Quotient of products

Average Error: 11.6 → 4.8

Time: 4.4s

Precision: binary64

\[[a1, a2]=\mathsf{sort}([a1, a2])\]

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(a2 \cdot \frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}\right) \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{b1}}}{b2}\\ \mathbf{if}\;b1 \cdot b2 \leq -2.3720194052494554 \cdot 10^{+232}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq -5.062919740973674 \cdot 10^{-112}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b1 \cdot b2 \leq 7.40266575556476 \cdot 10^{+200}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a1}{b1}}{\sqrt[3]{b2}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (* a2 (/ (* (cbrt a1) (cbrt a1)) (* (cbrt b1) (cbrt b1))))
           (/ (cbrt a1) (cbrt b1)))
          b2)))
   (if (<= (* b1 b2) -2.3720194052494554e+232)
     t_0
     (if (<= (* b1 b2) -5.062919740973674e-112)
       (/ (* a2 a1) (* b1 b2))
       (if (<= (* b1 b2) 0.0)
         t_0
         (if (<= (* b1 b2) 7.40266575556476e+200)
           (* a2 (/ a1 (* b1 b2)))
           (* (/ a2 (* (cbrt b2) (cbrt b2))) (/ (/ a1 b1) (cbrt b2)))))))))

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 = ((a2 * ((cbrt(a1) * cbrt(a1)) / (cbrt(b1) * cbrt(b1)))) * (cbrt(a1) / cbrt(b1))) / b2;
	double tmp;
	if ((b1 * b2) <= -2.3720194052494554e+232) {
		tmp = t_0;
	} else if ((b1 * b2) <= -5.062919740973674e-112) {
		tmp = (a2 * a1) / (b1 * b2);
	} else if ((b1 * b2) <= 0.0) {
		tmp = t_0;
	} else if ((b1 * b2) <= 7.40266575556476e+200) {
		tmp = a2 * (a1 / (b1 * b2));
	} else {
		tmp = (a2 / (cbrt(b2) * cbrt(b2))) * ((a1 / b1) / cbrt(b2));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.6
Target	11.1
Herbie	4.8

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 b1 b2) < -2.3720194052494554e232 or -5.06291974097367404e-112 < (*.f64 b1 b2) < -0.0

   1. Initial program 24.8

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Using strategy rm

   3. Applied associate-/r*_binary64 12.9

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

   4. Simplified 8.2

      \[\leadsto \frac{\color{blue}{a2 \cdot \frac{a1}{b1}}}{b2} \]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64 8.8

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

   7. Applied add-cube-cbrt_binary64 8.9

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

   8. Applied times-frac_binary64 8.9

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

   9. Applied associate-*r*_binary64 5.8

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

   if -2.3720194052494554e232 < (*.f64 b1 b2) < -5.06291974097367404e-112

   1. Initial program 3.8

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Using strategy rm

   3. Applied associate-/r*_binary64 11.2

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

   4. Simplified 14.6

      \[\leadsto \frac{\color{blue}{a2 \cdot \frac{a1}{b1}}}{b2} \]
   5. Using strategy rm

   6. Applied *-un-lft-identity_binary64 14.6

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

   7. Applied times-frac_binary64 10.6

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

   8. Simplified 10.6

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

   9. Simplified 3.6

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

   10. Taylor expanded around 0 3.8

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

   if -0.0 < (*.f64 b1 b2) < 7.40266575556476003e200

   1. Initial program 5.7

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Using strategy rm

   3. Applied associate-/r*_binary64 12.5

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

   4. Simplified 14.4

      \[\leadsto \frac{\color{blue}{a2 \cdot \frac{a1}{b1}}}{b2} \]
   5. Using strategy rm

   6. Applied *-un-lft-identity_binary64 14.4

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

   7. Applied times-frac_binary64 11.8

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

   8. Simplified 11.8

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

   9. Simplified 5.2

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

   if 7.40266575556476003e200 < (*.f64 b1 b2)

   1. Initial program 15.6

      \[\frac{a1 \cdot a2}{b1 \cdot b2} \]
   2. Using strategy rm

   3. Applied associate-/r*_binary64 7.7

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

   4. Simplified 4.6

      \[\leadsto \frac{\color{blue}{a2 \cdot \frac{a1}{b1}}}{b2} \]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64 4.9

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

   7. Applied times-frac_binary64 4.2

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

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \leq -2.3720194052494554 \cdot 10^{+232}:\\ \;\;\;\;\frac{\left(a2 \cdot \frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}\right) \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq -5.062919740973674 \cdot 10^{-112}:\\ \;\;\;\;\frac{a2 \cdot a1}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 0:\\ \;\;\;\;\frac{\left(a2 \cdot \frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}\right) \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{b1}}}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \leq 7.40266575556476 \cdot 10^{+200}:\\ \;\;\;\;a2 \cdot \frac{a1}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a1}{b1}}{\sqrt[3]{b2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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