Average Error: 52.1 → 47.4
Time: 20.1s
Precision: binary64
\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
↓
\[\begin{array}{l} \mathbf{if}\;C \leq 1.6585841150419772 \cdot 10^{+59}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(C + A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(C + A\right) + \left(C - A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]
\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}\;C \leq 1.6585841150419772 \cdot 10^{+59}:\\
\;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(C + A\right)}}{{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(C + A\right) + \left(C - A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\

\end{array}
(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 (<= C 1.6585841150419772e+59)
   (/
    (-
     (*
      (sqrt (* 2.0 (* (- (* B B) (* C (* A 4.0))) F)))
      (sqrt (+ (sqrt (+ (* B B) (pow (- A C) 2.0))) (+ C A)))))
    (- (pow B 2.0) (* C (* A 4.0))))
   (/
    (-
     (*
      (sqrt (* 2.0 (* (- (* B B) (* C (* A 4.0))) F)))
      (sqrt (+ (+ C A) (- C A)))))
    (- (pow B 2.0) (* C (* A 4.0))))))
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 (C <= 1.6585841150419772e+59) {
		tmp = -(sqrt(2.0 * (((B * B) - (C * (A * 4.0))) * F)) * sqrt(sqrt((B * B) + pow((A - C), 2.0)) + (C + A))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = -(sqrt(2.0 * (((B * B) - (C * (A * 4.0))) * F)) * sqrt((C + A) + (C - A))) / (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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if C < 1.65858411504197724e59

    1. Initial program 51.3

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

    3. Applied sqrt-prod_binary64_589149.4

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

    4. Simplified49.4

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

    5. Simplified49.4

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

    if 1.65858411504197724e59 < C

    1. Initial program 54.9

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

    3. Applied sqrt-prod_binary64_589153.0

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

    4. Simplified53.0

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

    5. Simplified53.0

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

    6. Taylor expanded around -inf 40.0

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

  4. Final simplification47.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.6585841150419772 \cdot 10^{+59}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\sqrt{B \cdot B + {\left(A - C\right)}^{2}} + \left(C + A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B \cdot B - C \cdot \left(A \cdot 4\right)\right) \cdot F\right)} \cdot \sqrt{\left(C + A\right) + \left(C - A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(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))))