Henrywood and Agarwal, Equation (12)

Average Error: 26.7 → 18.2

Time: 18.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.5}\\ t_1 := \frac{D \cdot M}{d \cdot 2}\\ t_2 := {t_1}^{2}\\ t_3 := {\left(-d\right)}^{0.5}\\ \mathbf{if}\;d \leq -1.5325362465457807 \cdot 10^{-77}:\\ \;\;\;\;\left(\left(t_3 \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right) \cdot t_0\right) \cdot \left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot t_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := {\left(\frac{d}{h}\right)}^{0.5}\\ \mathbf{if}\;d \leq -2.941112501210015 \cdot 10^{-305}:\\ \;\;\;\;\left(t_4 \cdot \left(t_3 \cdot {\left(\frac{-1}{\ell}\right)}^{0.5}\right)\right) \cdot \left(1 - \left(0.5 \cdot t_2\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 3.5292094803642007 \cdot 10^{+121}:\\ \;\;\;\;\begin{array}{l} t_5 := \left|t_1 \cdot \sqrt{\frac{0.5}{\ell}}\right| \cdot \sqrt{h}\\ \left(t_0 \cdot t_4\right) \cdot \left(1 - t_5 \cdot t_5\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array}\\ \end{array} \]

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 l) 0.5))
        (t_1 (/ (* D M) (* d 2.0)))
        (t_2 (pow t_1 2.0))
        (t_3 (pow (- d) 0.5)))
   (if (<= d -1.5325362465457807e-77)
     (* (* (* t_3 (pow (/ -1.0 h) 0.5)) t_0) (- 1.0 (* h (* (/ 0.5 l) t_2))))
     (let* ((t_4 (pow (/ d h) 0.5)))
       (if (<= d -2.941112501210015e-305)
         (*
          (* t_4 (* t_3 (pow (/ -1.0 l) 0.5)))
          (- 1.0 (* (* 0.5 t_2) (/ h l))))
         (if (<= d 3.5292094803642007e+121)
           (let* ((t_5 (* (fabs (* t_1 (sqrt (/ 0.5 l)))) (sqrt h))))
             (* (* t_0 t_4) (- 1.0 (* t_5 t_5))))
           (* 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 = pow((d / l), 0.5);
	double t_1 = (D * M) / (d * 2.0);
	double t_2 = pow(t_1, 2.0);
	double t_3 = pow(-d, 0.5);
	double tmp;
	if (d <= -1.5325362465457807e-77) {
		tmp = ((t_3 * pow((-1.0 / h), 0.5)) * t_0) * (1.0 - (h * ((0.5 / l) * t_2)));
	} else {
		double t_4 = pow((d / h), 0.5);
		double tmp_1;
		if (d <= -2.941112501210015e-305) {
			tmp_1 = (t_4 * (t_3 * pow((-1.0 / l), 0.5))) * (1.0 - ((0.5 * t_2) * (h / l)));
		} else if (d <= 3.5292094803642007e+121) {
			double t_5 = fabs(t_1 * sqrt(0.5 / l)) * sqrt(h);
			tmp_1 = (t_0 * t_4) * (1.0 - (t_5 * t_5));
		} else {
			tmp_1 = d * sqrt(1.0 / (h * l));
		}
		tmp = tmp_1;
	}
	return tmp;
}

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 d < -1.53253624654578069e-77

   1. Initial program 22.0

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

   3. Applied clear-num_binary64 22.0

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

   4. Applied un-div-inv_binary64 21.9

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

   5. Applied associate-/r/_binary64 20.2

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

   6. Simplified 20.2

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

   7. Taylor expanded around -inf 15.5

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

   8. Simplified 11.1

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

   if -1.53253624654578069e-77 < d < -2.941112501210015e-305

   1. Initial program 34.0

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

   2. Taylor expanded around -inf 30.0

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

   3. Simplified 27.3

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

   if -2.941112501210015e-305 < d < 3.5292094803642007e121

   1. Initial program 27.1

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

   3. Applied clear-num_binary64 27.1

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

   4. Applied un-div-inv_binary64 26.9

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

   5. Applied associate-/r/_binary64 25.6

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

   6. Simplified 25.6

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

   8. Applied add-sqr-sqrt_binary64 25.7

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

   9. Applied add-sqr-sqrt_binary64 25.8

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

   10. Applied unswap-sqr_binary64 25.8

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

   11. Simplified 25.7

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

   12. Simplified 21.4

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

   if 3.5292094803642007e121 < d

   1. Initial program 27.9

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

   2. Taylor expanded around inf 15.7

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

   3. Simplified 15.7

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

4. Final simplification 18.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5325362465457807 \cdot 10^{-77}:\\ \;\;\;\;\left(\left({\left(-d\right)}^{0.5} \cdot {\left(\frac{-1}{h}\right)}^{0.5}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - h \cdot \left(\frac{0.5}{\ell} \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{elif}\;d \leq -2.941112501210015 \cdot 10^{-305}:\\ \;\;\;\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot \left({\left(-d\right)}^{0.5} \cdot {\left(\frac{-1}{\ell}\right)}^{0.5}\right)\right) \cdot \left(1 - \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{elif}\;d \leq 3.5292094803642007 \cdot 10^{+121}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{0.5} \cdot {\left(\frac{d}{h}\right)}^{0.5}\right) \cdot \left(1 - \left(\left|\frac{D \cdot M}{d \cdot 2} \cdot \sqrt{\frac{0.5}{\ell}}\right| \cdot \sqrt{h}\right) \cdot \left(\left|\frac{D \cdot M}{d \cdot 2} \cdot \sqrt{\frac{0.5}{\ell}}\right| \cdot \sqrt{h}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{1}{h \cdot \ell}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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)))))