ABCF->ab-angle b

Average Error: 52.5 → 34.9

Time: 22.0s

Precision: binary64

\[[A, C]=\mathsf{sort}([A, C])\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.9268697537850326 \cdot 10^{+186}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{F \cdot \left(C \cdot -16\right)}}{C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.107057733816843 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{F} \cdot \frac{\sqrt{C \cdot -16}}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{{\left(2 \cdot \left(-8 \cdot \left(C \cdot F\right)\right)\right)}^{0.5}}{C}\\ \end{array}\]

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)))
      -1.9268697537850326e+186)
   (* -0.25 (/ (sqrt (* F (* C -16.0))) 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)))
        -1.107057733816843e-148)
     (/
      (-
       (sqrt
        (*
         (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
         (-
          (+ A C)
          (*
           (sqrt (sqrt (+ (pow (- A C) 2.0) (* B B))))
           (sqrt (sqrt (+ (pow (- A C) 2.0) (* B B)))))))))
      (- (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 (* -0.5 (/ F C))) (sqrt 2.0)))
       (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)
         (* -0.25 (* (sqrt F) (/ (sqrt (* C -16.0)) C)))
         (* -0.25 (/ (pow (* 2.0 (* -8.0 (* C F))) 0.5) 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 ((-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))) <= -1.9268697537850326e+186) {
		tmp = -0.25 * (sqrt(F * (C * -16.0)) / 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))) <= -1.107057733816843e-148) {
		tmp = -sqrt((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - (sqrt(sqrt(pow((A - C), 2.0) + (B * B))) * sqrt(sqrt(pow((A - C), 2.0) + (B * B)))))) / (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(-0.5 * (F / C)) * sqrt(2.0));
	} 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 = -0.25 * (sqrt(F) * (sqrt(C * -16.0) / C));
	} else {
		tmp = -0.25 * (pow((2.0 * (-8.0 * (C * F))), 0.5) / 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 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))) < -1.9268697537850326e186

   1. Initial program 62.0

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

   2. Taylor expanded around -inf 41.1

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

   3. Simplified 41.1

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

   4. Taylor expanded around inf 34.2

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{C}}\]

   5. Simplified 34.2

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot C\right)}}{C}}\]
   6. Using strategy rm

   7. Applied sqrt-unprod_binary64 34.1

      \[\leadsto -0.25 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot C\right)\right)}}}{C}\]

   8. Simplified 34.3

      \[\leadsto -0.25 \cdot \frac{\sqrt{\color{blue}{F \cdot \left(C \cdot -16\right)}}}{C}\]

   if -1.9268697537850326e186 < (/.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))) < -1.107057733816843e-148

   1. Initial program 1.6

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

   3. Applied add-sqr-sqrt_binary64 1.7

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

   4. Simplified 1.7

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

   5. Simplified 1.7

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

   if -1.107057733816843e-148 < (/.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 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. Taylor expanded around -inf 37.1

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

   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 38.9

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

   2. Taylor expanded around -inf 15.2

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

   3. Simplified 15.2

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

   4. Taylor expanded around inf 10.6

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{C}}\]

   5. Simplified 10.6

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot C\right)}}{C}}\]
   6. Using strategy rm

   7. Applied sqrt-unprod_binary64 10.5

      \[\leadsto -0.25 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(-8 \cdot \left(F \cdot C\right)\right)}}}{C}\]

   8. Simplified 10.3

      \[\leadsto -0.25 \cdot \frac{\sqrt{\color{blue}{F \cdot \left(C \cdot -16\right)}}}{C}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity_binary64 10.3

       \[\leadsto -0.25 \cdot \frac{\sqrt{F \cdot \left(C \cdot -16\right)}}{\color{blue}{1 \cdot C}}\]

   11. Applied sqrt-prod_binary64 2.9

       \[\leadsto -0.25 \cdot \frac{\color{blue}{\sqrt{F} \cdot \sqrt{C \cdot -16}}}{1 \cdot C}\]

   12. Applied times-frac_binary64 2.8

       \[\leadsto -0.25 \cdot \color{blue}{\left(\frac{\sqrt{F}}{1} \cdot \frac{\sqrt{C \cdot -16}}{C}\right)}\]

   13. Simplified 2.8

       \[\leadsto -0.25 \cdot \left(\color{blue}{\sqrt{F}} \cdot \frac{\sqrt{C \cdot -16}}{C}\right)\]

   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 63.4

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

   3. Simplified 63.4

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

   4. Taylor expanded around inf 50.4

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}}{C}}\]

   5. Simplified 50.4

      \[\leadsto \color{blue}{-0.25 \cdot \frac{\sqrt{2} \cdot \sqrt{-8 \cdot \left(F \cdot C\right)}}{C}}\]
   6. Using strategy rm

   7. Applied pow1/2_binary64 50.4

      \[\leadsto -0.25 \cdot \frac{\sqrt{2} \cdot \color{blue}{{\left(-8 \cdot \left(F \cdot C\right)\right)}^{0.5}}}{C}\]

   8. Applied pow1/2_binary64 50.4

      \[\leadsto -0.25 \cdot \frac{\color{blue}{{2}^{0.5}} \cdot {\left(-8 \cdot \left(F \cdot C\right)\right)}^{0.5}}{C}\]

   9. Applied pow-prod-down_binary64 50.4

      \[\leadsto -0.25 \cdot \frac{\color{blue}{{\left(2 \cdot \left(-8 \cdot \left(F \cdot C\right)\right)\right)}^{0.5}}}{C}\]
3. Recombined 5 regimes into one program.

4. Final simplification 34.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.9268697537850326 \cdot 10^{+186}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{F \cdot \left(C \cdot -16\right)}}{C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -1.107057733816843 \cdot 10^{-148}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}} \cdot \sqrt{\sqrt{{\left(A - C\right)}^{2} + B \cdot B}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{F} \cdot \frac{\sqrt{C \cdot -16}}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{{\left(2 \cdot \left(-8 \cdot \left(C \cdot F\right)\right)\right)}^{0.5}}{C}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021110 
(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))))