Henrywood and Agarwal, Equation (12)

Average Error: 26.6 → 19.3

Time: 15.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ t_1 := \sqrt{\frac{h}{\ell}}\\ t_2 := \frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}}\\ t_3 := \sqrt{\frac{{h}^{-1}}{\ell}}\\ \mathbf{if}\;t_0 \leq -7.283059180255588 \cdot 10^{-249}:\\ \;\;\;\;t_2 \cdot \left(1 - 0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot t_1\right)}^{2}\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(d, t_3, -0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{\frac{\frac{d}{M}}{D \cdot \left(M \cdot D\right)}}\right)\\ \mathbf{elif}\;t_0 \leq 7.116611040597342 \cdot 10^{+245}:\\ \;\;\;\;t_2 \cdot \left(1 - 0.5 \cdot {\left(0.5 \cdot \left(t_1 \cdot \frac{M \cdot D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot t_3\\ \end{array} \]

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
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)) (/ h l)))))
        (t_1 (sqrt (/ h l)))
        (t_2 (/ (sqrt (/ d l)) (sqrt (/ h d))))
        (t_3 (sqrt (/ (pow h -1.0) l))))
   (if (<= t_0 -7.283059180255588e-249)
     (* t_2 (- 1.0 (* 0.5 (pow (* (* (* 0.5 M) (/ D d)) t_1) 2.0))))
     (if (<= t_0 0.0)
       (fma
        d
        t_3
        (* -0.125 (/ (sqrt (/ h (pow l 3.0))) (/ (/ d M) (* D (* M D))))))
       (if (<= t_0 7.116611040597342e+245)
         (* t_2 (- 1.0 (* 0.5 (pow (* 0.5 (* t_1 (/ (* M D) d))) 2.0))))
         (* d t_3))))))

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 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((0.5 * pow(((M * D) / (d * 2.0)), 2.0)) * (h / l)));
	double t_1 = sqrt((h / l));
	double t_2 = sqrt((d / l)) / sqrt((h / d));
	double t_3 = sqrt((pow(h, -1.0) / l));
	double tmp;
	if (t_0 <= -7.283059180255588e-249) {
		tmp = t_2 * (1.0 - (0.5 * pow((((0.5 * M) * (D / d)) * t_1), 2.0)));
	} else if (t_0 <= 0.0) {
		tmp = fma(d, t_3, (-0.125 * (sqrt((h / pow(l, 3.0))) / ((d / M) / (D * (M * D))))));
	} else if (t_0 <= 7.116611040597342e+245) {
		tmp = t_2 * (1.0 - (0.5 * pow((0.5 * (t_1 * ((M * D) / d))), 2.0)));
	} else {
		tmp = d * t_3;
	}
	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(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)) * Float64(h / l))))
	t_1 = sqrt(Float64(h / l))
	t_2 = Float64(sqrt(Float64(d / l)) / sqrt(Float64(h / d)))
	t_3 = sqrt(Float64((h ^ -1.0) / l))
	tmp = 0.0
	if (t_0 <= -7.283059180255588e-249)
		tmp = Float64(t_2 * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(0.5 * M) * Float64(D / d)) * t_1) ^ 2.0))));
	elseif (t_0 <= 0.0)
		tmp = fma(d, t_3, Float64(-0.125 * Float64(sqrt(Float64(h / (l ^ 3.0))) / Float64(Float64(d / M) / Float64(D * Float64(M * D))))));
	elseif (t_0 <= 7.116611040597342e+245)
		tmp = Float64(t_2 * Float64(1.0 - Float64(0.5 * (Float64(0.5 * Float64(t_1 * Float64(Float64(M * D) / d))) ^ 2.0))));
	else
		tmp = Float64(d * t_3);
	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[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(h / d), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[h, -1.0], $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -7.283059180255588e-249], N[(t$95$2 * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(d * t$95$3 + N[(-0.125 * N[(N[Sqrt[N[(h / N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(d / M), $MachinePrecision] / N[(D * N[(M * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 7.116611040597342e+245], N[(t$95$2 * N[(1.0 - N[(0.5 * N[Power[N[(0.5 * N[(t$95$1 * N[(N[(M * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(d * t$95$3), $MachinePrecision]]]]]]]]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -7.28305918025558773e-249

   1. Initial program 28.0

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

   2. Applied egg-rr 28.3

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

   3. Applied egg-rr 29.0

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

   4. Applied egg-rr 29.1

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

   5. Applied egg-rr 21.9

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

   if -7.28305918025558773e-249 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < -0.0

   1. Initial program 40.7

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

   2. Taylor expanded in d around 0 29.8

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{\ell \cdot h}} - 0.125 \cdot \left(\frac{{D}^{2} \cdot {M}^{2}}{d} \cdot \sqrt{\frac{h}{{\ell}^{3}}}\right)} \]

   3. Simplified 28.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(d, \sqrt{\frac{{h}^{-1}}{\ell}}, -0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{\frac{\frac{d}{M}}{D \cdot \left(D \cdot M\right)}}\right)} \]

   if -0.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 7.11661104059734197e245

   1. Initial program 1.0

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

   2. Applied egg-rr 1.1

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

   3. Applied egg-rr 1.4

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

   4. Applied egg-rr 1.4

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

   5. Taylor expanded in M around 0 1.0

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

   if 7.11661104059734197e245 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 61.6

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

   2. Taylor expanded in d around inf 43.7

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

   3. Simplified 43.7

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{{h}^{-1}}{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 19.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq -7.283059180255588 \cdot 10^{-249}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(d, \sqrt{\frac{{h}^{-1}}{\ell}}, -0.125 \cdot \frac{\sqrt{\frac{h}{{\ell}^{3}}}}{\frac{\frac{d}{M}}{D \cdot \left(M \cdot D\right)}}\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \leq 7.116611040597342 \cdot 10^{+245}:\\ \;\;\;\;\frac{\sqrt{\frac{d}{\ell}}}{\sqrt{\frac{h}{d}}} \cdot \left(1 - 0.5 \cdot {\left(0.5 \cdot \left(\sqrt{\frac{h}{\ell}} \cdot \frac{M \cdot D}{d}\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{{h}^{-1}}{\ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(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)))))