Average Error: 52.6 → 30.9
Time: 36.9s
Precision: binary64
[A C]: =sort([A C])
\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;-\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.586859497683581 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\sqrt{\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \sqrt{\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\left(A \cdot \sqrt{2}\right) \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\sqrt{2}} \cdot \left(\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}} \cdot \sqrt{\sqrt{2}}\right)\\ \end{array}\]
\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}\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\
\;\;\;\;-\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\\

\mathbf{elif}\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.586859497683581 \cdot 10^{-197}:\\
\;\;\;\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\

\mathbf{elif}\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\
\;\;\;\;-\sqrt{\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \sqrt{\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}}\\

\mathbf{elif}\;\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{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\
\;\;\;\;\frac{\left(A \cdot \sqrt{2}\right) \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\

\mathbf{else}:\\
\;\;\;\;-\sqrt{\sqrt{2}} \cdot \left(\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}} \cdot \sqrt{\sqrt{2}}\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 (<=
      (/
       (-
        (sqrt
         (*
          (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
          (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
       (- (pow B 2.0) (* (* 4.0 A) C)))
      (- INFINITY))
   (- (* (sqrt 2.0) (/ (sqrt (* F -0.5)) (sqrt C))))
   (if (<=
        (/
         (-
          (sqrt
           (*
            (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
            (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
         (- (pow B 2.0) (* (* 4.0 A) C)))
        -1.586859497683581e-197)
     (/
      (-
       (sqrt
        (*
         (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
         (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
      (- (pow B 2.0) (* (* 4.0 A) C)))
     (if (<=
          (/
           (-
            (sqrt
             (*
              (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
              (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
           (- (pow B 2.0) (* (* 4.0 A) C)))
          0.0)
       (-
        (*
         (sqrt (* (sqrt 2.0) (/ (sqrt (* F -0.5)) (sqrt C))))
         (sqrt (* (sqrt 2.0) (/ (sqrt (* F -0.5)) (sqrt C))))))
       (if (<=
            (/
             (-
              (sqrt
               (*
                (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
                (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
             (- (pow B 2.0) (* (* 4.0 A) C)))
            INFINITY)
         (/
          (* (* A (sqrt 2.0)) (sqrt (* -8.0 (* C F))))
          (- (pow B 2.0) (* (* 4.0 A) C)))
         (-
          (*
           (sqrt (sqrt 2.0))
           (* (/ (sqrt (* F -0.5)) (sqrt C)) (sqrt (sqrt 2.0))))))))))
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 ((-sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C))) <= -((double) INFINITY)) {
		tmp = -(sqrt(2.0) * (sqrt(F * -0.5) / sqrt(C)));
	} else if ((-sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C))) <= -1.586859497683581e-197) {
		tmp = -sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else if ((-sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C))) <= 0.0) {
		tmp = -(sqrt(sqrt(2.0) * (sqrt(F * -0.5) / sqrt(C))) * sqrt(sqrt(2.0) * (sqrt(F * -0.5) / sqrt(C))));
	} else if ((-sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / (pow(B, 2.0) - ((4.0 * A) * C))) <= ((double) INFINITY)) {
		tmp = ((A * sqrt(2.0)) * sqrt(-8.0 * (C * F))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -(sqrt(sqrt(2.0)) * ((sqrt(F * -0.5) / sqrt(C)) * sqrt(sqrt(2.0))));
	}
	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 5 regimes

  2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -inf.0

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 33.5

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

    3. Simplified33.5

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}}\]
    4. Using strategy rm

    5. Applied associate-*r/_binary64_206633.5

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

    6. Applied sqrt-div_binary64_214121.9

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

    if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1.5868594976835809e-197

    1. Initial program 1.3

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

    if -1.5868594976835809e-197 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 0.0

    1. Initial program 59.9

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

    2. Taylor expanded around -inf 33.6

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

    3. Simplified33.6

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}}\]
    4. Using strategy rm

    5. Applied associate-*r/_binary64_206633.6

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

    6. Applied sqrt-div_binary64_214130.3

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt_binary64_214630.4

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

    if 0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

    1. Initial program 38.7

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

    2. Taylor expanded around -inf 12.6

      \[\leadsto \frac{-\color{blue}{-1 \cdot \left(\sqrt{-8 \cdot \left(C \cdot F\right)} \cdot \left(A \cdot \sqrt{2}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

    3. Simplified12.6

      \[\leadsto \frac{-\color{blue}{\left(-\left(A \cdot \sqrt{2}\right) \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\]

    if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 52.6

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

    3. Simplified52.6

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}}\]
    4. Using strategy rm

    5. Applied associate-*r/_binary64_206652.6

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

    6. Applied sqrt-div_binary64_214148.3

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt_binary64_214648.3

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

    9. Applied associate-*l*_binary64_206548.3

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

    10. Simplified48.3

      \[\leadsto -\sqrt{\sqrt{2}} \cdot \color{blue}{\left(\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}} \cdot \sqrt{\sqrt{2}}\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification30.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;-\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.586859497683581 \cdot 10^{-197}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\sqrt{\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \sqrt{\sqrt{2} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\left(A \cdot \sqrt{2}\right) \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\sqrt{2}} \cdot \left(\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}} \cdot \sqrt{\sqrt{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021007 
(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))))