Henrywood and Agarwal, Equation (9a)

Average Error: 13.9 → 8.5

Time: 15.6s

Precision: binary64

Cost: 7744

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{M}{d} \cdot D\\ w0 \cdot \sqrt{1 - t_0 \cdot \left(\frac{t_0}{\ell \cdot 4} \cdot h\right)} \end{array} \]

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 (* (/ M d) D)))
   (* w0 (sqrt (- 1.0 (* t_0 (* (/ t_0 (* l 4.0)) 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 = (M / d) * D;
	return w0 * sqrt((1.0 - (t_0 * ((t_0 / (l * 4.0)) * 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 = (m / d_1) * d
    code = w0 * sqrt((1.0d0 - (t_0 * ((t_0 / (l * 4.0d0)) * 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 = (M / d) * D;
	return w0 * Math.sqrt((1.0 - (t_0 * ((t_0 / (l * 4.0)) * 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 = (M / d) * D
	return w0 * math.sqrt((1.0 - (t_0 * ((t_0 / (l * 4.0)) * 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(M / d) * D)
	return Float64(w0 * sqrt(Float64(1.0 - Float64(t_0 * Float64(Float64(t_0 / Float64(l * 4.0)) * 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 = (M / d) * D;
	tmp = w0 * sqrt((1.0 - (t_0 * ((t_0 / (l * 4.0)) * 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[(M / d), $MachinePrecision] * D), $MachinePrecision]}, N[(w0 * N[Sqrt[N[(1.0 - N[(t$95$0 * N[(N[(t$95$0 / N[(l * 4.0), $MachinePrecision]), $MachinePrecision] * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 13.9

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

2. Simplified 13.9

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

   [Start] 13.9	\[ w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
   times-frac [=>] 13.9	\[ w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}} \]

3. Applied egg-rr 13.5

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{M}{d} \cdot D\right)}{\frac{\ell}{h} \cdot 4}}} \]

4. Applied egg-rr 12.1

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{\frac{M}{d} \cdot D}{\frac{\ell}{h} \cdot 4} \cdot \left(\frac{M}{d} \cdot D\right)}} \]

5. Applied egg-rr 8.5

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{\frac{M}{d} \cdot D}{\ell \cdot 4} \cdot h\right)} \cdot \left(\frac{M}{d} \cdot D\right)} \]

6. Final simplification 8.5

   \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M}{d} \cdot D\right) \cdot \left(\frac{\frac{M}{d} \cdot D}{\ell \cdot 4} \cdot h\right)} \]

Alternatives

Alternative 1
Error	8.9
Cost	8264

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \left(\ell \cdot 4\right)}\right)}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -2 \cdot 10^{-304}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d}\right) \cdot \frac{\left(M \cdot D\right) \cdot 0.25}{d}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 2
Error	12.2
Cost	8141

\[\begin{array}{l} \mathbf{if}\;D \leq -3 \cdot 10^{-92} \lor \neg \left(D \leq 1.9 \cdot 10^{-81}\right) \land D \leq 1.8 \cdot 10^{+163}:\\ \;\;\;\;w0 \cdot \sqrt{1 - D \cdot \left(\frac{M}{d} \cdot \frac{D \cdot \left(M \cdot h\right)}{d \cdot \left(\ell \cdot 4\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Alternative 3
Error	13.8
Cost	64

\[w0 \]

Reproduce

herbie shell --seed 2023010 
(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))))))