Quotient of products

Average Error: 11.3 → 2.5

Time: 5.0s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -6.797433893807113 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.1276937038153 \cdot 10^{-319}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.0608602275507985 \cdot 10^{+270}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

FPCore

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

↓

(FPCore (a1 a2 b1 b2)
 :precision binary64
 (if (<= (/ (* a1 a2) (* b1 b2)) (- INFINITY))
   (/ (* a1 (/ a2 b1)) b2)
   (if (<= (/ (* a1 a2) (* b1 b2)) -6.797433893807113e-294)
     (/ (* a1 a2) (* b1 b2))
     (if (<= (/ (* a1 a2) (* b1 b2)) 2.1276937038153e-319)
       (*
        (/ (/ a1 (* (cbrt b1) (cbrt b1))) (* (cbrt b2) (cbrt b2)))
        (/ (/ a2 (cbrt b1)) (cbrt b2)))
       (if (<= (/ (* a1 a2) (* b1 b2)) 1.0608602275507985e+270)
         (* (* a1 a2) (/ (/ 1.0 b1) b2))
         (* (/ a1 b1) (/ a2 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 tmp;
	if (((a1 * a2) / (b1 * b2)) <= -((double) INFINITY)) {
		tmp = (a1 * (a2 / b1)) / b2;
	} else if (((a1 * a2) / (b1 * b2)) <= -6.797433893807113e-294) {
		tmp = (a1 * a2) / (b1 * b2);
	} else if (((a1 * a2) / (b1 * b2)) <= 2.1276937038153e-319) {
		tmp = ((a1 / (cbrt(b1) * cbrt(b1))) / (cbrt(b2) * cbrt(b2))) * ((a2 / cbrt(b1)) / cbrt(b2));
	} else if (((a1 * a2) / (b1 * b2)) <= 1.0608602275507985e+270) {
		tmp = (a1 * a2) * ((1.0 / b1) / b2);
	} else {
		tmp = (a1 / b1) * (a2 / b2);
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.2
Herbie	2.5

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

Derivation

1. Split input into 5 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/r*_binary64 34.9

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

   4. Simplified 17.3

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

   6. Applied pow1_binary64 17.3

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

   7. Applied pow1_binary64 17.3

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

   8. Applied pow-prod-down_binary64 17.3

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

   9. Simplified 34.9

      \[\leadsto \frac{{\color{blue}{\left(\frac{a1 \cdot a2}{b1}\right)}}^{1}}{b2}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity_binary64 34.9

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

   12. Applied times-frac_binary64 16.2

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

   13. Simplified 16.2

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -6.79743389380711298e-294

   1. Initial program 0.7

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

   if -6.79743389380711298e-294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 2.12769e-319

   1. Initial program 12.9

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

   3. Applied associate-/r*_binary64 6.3

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

   4. Simplified 4.1

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

   6. Applied pow1_binary64 4.1

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

   7. Applied pow1_binary64 4.1

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

   8. Applied pow-prod-down_binary64 4.1

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

   9. Simplified 6.3

      \[\leadsto \frac{{\color{blue}{\left(\frac{a1 \cdot a2}{b1}\right)}}^{1}}{b2}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt_binary64 6.5

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

   12. Applied add-cube-cbrt_binary64 6.5

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

   13. Applied times-frac_binary64 3.8

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

   14. Applied unpow-prod-down_binary64 3.8

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

   15. Applied times-frac_binary64 2.3

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

   if 2.12769e-319 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.0608602275507985e270

   1. Initial program 0.7

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

   3. Applied associate-/r*_binary64 7.7

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

   4. Simplified 14.3

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

   6. Applied pow1_binary64 14.3

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

   7. Applied pow1_binary64 14.3

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

   8. Applied pow-prod-down_binary64 14.3

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

   9. Simplified 7.7

      \[\leadsto \frac{{\color{blue}{\left(\frac{a1 \cdot a2}{b1}\right)}}^{1}}{b2}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity_binary64 7.7

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

   12. Applied div-inv_binary64 7.8

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

   13. Applied unpow-prod-down_binary64 7.8

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

   14. Applied times-frac_binary64 1.0

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

   if 1.0608602275507985e270 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 55.3

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

   3. Applied times-frac_binary64 9.1

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -6.797433893807113 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 2.1276937038153 \cdot 10^{-319}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.0608602275507985 \cdot 10^{+270}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021139 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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