Henrywood and Agarwal, Equation (13)

Average Error: 59.7 → 24.9

Time: 14.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{0.5}{c0}, \frac{D}{\frac{d}{M \cdot \left(M \cdot \left(w \cdot h\right)\right)} \cdot \frac{d}{D}}, 0\right)\\ \mathbf{if}\;D \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;D \leq 4.4 \cdot 10^{+145}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{D \cdot D}{\frac{d}{M} \cdot \frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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
         (*
          (/ c0 (* 2.0 w))
          (fma (/ 0.5 c0) (/ D (* (/ d (* M (* M (* w h)))) (/ d D))) 0.0))))
   (if (<= D -1.35e+154)
     t_0
     (if (<= D 4.4e+145) (* 0.25 (* h (/ (* D D) (* (/ d M) (/ d M))))) t_0))))

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 = (c0 / (2.0 * w)) * fma((0.5 / c0), (D / ((d / (M * (M * (w * h)))) * (d / D))), 0.0);
	double tmp;
	if (D <= -1.35e+154) {
		tmp = t_0;
	} else if (D <= 4.4e+145) {
		tmp = 0.25 * (h * ((D * D) / ((d / M) * (d / M))));
	} else {
		tmp = t_0;
	}
	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(c0 / Float64(2.0 * w)) * fma(Float64(0.5 / c0), Float64(D / Float64(Float64(d / Float64(M * Float64(M * Float64(w * h)))) * Float64(d / D))), 0.0))
	tmp = 0.0
	if (D <= -1.35e+154)
		tmp = t_0;
	elseif (D <= 4.4e+145)
		tmp = Float64(0.25 * Float64(h * Float64(Float64(D * D) / Float64(Float64(d / M) * Float64(d / M)))));
	else
		tmp = t_0;
	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[(c0 / N[(2.0 * w), $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / c0), $MachinePrecision] * N[(D / N[(N[(d / N[(M * N[(M * N[(w * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(d / D), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[D, -1.35e+154], t$95$0, If[LessEqual[D, 4.4e+145], N[(0.25 * N[(h * N[(N[(D * D), $MachinePrecision] / N[(N[(d / M), $MachinePrecision] * N[(d / M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if D < -1.35000000000000003e154 or 4.40000000000000017e145 < D

   1. Initial program 62.7

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

   2. Taylor expanded in c0 around -inf 63.8

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

   3. Simplified 63.2

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

   4. Applied egg-rr 42.6

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

   if -1.35000000000000003e154 < D < 4.40000000000000017e145

   1. Initial program 59.4

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

   2. Taylor expanded in c0 around -inf 59.3

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

   3. Simplified 37.3

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

   4. Taylor expanded in c0 around 0 31.5

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

   5. Simplified 22.7

      \[\leadsto \color{blue}{0.25 \cdot \left(h \cdot \frac{D \cdot D}{\frac{d}{M} \cdot \frac{d}{M}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{0.5}{c0}, \frac{D}{\frac{d}{M \cdot \left(M \cdot \left(w \cdot h\right)\right)} \cdot \frac{d}{D}}, 0\right)\\ \mathbf{elif}\;D \leq 4.4 \cdot 10^{+145}:\\ \;\;\;\;0.25 \cdot \left(h \cdot \frac{D \cdot D}{\frac{d}{M} \cdot \frac{d}{M}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \mathsf{fma}\left(\frac{0.5}{c0}, \frac{D}{\frac{d}{M \cdot \left(M \cdot \left(w \cdot h\right)\right)} \cdot \frac{d}{D}}, 0\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022181 
(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))))))