Henrywood and Agarwal, Equation (12)

Average Error: 26.7 → 21.1

Time: 11.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, 1\right)\\ \mathbf{if}\;d \leq 0:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+147}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+205}:\\ \;\;\;\;t_0 \cdot \frac{1}{\frac{\sqrt{h}}{\sqrt{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

↓

(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d l))
          (fma (/ h l) (* (pow (* M (/ (/ D d) 2.0)) 2.0) -0.5) 1.0))))
   (if (<= d 0.0)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (if (<= d 3e+90)
       (*
        (* (* (sqrt d) (pow (/ 1.0 h) 0.5)) (pow (/ d l) 0.5))
        (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)))))
       (if (<= d 1.45e+147)
         (* d (sqrt (/ (/ 1.0 h) l)))
         (if (<= d 3.4e+205)
           (* t_0 (/ 1.0 (/ (sqrt h) (sqrt d))))
           (* d (sqrt (/ 1.0 (* h l))))))))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

↓

double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l)) * fma((h / l), (pow((M * ((D / d) / 2.0)), 2.0) * -0.5), 1.0);
	double tmp;
	if (d <= 0.0) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else if (d <= 3e+90) {
		tmp = ((sqrt(d) * pow((1.0 / h), 0.5)) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	} else if (d <= 1.45e+147) {
		tmp = d * sqrt(((1.0 / h) / l));
	} else if (d <= 3.4e+205) {
		tmp = t_0 * (1.0 / (sqrt(h) / sqrt(d)));
	} else {
		tmp = d * sqrt((1.0 / (h * l)));
	}
	return tmp;
}

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

↓

function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / l)) * fma(Float64(h / l), Float64((Float64(M * Float64(Float64(D / d) / 2.0)) ^ 2.0) * -0.5), 1.0))
	tmp = 0.0
	if (d <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	elseif (d <= 3e+90)
		tmp = Float64(Float64(Float64(sqrt(d) * (Float64(1.0 / h) ^ 0.5)) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))));
	elseif (d <= 1.45e+147)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / h) / l)));
	elseif (d <= 3.4e+205)
		tmp = Float64(t_0 * Float64(1.0 / Float64(sqrt(h) / sqrt(d))));
	else
		tmp = Float64(d * sqrt(Float64(1.0 / Float64(h * l))));
	end
	return tmp
end

Wolfram

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

↓

code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(M * N[(N[(D / d), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, 0.0], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[d, 3e+90], N[(N[(N[(N[Sqrt[d], $MachinePrecision] * N[Power[N[(1.0 / h), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.45e+147], N[(d * N[Sqrt[N[(N[(1.0 / h), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.4e+205], N[(t$95$0 * N[(1.0 / N[(N[Sqrt[h], $MachinePrecision] / N[Sqrt[d], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * N[Sqrt[N[(1.0 / N[(h * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < 0.0

   1. Initial program 26.3

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

   2. Simplified 26.7

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

   3. Applied egg-rr 20.5

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

   if 0.0 < d < 2.99999999999999979e90

   1. Initial program 27.0

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

   2. Applied egg-rr 24.9

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

   if 2.99999999999999979e90 < d < 1.4499999999999999e147

   1. Initial program 20.3

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

   2. Applied egg-rr 20.5

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

   3. Taylor expanded in d around inf 16.7

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]

   4. Simplified 16.5

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}} \]

   if 1.4499999999999999e147 < d < 3.4e205

   1. Initial program 27.5

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

   2. Simplified 27.8

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

   3. Applied egg-rr 15.3

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

   if 3.4e205 < d

   1. Initial program 33.0

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

   2. Applied egg-rr 33.1

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

   3. Taylor expanded in d around inf 15.0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\ell \cdot h}} \cdot d} \]
3. Recombined 5 regimes into one program.

4. Final simplification 21.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 0:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, 1\right)\right)\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+90}:\\ \;\;\;\;\left(\left(\sqrt{d} \cdot {\left(\frac{1}{h}\right)}^{0.5}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+147}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{h}}{\ell}}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+205}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \mathsf{fma}\left(\frac{h}{\ell}, {\left(M \cdot \frac{\frac{D}{d}}{2}\right)}^{2} \cdot -0.5, 1\right)\right) \cdot \frac{1}{\frac{\sqrt{h}}{\sqrt{d}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))