Henrywood and Agarwal, Equation (9a)

Average Error: 14.3 → 8.7

Time: 14.3s

Precision: binary64

Cost: 8000

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(0.5 \cdot D\right) \cdot M}{d}\\ w0 \cdot \sqrt{1 - \frac{t_0}{\ell} \cdot \frac{t_0}{\frac{1}{h}}} \end{array} \]

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
 (let* ((t_0 (/ (* (* 0.5 D) M) d)))
   (* w0 (sqrt (- 1.0 (* (/ t_0 l) (/ t_0 (/ 1.0 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) {
	double t_0 = ((0.5 * D) * M) / d;
	return w0 * sqrt((1.0 - ((t_0 / l) * (t_0 / (1.0 / 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
    real(8) :: t_0
    t_0 = ((0.5d0 * d) * m) / d_1
    code = w0 * sqrt((1.0d0 - ((t_0 / l) * (t_0 / (1.0d0 / 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) {
	double t_0 = ((0.5 * D) * M) / d;
	return w0 * Math.sqrt((1.0 - ((t_0 / l) * (t_0 / (1.0 / 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):
	t_0 = ((0.5 * D) * M) / d
	return w0 * math.sqrt((1.0 - ((t_0 / l) * (t_0 / (1.0 / 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)
	t_0 = Float64(Float64(Float64(0.5 * D) * M) / d)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(Float64(t_0 / l) * Float64(t_0 / Float64(1.0 / 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)
	t_0 = ((0.5 * D) * M) / d;
	tmp = w0 * sqrt((1.0 - ((t_0 / l) * (t_0 / (1.0 / 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_] := Block[{t$95$0 = N[(N[(N[(0.5 * D), $MachinePrecision] * M), $MachinePrecision] / d), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(N[(t$95$0 / l), $MachinePrecision] * N[(t$95$0 / N[(1.0 / h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 14.3

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

2. Applied egg-rr 14.2

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

3. Applied egg-rr 13.9

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

4. Applied egg-rr 8.7

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

5. Final simplification 8.7

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

Alternatives

Alternative 1
Error	9.8
Cost	8132

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -2 \cdot 10^{-205}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\left(0.5 \cdot D\right) \cdot M}{d}}{\ell} \cdot \left(h \cdot \left(M \cdot \frac{0.5 \cdot D}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	10.1
Cost	8004

\[\begin{array}{l} t_0 := \frac{\left(0.5 \cdot D\right) \cdot M}{d}\\ \mathbf{if}\;D \leq 4.194907983539999 \cdot 10^{-127}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0}{\ell} \cdot \left(h \cdot \left(M \cdot \frac{0.5 \cdot D}{d}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(t_0 \cdot \frac{h}{\ell}\right)}\\ \end{array} \]

Alternative 3
Error	13.7
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2022325 
(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))))))