Average Error: 52.6 → 51.2
Time: 38.7s
Precision: binary64
\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
↓
\[\begin{array}{l} \mathbf{if}\;C \leq -8.580045937962391 \cdot 10^{+29}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(C + A\right) - \left(A - C\right)\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\left|\sqrt[3]{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt[3]{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}}\right| \cdot \sqrt{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}}\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\right)}\\ \end{array}\]
\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 -8.580045937962391 \cdot 10^{+29}:\\
\;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(C + A\right) - \left(A - C\right)\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\left|\sqrt[3]{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt[3]{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}}\right| \cdot \sqrt{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}}\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\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 -8.580045937962391e+29)
   (/
    (-
     (sqrt
      (* (* 2.0 (* (- (pow B 2.0) (* C (* 4.0 A))) F)) (- (+ C A) (- A C)))))
    (- (pow B 2.0) (* C (* 4.0 A))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* (- (pow B 2.0) (* C (* 4.0 A))) F))
       (-
        (+ C A)
        (*
         (sqrt (sqrt (+ (pow (- A C) 2.0) (* B B))))
         (sqrt
          (*
           (fabs
            (*
             (cbrt (sqrt (+ (pow (- A C) 2.0) (* B B))))
             (cbrt (sqrt (+ (pow (- A C) 2.0) (* B B))))))
           (sqrt (cbrt (+ (pow (- A C) 2.0) (* B B)))))))))))
    (- (pow B 2.0) (* C (* 4.0 A))))))
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 <= -8.580045937962391e+29) {
		tmp = -sqrt((2.0 * ((pow(B, 2.0) - (C * (4.0 * A))) * F)) * ((C + A) - (A - C))) / (pow(B, 2.0) - (C * (4.0 * A)));
	} else {
		tmp = -sqrt((2.0 * ((pow(B, 2.0) - (C * (4.0 * A))) * F)) * ((C + A) - (sqrt(sqrt(pow((A - C), 2.0) + (B * B))) * sqrt(fabs(cbrt(sqrt(pow((A - C), 2.0) + (B * B))) * cbrt(sqrt(pow((A - C), 2.0) + (B * B)))) * sqrt(cbrt(pow((A - C), 2.0) + (B * B))))))) / (pow(B, 2.0) - (C * (4.0 * A)));
	}
	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 < -8.5800459379623908e29

    1. Initial program 54.6

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

    2. Taylor expanded around inf 48.2

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

    if -8.5800459379623908e29 < C

    1. Initial program 52.0

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

    3. Applied add-sqr-sqrt_binary64_248752.1

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

    4. Simplified52.1

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

    5. Simplified52.1

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

    7. Applied add-cube-cbrt_binary64_250052.1

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

    8. Applied sqrt-prod_binary64_248152.1

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

    9. Simplified52.1

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

    11. Applied add-sqr-sqrt_binary64_248752.1

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

    12. Applied cbrt-prod_binary64_249652.1

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

  4. Final simplification51.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -8.580045937962391 \cdot 10^{+29}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(C + A\right) - \left(A - C\right)\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - C \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right) \cdot \left(\left(C + A\right) - \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\left|\sqrt[3]{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt[3]{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}}\right| \cdot \sqrt{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}}\right)}}{{B}^{2} - C \cdot \left(4 \cdot A\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))