Quotient of products

Average Error: 11.3 → 2.1

Time: 4.2s

Precision: binary64

\[[b1, b2] = \mathsf{sort}([b1, b2]) \\]

Math

\[\frac{a1 \cdot a2}{b1 \cdot b2} \]

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -9.070334479927686 \cdot 10^{+287}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1}} \cdot \frac{1}{\frac{b2}{a2}}\\ \mathbf{elif}\;t_0 \leq -1.2861353814587239 \cdot 10^{-303}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\frac{b1}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}} \cdot \frac{\sqrt[3]{a2}}{\frac{b2}{\sqrt[3]{a1}}}\\ \mathbf{elif}\;t_0 \leq 8.508576472122102 \cdot 10^{+294}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \]

FPCore

(FPCore (a1 a2 b1 b2) :precision binary64 (/ (* a1 a2) (* b1 b2)))

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0 (/ (* a1 a2) (* b1 b2))))
   (if (<= t_0 -9.070334479927686e+287)
     (* (/ 1.0 (/ b1 a1)) (/ 1.0 (/ b2 a2)))
     (if (<= t_0 -1.2861353814587239e-303)
       t_0
       (if (<= t_0 0.0)
         (*
          (/ (* (cbrt a2) (cbrt a2)) (/ b1 (* (cbrt a1) (cbrt a1))))
          (/ (cbrt a2) (/ b2 (cbrt a1))))
         (if (<= t_0 8.508576472122102e+294) t_0 (/ (/ a2 b2) (/ b1 a1))))))))

C

double code(double a1, double a2, double b1, double b2) {
	return (a1 * a2) / (b1 * b2);
}

↓

double code(double a1, double a2, double b1, double b2) {
	double t_0 = (a1 * a2) / (b1 * b2);
	double tmp;
	if (t_0 <= -9.070334479927686e+287) {
		tmp = (1.0 / (b1 / a1)) * (1.0 / (b2 / a2));
	} else if (t_0 <= -1.2861353814587239e-303) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((cbrt(a2) * cbrt(a2)) / (b1 / (cbrt(a1) * cbrt(a1)))) * (cbrt(a2) / (b2 / cbrt(a1)));
	} else if (t_0 <= 8.508576472122102e+294) {
		tmp = t_0;
	} else {
		tmp = (a2 / b2) / (b1 / a1);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.4
Herbie	2.1

\[\frac{a1}{b1} \cdot \frac{a2}{b2} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -9.07033447992768629e287

   1. Initial program 54.0

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]

   2. Applied clear-num_binary64 54.0

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}} \]

   3. Applied times-frac_binary64 13.5

      \[\leadsto \frac{1}{\color{blue}{\frac{b1}{a1} \cdot \frac{b2}{a2}}} \]

   4. Applied *-un-lft-identity_binary64 13.5

      \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\frac{b1}{a1} \cdot \frac{b2}{a2}} \]

   5. Applied times-frac_binary64 14.3

      \[\leadsto \color{blue}{\frac{1}{\frac{b1}{a1}} \cdot \frac{1}{\frac{b2}{a2}}} \]

   if -9.07033447992768629e287 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -1.2861353814587239e-303 or 0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 8.5085764721221019e294

   1. Initial program 0.7

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]

   if -1.2861353814587239e-303 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 0.0

   1. Initial program 13.3

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]

   2. Applied clear-num_binary64 13.7

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}} \]

   3. Applied associate-/r*_binary64 8.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{b1 \cdot b2}{a1}}{a2}}} \]

   4. Applied add-cube-cbrt_binary64 8.3

      \[\leadsto \frac{1}{\frac{\frac{b1 \cdot b2}{a1}}{\color{blue}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}}} \]

   5. Applied add-cube-cbrt_binary64 8.3

      \[\leadsto \frac{1}{\frac{\frac{b1 \cdot b2}{\color{blue}{\left(\sqrt[3]{a1} \cdot \sqrt[3]{a1}\right) \cdot \sqrt[3]{a1}}}}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}} \]

   6. Applied times-frac_binary64 5.0

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{b1}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}} \cdot \frac{b2}{\sqrt[3]{a1}}}}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}} \]

   7. Applied times-frac_binary64 3.2

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{b1}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{\frac{b2}{\sqrt[3]{a1}}}{\sqrt[3]{a2}}}} \]

   8. Applied add-cube-cbrt_binary64 3.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{\frac{b1}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{\frac{b2}{\sqrt[3]{a1}}}{\sqrt[3]{a2}}} \]

   9. Applied times-frac_binary64 2.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{\frac{b1}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{b2}{\sqrt[3]{a1}}}{\sqrt[3]{a2}}}} \]

   10. Simplified 2.4

       \[\leadsto \color{blue}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\frac{b1}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}}} \cdot \frac{\sqrt[3]{1}}{\frac{\frac{b2}{\sqrt[3]{a1}}}{\sqrt[3]{a2}}} \]

   11. Simplified 2.4

       \[\leadsto \frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\frac{b1}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}} \cdot \color{blue}{\frac{\sqrt[3]{a2}}{\frac{b2}{\sqrt[3]{a1}}}} \]

   if 8.5085764721221019e294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 61.2

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]

   2. Applied clear-num_binary64 61.2

      \[\leadsto \color{blue}{\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}} \]

   3. Applied associate-/r*_binary64 48.0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{b1 \cdot b2}{a1}}{a2}}} \]

   4. Taylor expanded in b1 around 0 61.2

      \[\leadsto \color{blue}{\frac{a1 \cdot a2}{b2 \cdot b1}} \]

   5. Simplified 48.0

      \[\leadsto \color{blue}{\frac{a2}{\frac{b2 \cdot b1}{a1}}} \]

   6. Applied *-un-lft-identity_binary64 48.0

      \[\leadsto \frac{a2}{\frac{b2 \cdot b1}{\color{blue}{1 \cdot a1}}} \]

   7. Applied times-frac_binary64 12.7

      \[\leadsto \frac{a2}{\color{blue}{\frac{b2}{1} \cdot \frac{b1}{a1}}} \]

   8. Applied associate-/r*_binary64 5.4

      \[\leadsto \color{blue}{\frac{\frac{a2}{\frac{b2}{1}}}{\frac{b1}{a1}}} \]

   9. Simplified 5.4

      \[\leadsto \frac{\color{blue}{\frac{a2}{b2}}}{\frac{b1}{a1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -9.070334479927686 \cdot 10^{+287}:\\ \;\;\;\;\frac{1}{\frac{b1}{a1}} \cdot \frac{1}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -1.2861353814587239 \cdot 10^{-303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\frac{b1}{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}} \cdot \frac{\sqrt[3]{a2}}{\frac{b2}{\sqrt[3]{a1}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 8.508576472122102 \cdot 10^{+294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a2}{b2}}{\frac{b1}{a1}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))