Quotient of products

Average Error: 11.5 → 2.5

Time: 4.4s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ t_1 := \frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -5.228923257196031 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;t_0 \leq 1.4421320560907244 \cdot 10^{+306}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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)))
        (t_1
         (* (/ 1.0 (/ b1 (* (cbrt a2) (cbrt a2)))) (/ a1 (/ b2 (cbrt a2))))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -5.228923257196031e-300)
       t_0
       (if (<= t_0 0.0)
         (/ a1 (* b1 (/ b2 a2)))
         (if (<= t_0 1.4421320560907244e+306) t_0 t_1))))))

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 t_1 = (1.0 / (b1 / (cbrt(a2) * cbrt(a2)))) * (a1 / (b2 / cbrt(a2)));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -5.228923257196031e-300) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a1 / (b1 * (b2 / a2));
	} else if (t_0 <= 1.4421320560907244e+306) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.0
Herbie	2.5

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -inf.0 or 1.4421320560907244e306 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 63.5

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

   2. Applied associate-/l*_binary64 41.3

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

   3. Applied add-cube-cbrt_binary64 41.6

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

   4. Applied times-frac_binary64 13.4

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

   5. Applied *-un-lft-identity_binary64 13.4

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

   6. Applied times-frac_binary64 7.7

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.22892325719603109e-300 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.4421320560907244e306

   1. Initial program 0.8

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

   2. Applied associate-/l*_binary64 7.5

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

   3. Applied div-inv_binary64 7.6

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

   4. Applied *-un-lft-identity_binary64 7.6

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

   5. Applied times-frac_binary64 1.3

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

   6. Applied *-un-lft-identity_binary64 1.3

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

   7. Applied associate-*l*_binary64 1.3

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

   8. Simplified 0.8

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

   if -5.22892325719603109e-300 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.6

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

   2. Applied associate-/l*_binary64 7.5

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

   3. Applied *-un-lft-identity_binary64 7.5

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

   4. Applied times-frac_binary64 3.8

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

   5. Simplified 3.8

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.228923257196031 \cdot 10^{-300}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{a1}{b1 \cdot \frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.4421320560907244 \cdot 10^{+306}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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