ABCF->ab-angle b

Average Error: 52.5 → 50.3

Time: 1.0min

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \left(A - C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 7.214973498871856 \cdot 10^{+167}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

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 (<=
      (/
       (-
        (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 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (- A C)))))
    (- (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)))
        7.214973498871856e+167)
     (/
      (sqrt
       (*
        (* 2.0 (* F (- (* B B) (* (* 4.0 A) C))))
        (+ A (- C (sqrt (+ (pow (- A C) 2.0) (* B B)))))))
      (- (* (* 4.0 A) C) (* B B)))
     0.0)))

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 ((-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 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - (A - C))) / (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))) <= 7.214973498871856e+167) {
		tmp = sqrt((2.0 * (F * ((B * B) - ((4.0 * A) * C)))) * (A + (C - sqrt(pow((A - C), 2.0) + (B * B))))) / (((4.0 * A) * C) - (B * B));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Derivation

1. Split input into 3 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 59.0

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

   if -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))) < 7.2149734988718558e167

   1. Initial program 29.1

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

   3. Applied associate--l+_binary64_2402 28.2

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

   4. Applied distribute-rgt-in_binary64_2415 28.2

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

   5. Simplified 28.2

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

   6. Simplified 28.2

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

   8. Applied frac-2neg_binary64_2476 28.2

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

   9. Simplified 28.2

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

   10. Simplified 28.2

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

   if 7.2149734988718558e167 < (/.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 63.3

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

   3. Applied associate--l+_binary64_2402 63.3

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

   4. Applied distribute-rgt-in_binary64_2415 63.3

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

   5. Simplified 63.3

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

   6. Simplified 63.3

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

   7. Taylor expanded around inf 61.3

      \[\leadsto \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 50.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \left(A - C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 7.214973498871856 \cdot 10^{+167}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - \left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(A + \left(C - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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))))