left(k + 3\right) \cdot \left(\left(k + 1\right) \cdot \left(\left(k + 2\right) \cdot \left(k + 4\right)\right)\right)}}{w}} \]

  4. Applied egg-rr0.5

    \[\leadsto \color{blue}{\frac{\frac{0.0625}{w}}{\frac{w}{\sqrt{k + 3}}} \cdot \sqrt{\left(k + 2\right) \cdot \left(\left(k + 4\right) \cdot \left(k + 1\right)\right)}} \]

  5. Final simplification0.5

    \[\leadsto \frac{\frac{0.0625}{w}}{\frac{w}{\sqrt{k + 3}}} \cdot \sqrt{\left(k + 2\right) \cdot \left(\left(k + 4\right) \cdot \left(k + 1\right)\right)} \]

Alternatives

Alternative 1
Error0.6
Cost7744
\[\frac{0.0625}{w \cdot w} \cdot \sqrt{\left(k + 3\right) \cdot \left(\left(k + 4\right) \cdot \left(\left(k + 2\right) \cdot \left(k + 1\right)\right)\right)} \]
Alternative 2
Error1.9
Cost7616
\[\frac{0.0625}{w \cdot w} \cdot \sqrt{\left(k + 3\right) \cdot \left(\left(k + 4\right) \cdot \left(2 + k \cdot 3\right)\right)} \]
Alternative 3
Error1.9
Cost7360
\[\frac{\frac{0.0625}{w} \cdot \sqrt{\left(k + 3\right) \cdot \left(k \cdot 14 + 8\right)}}{w} \]
Alternative 4
Error1.9
Cost7104
\[\frac{0.0625}{w \cdot w} \cdot \sqrt{k \cdot 50 + 24} \]
Alternative 5
Error2.3
Cost6912
\[\frac{0.0625}{w \cdot \left(w \cdot {24}^{-0.5}\right)} \]
Alternative 6
Error2.3
Cost6912
\[\frac{0.0625}{\left(w \cdot w\right) \cdot {24}^{-0.5}} \]
Alternative 7
Error2.7
Cost6848
\[0.0625 \cdot \frac{\sqrt{24}}{w \cdot w} \]
Alternative 8
Error2.5
Cost6848
\[\frac{0.0625}{w \cdot w} \cdot \sqrt{24} \]
Alternative 9
Error2.5
Cost6848
\[\frac{0.0625 \cdot \sqrt{24}}{w \cdot w} \]
Alternative 10
Error61.3
Cost576
\[0.0625 \cdot \left(\frac{k}{w} \cdot \frac{k}{w}\right) \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (w k)
  :name " (1/(16*w^2))*sqrt((k+1)*(k+2)*(k+3)*(k+4))"
  :precision binary64
  :pre (and (and (<= 0.0 w) (<= w 10.0)) (and (<= 0.0 k) (<= k 10000.0)))
  (* (/ 1.0 (* 16.0 (pow w 2.0))) (sqrt (* (* (* (+ k 1.0) (+ k 2.0)) (+ k 3.0)) (+ k 4.0)))))