ABCF->ab-angle a

Average Error: 52.3 → 48.5

Time: 52.6s

Precision: binary64

Math

\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

↓

\[\begin{array}{l} \mathbf{if}\;A \leq -1.0923582101290374 \cdot 10^{-212}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left|\sqrt[3]{B \cdot B + {\left(A - C\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{B \cdot B + {\left(A - C\right)}^{2}} \cdot \sqrt[3]{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{B \cdot B + {\left(A - C\right)}^{2}}}} + \left(A + C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 1.0370206653323664 \cdot 10^{-290}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \left(C - A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 3.6253503641308704 \cdot 10^{+122}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(A - C\right) + \left(A + C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (if (<= A -1.0923582101290374e-212)
   (/
    (-
     (*
      (sqrt (* 2.0 (* (- (* B B) (* C (* A 4.0))) F)))
      (sqrt
       (+
        (*
         (fabs (cbrt (+ (* B B) (pow (- A C) 2.0))))
         (sqrt
          (*
           (cbrt
            (*
             (cbrt (+ (* B B) (pow (- A C) 2.0)))
             (cbrt (+ (* B B) (pow (- A C) 2.0)))))
           (cbrt (cbrt (+ (* B B) (pow (- A C) 2.0)))))))
        (+ A C)))))
    (- (pow B 2.0) (* C (* A 4.0))))
   (if (<= A 1.0370206653323664e-290)
     (/
      (-
       (*
        (sqrt (* 2.0 (* (- (* B B) (* C (* A 4.0))) F)))
        (sqrt (+ (+ A C) (- C A)))))
      (- (pow B 2.0) (* C (* A 4.0))))
     (if (<= A 3.6253503641308704e+122)
       (/
        (-
         (*
          (sqrt (* 2.0 (* (- (* B B) (* C (* A 4.0))) F)))
          (sqrt
           (+
            (+ A C)
            (*
             (sqrt (sqrt (+ (* B B) (pow (- A C) 2.0))))
             (sqrt (sqrt (+ (* B B) (pow (- A C) 2.0)))))))))
        (- (pow B 2.0) (* C (* A 4.0))))
       (/
        (-
         (*
          (sqrt (* 2.0 (* (- (* B B) (* C (* A 4.0))) F)))
          (sqrt (+ (- A C) (+ A C)))))
        (- (pow B 2.0) (* C (* A 4.0))))))))

C

double code(double A, double B, double C, double F) {
	return -sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt(pow((A - C), 2.0) + pow(B, 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -1.0923582101290374e-212) {
		tmp = -(sqrt(2.0 * (((B * B) - (C * (A * 4.0))) * F)) * sqrt((fabs(cbrt((B * B) + pow((A - C), 2.0))) * sqrt(cbrt(cbrt((B * B) + pow((A - C), 2.0)) * cbrt((B * B) + pow((A - C), 2.0))) * cbrt(cbrt((B * B) + pow((A - C), 2.0))))) + (A + C))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (A <= 1.0370206653323664e-290) {
		tmp = -(sqrt(2.0 * (((B * B) - (C * (A * 4.0))) * F)) * sqrt((A + C) + (C - A))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (A <= 3.6253503641308704e+122) {
		tmp = -(sqrt(2.0 * (((B * B) - (C * (A * 4.0))) * F)) * sqrt((A + C) + (sqrt(sqrt((B * B) + pow((A - C), 2.0))) * sqrt(sqrt((B * B) + pow((A - C), 2.0)))))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = -(sqrt(2.0 * (((B * B) - (C * (A * 4.0))) * F)) * sqrt((A - C) + (A + C))) / (pow(B, 2.0) - (C * (A * 4.0)));
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Derivation

1. Split input into 4 regimes

2. if A < -1.09235821012903745e-212

   1. Initial program 55.8

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
   2. Using strategy rm

   3. Applied sqrt-prod_binary64_5891 54.9

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   4. Simplified 54.9

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   5. Simplified 54.9

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \color{blue}{\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt_binary64_5910 54.9

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B} \cdot \sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}\right) \cdot \sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}} + \left(A + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   8. Applied sqrt-prod_binary64_5891 54.9

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B} \cdot \sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}} + \left(A + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   9. Simplified 54.9

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\left|\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}\right|} \cdot \sqrt{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}} + \left(A + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt_binary64_5910 54.9

       \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left|\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}\right| \cdot \sqrt{\sqrt[3]{\color{blue}{\left(\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B} \cdot \sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}\right) \cdot \sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}}} + \left(A + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   12. Applied cbrt-prod_binary64_5906 54.9

       \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left|\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}\right| \cdot \sqrt{\color{blue}{\sqrt[3]{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B} \cdot \sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt[3]{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}}} + \left(A + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   if -1.09235821012903745e-212 < A < 1.03702066533236635e-290

   1. Initial program 50.1

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
   2. Using strategy rm

   3. Applied sqrt-prod_binary64_5891 48.0

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   4. Simplified 48.0

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   5. Simplified 48.0

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \color{blue}{\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   6. Taylor expanded around -inf 55.1

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\left(C - A\right)} + \left(A + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   if 1.03702066533236635e-290 < A < 3.6253503641308704e122

   1. Initial program 44.5

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
   2. Using strategy rm

   3. Applied sqrt-prod_binary64_5891 41.6

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   4. Simplified 41.6

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   5. Simplified 41.6

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \color{blue}{\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt_binary64_5897 41.7

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}}} + \left(A + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   if 3.6253503641308704e122 < A

   1. Initial program 61.0

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
   2. Using strategy rm

   3. Applied sqrt-prod_binary64_5891 60.0

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   4. Simplified 60.0

      \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)}} \cdot \sqrt{\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   5. Simplified 60.0

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \color{blue}{\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

   6. Taylor expanded around inf 41.4

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\color{blue}{\left(A - C\right)} + \left(A + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
3. Recombined 4 regimes into one program.

4. Final simplification 48.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.0923582101290374 \cdot 10^{-212}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left|\sqrt[3]{B \cdot B + {\left(A - C\right)}^{2}}\right| \cdot \sqrt{\sqrt[3]{\sqrt[3]{B \cdot B + {\left(A - C\right)}^{2}} \cdot \sqrt[3]{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{B \cdot B + {\left(A - C\right)}^{2}}}} + \left(A + C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 1.0370206653323664 \cdot 10^{-290}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \left(C - A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 3.6253503641308704 \cdot 10^{+122}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}}} \cdot \sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}}}}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(A - C\right) + \left(A + C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))