Average Error: 14.0 → 10.0
Time: 18.3s
Precision: binary64
Cost: 7744
\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[w0 \cdot \sqrt{1 + \frac{\left(\frac{h}{\frac{d}{D \cdot M}} \cdot \frac{D}{\frac{d}{M}}\right) \cdot -0.25}{\ell}} \]
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
↓
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (+ 1.0 (/ (* (* (/ h (/ d (* D M))) (/ D (/ d M))) -0.25) l)))))
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt((1.0 + ((((h / (d / (D * M))) * (D / (d / M))) * -0.25) / l)));
}

real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 - ((((m * d) / (2.0d0 * d_1)) ** 2.0d0) * (h / l))))
end function
↓
real(8) function code(w0, m, d, h, l, d_1)
    real(8), intent (in) :: w0
    real(8), intent (in) :: m
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: d_1
    code = w0 * sqrt((1.0d0 + ((((h / (d_1 / (d * m))) * (d / (d_1 / m))) * (-0.25d0)) / l)))
end function

public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 - (Math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))));
}
↓
public static double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * Math.sqrt((1.0 + ((((h / (d / (D * M))) * (D / (d / M))) * -0.25) / l)));
}

def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 - (math.pow(((M * D) / (2.0 * d)), 2.0) * (h / l))))
↓
def code(w0, M, D, h, l, d):
	return w0 * math.sqrt((1.0 + ((((h / (d / (D * M))) * (D / (d / M))) * -0.25) / l)))

function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64((Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0) * Float64(h / l)))))
end
↓
function code(w0, M, D, h, l, d)
	return Float64(w0 * sqrt(Float64(1.0 + Float64(Float64(Float64(Float64(h / Float64(d / Float64(D * M))) * Float64(D / Float64(d / M))) * -0.25) / l))))
end

function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 - ((((M * D) / (2.0 * d)) ^ 2.0) * (h / l))));
end
↓
function tmp = code(w0, M, D, h, l, d)
	tmp = w0 * sqrt((1.0 + ((((h / (d / (D * M))) * (D / (d / M))) * -0.25) / l)));
end

code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 - N[(N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[w0_, M_, D_, h_, l_, d_] := N[(w0 * N[Sqrt[N[(1.0 + N[(N[(N[(N[(h / N[(d / N[(D * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(D / N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 + \frac{\left(\frac{h}{\frac{d}{D \cdot M}} \cdot \frac{D}{\frac{d}{M}}\right) \cdot -0.25}{\ell}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.0

    \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]

  2. Applied egg-rr10.8

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2} \cdot h}{\ell}}} \]

  3. Taylor expanded in M around 0 31.2

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{0.25 \cdot \frac{{D}^{2} \cdot \left({M}^{2} \cdot h\right)}{{d}^{2}}}}{\ell}} \]

  4. Simplified16.0

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{0.25 \cdot \left(\frac{h}{d} \cdot \frac{\left(D \cdot M\right) \cdot \left(D \cdot M\right)}{d}\right)}}{\ell}} \]
    Proof
    (*.f64 1/4 (*.f64 (/.f64 h d) (/.f64 (*.f64 (*.f64 D M) (*.f64 D M)) d))): 0 points increase in error, 0 points decrease in error
    (*.f64 1/4 (*.f64 (/.f64 h d) (/.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 D D) (*.f64 M M))) d))): 58 points increase in error, 12 points decrease in error
    (*.f64 1/4 (*.f64 (/.f64 h d) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 D 2)) (*.f64 M M)) d))): 0 points increase in error, 0 points decrease in error
    (*.f64 1/4 (*.f64 (/.f64 h d) (/.f64 (*.f64 (pow.f64 D 2) (Rewrite<= unpow2_binary64 (pow.f64 M 2))) d))): 0 points increase in error, 0 points decrease in error
    (*.f64 1/4 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) d) (/.f64 h d)))): 0 points increase in error, 0 points decrease in error
    (*.f64 1/4 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 (pow.f64 D 2) (pow.f64 M 2)) h) (*.f64 d d)))): 20 points increase in error, 7 points decrease in error
    (*.f64 1/4 (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h))) (*.f64 d d))): 9 points increase in error, 5 points decrease in error
    (*.f64 1/4 (/.f64 (*.f64 (pow.f64 D 2) (*.f64 (pow.f64 M 2) h)) (Rewrite<= unpow2_binary64 (pow.f64 d 2)))): 0 points increase in error, 0 points decrease in error

  5. Applied egg-rr12.1

    \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \color{blue}{\frac{\left(D \cdot M\right) \cdot h}{\frac{d}{D \cdot M} \cdot d}}}{\ell}} \]

  6. Applied egg-rr10.0

    \[\leadsto w0 \cdot \sqrt{1 - \frac{0.25 \cdot \color{blue}{\left(\frac{h}{\frac{d}{D \cdot M}} \cdot \frac{D}{\frac{d}{M}}\right)}}{\ell}} \]

  7. Final simplification10.0

    \[\leadsto w0 \cdot \sqrt{1 + \frac{\left(\frac{h}{\frac{d}{D \cdot M}} \cdot \frac{D}{\frac{d}{M}}\right) \cdot -0.25}{\ell}} \]

Alternatives

Alternative 1
Error13.8
Cost64
\[w0 \]

Error

Reproduce

herbie shell --seed 2022310 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))