Henrywood and Agarwal, Equation (13)

Average Error: 47.9 → 28.5

Time: 18.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{c0}{2 \cdot w}\\ t_1 := \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)}\\ t_2 := t_0 \cdot \left(t_1 + \sqrt{t_1 \cdot t_1 - M \cdot M}\right)\\ t_3 := \left(w \cdot h\right) \cdot D\\ \mathbf{if}\;t_2 \leq -1.5156684725312666 \cdot 10^{-127}:\\ \;\;\;\;t_0 \cdot \left(2 \cdot \frac{\frac{d \cdot \left(c0 \cdot d\right)}{D}}{t_3}\right)\\ \mathbf{elif}\;t_2 \leq 0 \lor \neg \left(t_2 \leq \infty\right):\\ \;\;\;\;0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{D}}{t_3}\right)}{2 \cdot w}\\ \end{array} \]

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 (/ c0 (* 2.0 w)))
        (t_1 (/ (* c0 (* d d)) (* (* w h) (* D D))))
        (t_2 (* t_0 (+ t_1 (sqrt (- (* t_1 t_1) (* M M))))))
        (t_3 (* (* w h) D)))
   (if (<= t_2 -1.5156684725312666e-127)
     (* t_0 (* 2.0 (/ (/ (* d (* c0 d)) D) t_3)))
     (if (or (<= t_2 0.0) (not (<= t_2 INFINITY)))
       (* 0.25 (/ (* (pow D 2.0) (* h (pow M 2.0))) (pow d 2.0)))
       (/ (* c0 (* 2.0 (/ (/ (* c0 (pow d 2.0)) D) t_3))) (* 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 = c0 / (2.0 * w);
	double t_1 = (c0 * (d * d)) / ((w * h) * (D * D));
	double t_2 = t_0 * (t_1 + sqrt((t_1 * t_1) - (M * M)));
	double t_3 = (w * h) * D;
	double tmp;
	if (t_2 <= -1.5156684725312666e-127) {
		tmp = t_0 * (2.0 * (((d * (c0 * d)) / D) / t_3));
	} else if ((t_2 <= 0.0) || !(t_2 <= ((double) INFINITY))) {
		tmp = 0.25 * ((pow(D, 2.0) * (h * pow(M, 2.0))) / pow(d, 2.0));
	} else {
		tmp = (c0 * (2.0 * (((c0 * pow(d, 2.0)) / D) / t_3))) / (2.0 * w);
	}
	return tmp;
}

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -1.5156684725312666e-127

   1. Initial program 14.4

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

   2. Taylor expanded in c0 around inf 12.6

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

   3. Applied unpow2_binary64 12.6

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

   4. Applied associate-*l*_binary64 11.7

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

   5. Applied associate-/r*_binary64 10.8

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

   6. Applied unpow2_binary64 10.8

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

   7. Applied associate-*l*_binary64 9.7

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

   if -1.5156684725312666e-127 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < -0.0 or +inf.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M)))))

   1. Initial program 61.4

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

   2. Taylor expanded in c0 around -inf 36.0

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

   if -0.0 < (*.f64 (/.f64 c0 (*.f64 2 w)) (+.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (sqrt.f64 (-.f64 (*.f64 (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D))) (/.f64 (*.f64 c0 (*.f64 d d)) (*.f64 (*.f64 w h) (*.f64 D D)))) (*.f64 M M))))) < +inf.0

   1. Initial program 14.5

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

   2. Taylor expanded in c0 around inf 12.5

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

   3. Applied unpow2_binary64 12.5

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

   4. Applied associate-*l*_binary64 11.1

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

   5. Applied associate-/r*_binary64 10.2

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

   6. Applied associate-*l/_binary64 10.4

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

4. Final simplification 28.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq -1.5156684725312666 \cdot 10^{-127}:\\ \;\;\;\;\frac{c0}{2 \cdot w} \cdot \left(2 \cdot \frac{\frac{d \cdot \left(c0 \cdot d\right)}{D}}{\left(w \cdot h\right) \cdot D}\right)\\ \mathbf{elif}\;\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq 0 \lor \neg \left(\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right) \leq \infty\right):\\ \;\;\;\;0.25 \cdot \frac{{D}^{2} \cdot \left(h \cdot {M}^{2}\right)}{{d}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \left(2 \cdot \frac{\frac{c0 \cdot {d}^{2}}{D}}{\left(w \cdot h\right) \cdot D}\right)}{2 \cdot w}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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))))))