Henrywood and Agarwal, Equation (9a)

Average Error: 14.2 → 8.8

Time: 16.4s

Precision: binary64

Cost: 7872

Math

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

↓

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

FPCore

(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 (/ (* M (* (/ D d) -0.5)) (/ (* l (* 2.0 (/ (/ d D) M))) h))))))

C

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 + ((M * ((D / d) * -0.5)) / ((l * (2.0 * ((d / D) / M))) / h))));
}

Fortran

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 + ((m * ((d / d_1) * (-0.5d0))) / ((l * (2.0d0 * ((d_1 / d) / m))) / h))))
end function

Java

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 + ((M * ((D / d) * -0.5)) / ((l * (2.0 * ((d / D) / M))) / h))));
}

Python

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 + ((M * ((D / d) * -0.5)) / ((l * (2.0 * ((d / D) / M))) / h))))

Julia

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(M * Float64(Float64(D / d) * -0.5)) / Float64(Float64(l * Float64(2.0 * Float64(Float64(d / D) / M))) / h)))))
end

MATLAB

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 + ((M * ((D / d) * -0.5)) / ((l * (2.0 * ((d / D) / M))) / h))));
end

Wolfram

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[(M * N[(N[(D / d), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(N[(l * N[(2.0 * N[(N[(d / D), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.2

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

2. Simplified 14.2

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

   [Start] 14.2	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   associate-/l* [=>] 14.2	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}}^{2} \cdot \frac{h}{\ell}} \]

3. Applied egg-rr 12.2

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

4. Applied egg-rr 8.8

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

5. Final simplification 8.8

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

Alternatives

Alternative 1
Error	12.7
Cost	8136

\[\begin{array}{l} \mathbf{if}\;h \leq -5.4 \cdot 10^{+101}:\\ \;\;\;\;w0\\ \mathbf{elif}\;h \leq 2 \cdot 10^{-51}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\frac{d}{D}} \cdot \frac{M \cdot \left(\frac{h}{\ell} \cdot -0.25\right)}{\frac{d}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M}{\frac{\ell}{h} \cdot \left(\left(\frac{2}{D} \cdot \frac{d}{M}\right) \cdot \left(2 \cdot \frac{d}{D}\right)\right)}}\\ \end{array} \]

Alternative 2
Error	10.7
Cost	8132

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{\left(\left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \left(M \cdot \left(M \cdot h\right)\right)\right) \cdot -0.25}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M \cdot \left(\frac{D}{d} \cdot -0.5\right)}{\frac{\ell}{h} \cdot \frac{2}{M \cdot \frac{D}{d}}}}\\ \end{array} \]

Alternative 3
Error	12.5
Cost	8008

\[\begin{array}{l} \mathbf{if}\;h \leq -2.25 \cdot 10^{+102}:\\ \;\;\;\;w0\\ \mathbf{elif}\;h \leq 10^{-50}:\\ \;\;\;\;w0 \cdot \sqrt{1 + \frac{M}{\frac{d}{D}} \cdot \frac{M \cdot \left(\frac{h}{\ell} \cdot -0.25\right)}{\frac{d}{D}}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 4
Error	14.5
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023041 
(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))))))