Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 31.2

Time: 16.3s

Precision: binary64

Math

\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

↓

\[\begin{array}{l} t_0 := \frac{0.5 \cdot \frac{D \cdot D}{\frac{\frac{d \cdot \frac{d}{h}}{w \cdot M}}{M}}}{2 \cdot w}\\ t_1 := {\left(\frac{D}{d}\right)}^{2}\\ \mathbf{if}\;D \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(2 \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{elif}\;D \leq -5.2 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 2.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{c0 \cdot \mathsf{fma}\left(0.5, \frac{w \cdot \left(M \cdot t_1\right)}{\frac{\frac{c0}{h}}{M}}, 0\right)}{2 \cdot w}\\ \mathbf{elif}\;D \leq 12.2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \mathsf{fma}\left(0.5, w \cdot \left(t_1 \cdot \frac{M}{\frac{c0}{h \cdot M}}\right), 0\right)}{2 \cdot w}\\ \end{array} \]

FPCore

(FPCore (c0 w h D d M)
 :precision binary64
 (*
  (/ c0 (* 2.0 w))
  (+
   (/ (* c0 (* d d)) (* (* w h) (* D D)))
   (sqrt
    (-
     (*
      (/ (* c0 (* d d)) (* (* w h) (* D D)))
      (/ (* c0 (* d d)) (* (* w h) (* D D))))
     (* M M))))))

↓

(FPCore (c0 w h D d M)
 :precision binary64
 (let* ((t_0 (/ (* 0.5 (/ (* D D) (/ (/ (* d (/ d h)) (* w M)) M))) (* 2.0 w)))
        (t_1 (pow (/ D d) 2.0)))
   (if (<= D -2.8e+17)
     (* (/ c0 (* 2.0 w)) (* (* (/ d D) (/ d D)) (* 2.0 (/ c0 (* w h)))))
     (if (<= D -5.2e-218)
       t_0
       (if (<= D 2.05e-134)
         (/ (* c0 (fma 0.5 (/ (* w (* M t_1)) (/ (/ c0 h) M)) 0.0)) (* 2.0 w))
         (if (<= D 12.2)
           t_0
           (/
            (* c0 (fma 0.5 (* w (* t_1 (/ M (/ c0 (* h M))))) 0.0))
            (* 2.0 w))))))))

C

double code(double c0, double w, double h, double D, double d, double M) {
	return (c0 / (2.0 * w)) * (((c0 * (d * d)) / ((w * h) * (D * D))) + sqrt(((((c0 * (d * d)) / ((w * h) * (D * D))) * ((c0 * (d * d)) / ((w * h) * (D * D)))) - (M * M))));
}

↓

double code(double c0, double w, double h, double D, double d, double M) {
	double t_0 = (0.5 * ((D * D) / (((d * (d / h)) / (w * M)) / M))) / (2.0 * w);
	double t_1 = pow((D / d), 2.0);
	double tmp;
	if (D <= -2.8e+17) {
		tmp = (c0 / (2.0 * w)) * (((d / D) * (d / D)) * (2.0 * (c0 / (w * h))));
	} else if (D <= -5.2e-218) {
		tmp = t_0;
	} else if (D <= 2.05e-134) {
		tmp = (c0 * fma(0.5, ((w * (M * t_1)) / ((c0 / h) / M)), 0.0)) / (2.0 * w);
	} else if (D <= 12.2) {
		tmp = t_0;
	} else {
		tmp = (c0 * fma(0.5, (w * (t_1 * (M / (c0 / (h * M))))), 0.0)) / (2.0 * w);
	}
	return tmp;
}

Julia

function code(c0, w, h, D, d, M)
	return Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) + sqrt(Float64(Float64(Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D))) * Float64(Float64(c0 * Float64(d * d)) / Float64(Float64(w * h) * Float64(D * D)))) - Float64(M * M)))))
end

↓

function code(c0, w, h, D, d, M)
	t_0 = Float64(Float64(0.5 * Float64(Float64(D * D) / Float64(Float64(Float64(d * Float64(d / h)) / Float64(w * M)) / M))) / Float64(2.0 * w))
	t_1 = Float64(D / d) ^ 2.0
	tmp = 0.0
	if (D <= -2.8e+17)
		tmp = Float64(Float64(c0 / Float64(2.0 * w)) * Float64(Float64(Float64(d / D) * Float64(d / D)) * Float64(2.0 * Float64(c0 / Float64(w * h)))));
	elseif (D <= -5.2e-218)
		tmp = t_0;
	elseif (D <= 2.05e-134)
		tmp = Float64(Float64(c0 * fma(0.5, Float64(Float64(w * Float64(M * t_1)) / Float64(Float64(c0 / h) / M)), 0.0)) / Float64(2.0 * w));
	elseif (D <= 12.2)
		tmp = t_0;
	else
		tmp = Float64(Float64(c0 * fma(0.5, Float64(w * Float64(t_1 * Float64(M / Float64(c0 / Float64(h * M))))), 0.0)) / Float64(2.0 * w));
	end
	return tmp
end

