ABCF->ab-angle b

Average Error: 52.0 → 43.9

Time: 56.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;C \leq -1.3732001585011064 \cdot 10^{+135}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{\left(A \cdot F\right) \cdot -16}}{A}\\ \mathbf{elif}\;C \leq -1.144330577806148 \cdot 10^{-163}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) - \left|\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}\right| \cdot \sqrt{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 2.5219005088762548 \cdot 10^{-262}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{\left(A \cdot F\right) \cdot -16}}{A}\\ \mathbf{elif}\;C \leq 4.4863525056193624 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) - \left(\sqrt{B \cdot B + C \cdot C} - \left(C \cdot A\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{-2 \cdot \frac{C}{F}}}{\sqrt{2}}}\\ \end{array}\]

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 (<= C -1.3732001585011064e+135)
   (* -0.25 (/ (sqrt (* (* A F) -16.0)) A))
   (if (<= C -1.144330577806148e-163)
     (/
      (-
       (sqrt
        (*
         (* 2.0 (* F (- (pow B 2.0) (* C (* A 4.0)))))
         (-
          (+ C A)
          (*
           (fabs (cbrt (+ (pow (- A C) 2.0) (* B B))))
           (sqrt (cbrt (+ (pow (- A C) 2.0) (* B B)))))))))
      (- (pow B 2.0) (* C (* A 4.0))))
     (if (<= C 2.5219005088762548e-262)
       (* -0.25 (/ (sqrt (* (* A F) -16.0)) A))
       (if (<= C 4.4863525056193624e-18)
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* F (- (pow B 2.0) (* C (* A 4.0)))))
             (-
              (+ C A)
              (-
               (sqrt (+ (* B B) (* C C)))
               (* (* C A) (sqrt (/ 1.0 (+ (* B B) (* C C))))))))))
          (- (pow B 2.0) (* C (* A 4.0))))
         (/ -1.0 (/ (sqrt (* -2.0 (/ C F))) (sqrt 2.0))))))))

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 (C <= -1.3732001585011064e+135) {
		tmp = -0.25 * (sqrt((A * F) * -16.0) / A);
	} else if (C <= -1.144330577806148e-163) {
		tmp = -sqrt((2.0 * (F * (pow(B, 2.0) - (C * (A * 4.0))))) * ((C + A) - (fabs(cbrt(pow((A - C), 2.0) + (B * B))) * sqrt(cbrt(pow((A - C), 2.0) + (B * B)))))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (C <= 2.5219005088762548e-262) {
		tmp = -0.25 * (sqrt((A * F) * -16.0) / A);
	} else if (C <= 4.4863525056193624e-18) {
		tmp = -sqrt((2.0 * (F * (pow(B, 2.0) - (C * (A * 4.0))))) * ((C + A) - (sqrt((B * B) + (C * C)) - ((C * A) * sqrt(1.0 / ((B * B) + (C * C))))))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = -1.0 / (sqrt(-2.0 * (C / F)) / sqrt(2.0));
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Derivation

1. Split input into 4 regimes

2. if C < -1.37320015850110637e135 or -1.144330577806148e-163 < C < 2.5219005088762548e-262

   1. Initial program 54.5

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

   3. Applied associate--l+_binary64_3084 54.5

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

   4. Applied distribute-rgt-in_binary64_3097 54.5

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

   5. Simplified 54.5

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

   6. Simplified 54.5

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

   7. Taylor expanded around -inf 42.7

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

   8. Simplified 42.7

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

   if -1.37320015850110637e135 < C < -1.144330577806148e-163

   1. Initial program 42.9

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

   3. Applied add-cube-cbrt_binary64_3182 43.0

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

   4. Applied sqrt-prod_binary64_3163 43.0

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

   5. Simplified 43.0

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

   6. Simplified 43.0

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

   if 2.5219005088762548e-262 < C < 4.4863525056193624e-18

   1. Initial program 48.0

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

   2. Taylor expanded around 0 49.8

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

   3. Simplified 49.8

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

   if 4.4863525056193624e-18 < C

   1. Initial program 60.6

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

   3. Applied neg-mul-1_binary64_3143 60.6

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

   4. Applied associate-/l*_binary64_3092 60.6

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

   5. Simplified 60.6

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

   6. Taylor expanded around -inf 41.6

      \[\leadsto \frac{-1}{\color{blue}{\frac{\sqrt{-2 \cdot \frac{C}{F}}}{\sqrt{2}}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 43.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.3732001585011064 \cdot 10^{+135}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{\left(A \cdot F\right) \cdot -16}}{A}\\ \mathbf{elif}\;C \leq -1.144330577806148 \cdot 10^{-163}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) - \left|\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}\right| \cdot \sqrt{\sqrt[3]{{\left(A - C\right)}^{2} + B \cdot B}}\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;C \leq 2.5219005088762548 \cdot 10^{-262}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{\left(A \cdot F\right) \cdot -16}}{A}\\ \mathbf{elif}\;C \leq 4.4863525056193624 \cdot 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(C + A\right) - \left(\sqrt{B \cdot B + C \cdot C} - \left(C \cdot A\right) \cdot \sqrt{\frac{1}{B \cdot B + C \cdot C}}\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{-2 \cdot \frac{C}{F}}}{\sqrt{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020353 
(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))))