ABCF->ab-angle a

Average Error: 52.7 → 41.5

Time: 44.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;A \leq -1.6475797238506222 \cdot 10^{-74}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 7.275455770643732 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{\sqrt{F}}{\sqrt{B \cdot B - C \cdot \left(A \cdot 4\right)}} \cdot \left(-\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(A + C\right)}\right)\right)\\ \mathbf{elif}\;A \leq 3.46577574840479 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)}}{\frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(A + C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)}}{\frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{A + \left(A + C\right)}}}\\ \end{array}\]

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 -1.6475797238506222e-74)
   (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F A)))))
   (if (<= A 7.275455770643732e-294)
     (*
      (sqrt 2.0)
      (*
       (/ (sqrt F) (sqrt (- (* B B) (* C (* A 4.0)))))
       (- (sqrt (+ (sqrt (+ (* B B) (pow (- A C) 2.0))) (+ A C))))))
     (if (<= A 3.46577574840479e+82)
       (/
        (sqrt (* 2.0 (* F (- (pow B 2.0) (* C (* A 4.0))))))
        (/
         (- (* B B) (* C (* A 4.0)))
         (- (sqrt (+ (sqrt (+ (* B B) (pow (- A C) 2.0))) (+ A C))))))
       (*
        (sqrt 2.0)
        (/
         (sqrt (* F (- (* B B) (* C (* A 4.0)))))
         (/ (- (* B B) (* C (* A 4.0))) (- (sqrt (+ A (+ A C)))))))))))

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 <= -1.6475797238506222e-74) {
		tmp = -(sqrt(2.0) * sqrt(-0.5 * (F / A)));
	} else if (A <= 7.275455770643732e-294) {
		tmp = sqrt(2.0) * ((sqrt(F) / sqrt((B * B) - (C * (A * 4.0)))) * -sqrt(sqrt((B * B) + pow((A - C), 2.0)) + (A + C)));
	} else if (A <= 3.46577574840479e+82) {
		tmp = sqrt(2.0 * (F * (pow(B, 2.0) - (C * (A * 4.0))))) / (((B * B) - (C * (A * 4.0))) / -sqrt(sqrt((B * B) + pow((A - C), 2.0)) + (A + C)));
	} else {
		tmp = sqrt(2.0) * (sqrt(F * ((B * B) - (C * (A * 4.0)))) / (((B * B) - (C * (A * 4.0))) / -sqrt(A + (A + C))));
	}
	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 < -1.6475797238506222e-74

   1. Initial program 59.4

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

   2. Taylor expanded around inf 41.1

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{-0.5 \cdot \frac{F}{A}} \cdot \sqrt{2}\right)}\]

   3. Simplified 41.1

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}}\]

   if -1.6475797238506222e-74 < A < 7.27545577064373227e-294

   1. Initial program 49.6

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

   3. Applied sqrt-prod_binary64_2481 47.2

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

   4. Applied distribute-rgt-neg-in_binary64_2423 47.2

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

   5. Applied associate-/l*_binary64_2410 47.2

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

   6. Simplified 47.2

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

   8. Applied *-un-lft-identity_binary64_2465 47.2

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{\frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{\color{blue}{1 \cdot \left(-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}\right)}}}\]

   9. Applied *-un-lft-identity_binary64_2465 47.2

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{\frac{\color{blue}{1 \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)}}{1 \cdot \left(-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}\right)}}\]

   10. Applied times-frac_binary64_2471 47.2

       \[\leadsto \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{\color{blue}{\frac{1}{1} \cdot \frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}}\]

   11. Applied sqrt-prod_binary64_2481 47.3

       \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}}{\frac{1}{1} \cdot \frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}\]

   12. Applied times-frac_binary64_2471 47.3

       \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{1}{1}} \cdot \frac{\sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}{\frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}}\]

   13. Simplified 47.3

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

   14. Simplified 47.3

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

   16. Applied sqrt-prod_binary64_2481 42.2

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

   18. Applied associate-/r/_binary64_2411 42.2

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

   19. Simplified 42.2

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

   if 7.27545577064373227e-294 < A < 3.46577574840478983e82

   1. Initial program 44.7

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

   3. Applied sqrt-prod_binary64_2481 42.2

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

   4. Applied distribute-rgt-neg-in_binary64_2423 42.2

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

   5. Applied associate-/l*_binary64_2410 42.2

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

   6. Simplified 42.2

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

   if 3.46577574840478983e82 < A

   1. Initial program 57.3

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

   3. Applied sqrt-prod_binary64_2481 55.6

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

   4. Applied distribute-rgt-neg-in_binary64_2423 55.6

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

   5. Applied associate-/l*_binary64_2410 55.6

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

   6. Simplified 55.6

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

   8. Applied *-un-lft-identity_binary64_2465 55.6

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{\frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{\color{blue}{1 \cdot \left(-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}\right)}}}\]

   9. Applied *-un-lft-identity_binary64_2465 55.6

      \[\leadsto \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{\frac{\color{blue}{1 \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)}}{1 \cdot \left(-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}\right)}}\]

   10. Applied times-frac_binary64_2471 55.6

       \[\leadsto \frac{\sqrt{2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}}{\color{blue}{\frac{1}{1} \cdot \frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}}\]

   11. Applied sqrt-prod_binary64_2481 55.6

       \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}}{\frac{1}{1} \cdot \frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}\]

   12. Applied times-frac_binary64_2471 55.6

       \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{1}{1}} \cdot \frac{\sqrt{\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F}}{\frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B} + \left(A + C\right)}}}}\]

   13. Simplified 55.6

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

   14. Simplified 55.6

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

   15. Taylor expanded around inf 40.4

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

4. Final simplification 41.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.6475797238506222 \cdot 10^{-74}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \mathbf{elif}\;A \leq 7.275455770643732 \cdot 10^{-294}:\\ \;\;\;\;\sqrt{2} \cdot \left(\frac{\sqrt{F}}{\sqrt{B \cdot B - C \cdot \left(A \cdot 4\right)}} \cdot \left(-\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(A + C\right)}\right)\right)\\ \mathbf{elif}\;A \leq 3.46577574840479 \cdot 10^{+82}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)}}{\frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(A + C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)}}{\frac{B \cdot B - C \cdot \left(A \cdot 4\right)}{-\sqrt{A + \left(A + C\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020353 
(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))))