Quotient of products

Average Error: 11.2 → 2.5

Time: 6.2s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;t_0 \leq -5.57 \cdot 10^{-320}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;t_0 \leq 6.325887104397162 \cdot 10^{+303}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \end{array} \]

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 (- INFINITY))
     (* (/ a1 b2) (/ a2 b1))
     (if (<= t_0 -5.57e-320)
       t_0
       (if (<= t_0 0.0)
         (* a1 (/ (/ a2 b2) b1))
         (if (<= t_0 6.325887104397162e+303)
           t_0
           (* (/ (* (cbrt a2) (cbrt a2)) b1) (/ a1 (/ b2 (cbrt a2))))))))))

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 <= -((double) INFINITY)) {
		tmp = (a1 / b2) * (a2 / b1);
	} else if (t_0 <= -5.57e-320) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = a1 * ((a2 / b2) / b1);
	} else if (t_0 <= 6.325887104397162e+303) {
		tmp = t_0;
	} else {
		tmp = ((cbrt(a2) * cbrt(a2)) / b1) * (a1 / (b2 / cbrt(a2)));
	}
	return tmp;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	11.3
Herbie	2.5

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

Derivation

1. Split input into 4 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. Applied associate-/l*_binary64 33.0

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

   3. Applied div-inv_binary64 33.1

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

   4. Simplified 33.1

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

   5. Applied *-un-lft-identity_binary64 33.1

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

   6. Applied times-frac_binary64 19.7

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

   7. Applied associate-*r*_binary64 10.6

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

   8. Simplified 10.5

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.5701e-320 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 6.32588710439716204e303

   1. Initial program 0.9

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

   if -5.5701e-320 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.8

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

   2. Applied associate-/l*_binary64 8.1

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

   3. Applied div-inv_binary64 8.1

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

   4. Simplified 8.1

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

   5. Applied associate-/r*_binary64 4.0

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

   if 6.32588710439716204e303 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 63.5

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

   2. Applied associate-/l*_binary64 48.1

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

   3. Applied add-cube-cbrt_binary64 48.3

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

   4. Applied times-frac_binary64 14.5

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

   5. Applied *-un-lft-identity_binary64 14.5

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

   6. Applied times-frac_binary64 6.2

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

   7. Simplified 6.2

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.57 \cdot 10^{-320}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;a1 \cdot \frac{\frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 6.325887104397162 \cdot 10^{+303}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021273 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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