Wolfram

code[c0_, w_, h_, D_, d_, M_] := N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(c0 * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(w * h), $MachinePrecision] * N[(D * D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[c0_, w_, h_, D_, d_, M_] := Block[{t$95$0 = N[(N[(0.5 * N[(N[(D * D), $MachinePrecision] / N[(N[(N[(d * N[(d / h), $MachinePrecision]), $MachinePrecision] / N[(w * M), $MachinePrecision]), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(D / d), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[D, -2.8e+17], N[(N[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d / D), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(c0 / N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, -5.2e-218], t$95$0, If[LessEqual[D, 2.05e-134], N[(N[(c0 * N[(0.5 * N[(N[(w * N[(M * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(c0 / h), $MachinePrecision] / M), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision], If[LessEqual[D, 12.2], t$95$0, N[(N[(c0 * N[(0.5 * N[(w * N[(t$95$1 * N[(M / N[(c0 / N[(h * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * w), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if D < -2.8e17

   1. Initial program 58.2

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around inf 59.0

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(2 \cdot \frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(w \cdot h\right)}\right)} \]

   3. Simplified 49.6

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(\frac{c0}{w \cdot h} \cdot 2\right)\right)} \]

   if -2.8e17 < D < -5.19999999999999966e-218 or 2.0500000000000001e-134 < D < 12.199999999999999

   1. Initial program 57.3

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around -inf 57.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   3. Simplified 35.1

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w}{\frac{c0}{M \cdot \left(M \cdot h\right)}}, c0 \cdot 0\right)} \]

   4. Applied egg-rr 30.4

      \[\leadsto \color{blue}{\frac{c0 \cdot \mathsf{fma}\left(0.5, \frac{{\left(\frac{D}{d}\right)}^{2} \cdot w}{\frac{c0}{M \cdot \left(M \cdot h\right)}}, 0\right)}{w \cdot 2}} \]

   5. Taylor expanded in c0 around 0 33.5

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

   6. Simplified 27.6

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{D \cdot D}{\frac{\frac{d \cdot \frac{d}{h}}{w \cdot M}}{M}}}}{w \cdot 2} \]

   if -5.19999999999999966e-218 < D < 2.0500000000000001e-134

   1. Initial program 63.3

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around -inf 63.3

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   3. Simplified 32.5

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w}{\frac{c0}{M \cdot \left(M \cdot h\right)}}, c0 \cdot 0\right)} \]

   4. Applied egg-rr 27.0

      \[\leadsto \color{blue}{\frac{c0 \cdot \mathsf{fma}\left(0.5, \frac{{\left(\frac{D}{d}\right)}^{2} \cdot w}{\frac{c0}{M \cdot \left(M \cdot h\right)}}, 0\right)}{w \cdot 2}} \]

   5. Applied egg-rr 25.9

      \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(0.5, \color{blue}{w \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{M}{\frac{c0}{M \cdot h}}\right)}, 0\right)}{w \cdot 2} \]

   6. Applied egg-rr 25.9

      \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(0.5, \color{blue}{\frac{\left({\left(\frac{D}{d}\right)}^{2} \cdot M\right) \cdot w}{\frac{\frac{c0}{h}}{M}}}, 0\right)}{w \cdot 2} \]

   if 12.199999999999999 < D

   1. Initial program 58.9

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \]

   2. Taylor expanded in c0 around -inf 58.9

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\left(0.5 \cdot \frac{{D}^{2} \cdot \left(w \cdot \left({M}^{2} \cdot h\right)\right)}{{d}^{2} \cdot c0} + -1 \cdot \left(\left(\frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)} + -1 \cdot \frac{{d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right) \cdot c0\right)\right)} \]

   3. Simplified 38.7

      \[\leadsto \frac{c0}{2 \cdot w} \cdot \color{blue}{\mathsf{fma}\left(0.5, \left(\frac{D}{d} \cdot \frac{D}{d}\right) \cdot \frac{w}{\frac{c0}{M \cdot \left(M \cdot h\right)}}, c0 \cdot 0\right)} \]

   4. Applied egg-rr 35.5

      \[\leadsto \color{blue}{\frac{c0 \cdot \mathsf{fma}\left(0.5, \frac{{\left(\frac{D}{d}\right)}^{2} \cdot w}{\frac{c0}{M \cdot \left(M \cdot h\right)}}, 0\right)}{w \cdot 2}} \]

   5. Applied egg-rr 34.7

      \[\leadsto \frac{c0 \cdot \mathsf{fma}\left(0.5, \color{blue}{w \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{M}{\frac{c0}{M \cdot h}}\right)}, 0\right)}{w \cdot 2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \left(2 \cdot \frac{c0}{w \cdot h}\right)\right)\\ \mathbf{elif}\;D \leq -5.2 \cdot 10^{-218}:\\ \;\;\;\;\frac{0.5 \cdot \frac{D \cdot D}{\frac{\frac{d \cdot \frac{d}{h}}{w \cdot M}}{M}}}{2 \cdot w}\\ \mathbf{elif}\;D \leq 2.05 \cdot 10^{-134}:\\ \;\;\;\;\frac{c0 \cdot \mathsf{fma}\left(0.5, \frac{w \cdot \left(M \cdot {\left(\frac{D}{d}\right)}^{2}\right)}{\frac{\frac{c0}{h}}{M}}, 0\right)}{2 \cdot w}\\ \mathbf{elif}\;D \leq 12.2:\\ \;\;\;\;\frac{0.5 \cdot \frac{D \cdot D}{\frac{\frac{d \cdot \frac{d}{h}}{w \cdot M}}{M}}}{2 \cdot w}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \mathsf{fma}\left(0.5, w \cdot \left({\left(\frac{D}{d}\right)}^{2} \cdot \frac{M}{\frac{c0}{h \cdot M}}\right), 0\right)}{2 \cdot w}\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  :precision binary64
  (* (/ c0 (* 2.0 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))