Quotient of products

Average Error: 11.2 → 2.2

Time: 3.8s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;t_0 \leq -5.7467652850681114 \cdot 10^{-294}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;t_0 \leq 1.5148245907412011 \cdot 10^{+287}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b1} \cdot \frac{\sqrt[3]{a1}}{\frac{b2}{a2}}\\ \end{array} \]

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 -5.7467652850681114e-294)
       t_0
       (if (<= t_0 0.0)
         (* (/ 1.0 (/ b1 (* (cbrt a2) (cbrt a2)))) (/ a1 (/ b2 (cbrt a2))))
         (if (<= t_0 1.5148245907412011e+287)
           t_0
           (* (/ (* (cbrt a1) (cbrt a1)) b1) (/ (cbrt a1) (/ b2 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 <= -5.7467652850681114e-294) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / (b1 / (cbrt(a2) * cbrt(a2)))) * (a1 / (b2 / cbrt(a2)));
	} else if (t_0 <= 1.5148245907412011e+287) {
		tmp = t_0;
	} else {
		tmp = ((cbrt(a1) * cbrt(a1)) / b1) * (cbrt(a1) / (b2 / 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	2.2

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

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 26.1

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

   3. Applied div-inv_binary64 26.1

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

   4. Simplified 26.1

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

   5. Applied *-un-lft-identity_binary64 26.1

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

   6. Applied times-frac_binary64 13.5

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

   7. Applied associate-*r*_binary64 9.7

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

   8. Simplified 9.7

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

   if -inf.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -5.74676528506811142e-294 or -0.0 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < 1.51482459074120114e287

   1. Initial program 0.8

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

   2. Applied associate-/l*_binary64 8.0

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

   3. Applied div-inv_binary64 8.4

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

   4. Simplified 8.1

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

   5. Applied pow1_binary64 8.1

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

   6. Applied pow1_binary64 8.1

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

   7. Applied pow-prod-down_binary64 8.1

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

   8. Simplified 0.8

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

   if -5.74676528506811142e-294 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2)) < -0.0

   1. Initial program 13.4

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

   2. Applied associate-/l*_binary64 7.1

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

   3. Applied add-cube-cbrt_binary64 7.3

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

   4. Applied times-frac_binary64 3.8

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

   5. Applied *-un-lft-identity_binary64 3.8

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

   6. Applied times-frac_binary64 2.4

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

   if 1.51482459074120114e287 < (/.f64 (*.f64 a1 a2) (*.f64 b1 b2))

   1. Initial program 59.9

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

   2. Applied associate-/l*_binary64 45.3

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

   3. Applied *-un-lft-identity_binary64 45.3

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

   4. Applied times-frac_binary64 15.7

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

   5. Applied add-cube-cbrt_binary64 16.5

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

   6. Applied times-frac_binary64 8.7

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

   7. Simplified 8.7

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -\infty:\\ \;\;\;\;\frac{a1}{b2} \cdot \frac{a2}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq -5.7467652850681114 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 0:\\ \;\;\;\;\frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \leq 1.5148245907412011 \cdot 10^{+287}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b1} \cdot \frac{\sqrt[3]{a1}}{\frac{b2}{a2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022004 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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