Henrywood and Agarwal, Equation (9a)

Average Error: 13.8 → 8.3

Time: 14.1s

Precision: binary64

Cost: 7872

\[ \begin{array}{c}[M, D] = \mathsf{sort}([M, D])\\ \end{array} \]

Math

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

↓

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

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

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

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 * (0.5d0 * (d / d_1))) * h) * (((d / d_1) * (m * (-0.5d0))) / l))))
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 * (0.5 * (D / d))) * h) * (((D / d) * (M * -0.5)) / l))));
}

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

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(Float64(M * Float64(0.5 * Float64(D / d))) * h) * Float64(Float64(Float64(D / d) * Float64(M * -0.5)) / l)))))
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 * (0.5 * (D / d))) * h) * (((D / d) * (M * -0.5)) / l))));
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[(N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision] * N[(N[(N[(D / d), $MachinePrecision] * N[(M * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 13.8

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

2. Simplified 13.9

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

3. Applied egg-rr 13.5

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

4. Applied egg-rr 8.3

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

5. Applied egg-rr 8.3

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

6. Final simplification 8.3

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

Alternatives

Alternative 1
Error	11.8
Cost	8136

\[\begin{array}{l} t_0 := w0 \cdot \sqrt{1 + 0.25 \cdot \frac{\left(h \cdot \left(M \cdot D\right)\right) \cdot \left(\left(M \cdot D\right) \cdot \frac{-1}{d}\right)}{d \cdot \ell}}\\ \mathbf{if}\;w0 \leq -3.14108765434416 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w0 \leq 4.202006118738102 \cdot 10^{-83}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{\left(M \cdot h\right) \cdot \left(M \cdot \frac{D}{d}\right)}{\frac{d}{D}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	10.9
Cost	8004

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq 0:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{0.25 \cdot \frac{\left(M \cdot h\right) \cdot \left(M \cdot \frac{D}{d}\right)}{\frac{d}{D}}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	13.4
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2022221 
(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))))))