ABCF->ab-angle b

Average Error: 52.2 → 45.5

Time: 45.8s

Precision: binary64

Cost: 22210

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;A \leq -7.910126495996752 \cdot 10^{+18}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;A \leq 1.8416199243309805 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \end{array}\]

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 (<= A -7.910126495996752e+18)
   (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F C)))))
   (if (<= A 1.8416199243309805e-58)
     (/
      (-
       (sqrt
        (*
         (* 2.0 (* F (- (* B B) (* C (* A 4.0)))))
         (- (+ A C) (sqrt (+ (* B B) (pow (- A C) 2.0)))))))
      (- (* B B) (* C (* A 4.0))))
     (- (* (sqrt 2.0) (sqrt (* -0.5 (/ F A))))))))

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 (A <= -7.910126495996752e+18) {
		tmp = -(sqrt(2.0) * sqrt(-0.5 * (F / C)));
	} else if (A <= 1.8416199243309805e-58) {
		tmp = -sqrt((2.0 * (F * ((B * B) - (C * (A * 4.0))))) * ((A + C) - sqrt((B * B) + pow((A - C), 2.0)))) / ((B * B) - (C * (A * 4.0)));
	} else {
		tmp = -(sqrt(2.0) * sqrt(-0.5 * (F / A)));
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Bits error versus F

Alternatives

Alternative 1
Error	46.1
Cost	15746

\[\begin{array}{l} \mathbf{if}\;A \leq -7.419398851492627 \cdot 10^{+18}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;A \leq 7.90807920936131 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + C \cdot C}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \end{array}\]

Alternative 2
Error	47.4
Cost	16388

\[\begin{array}{l} \mathbf{if}\;A \leq -7.910126495996752 \cdot 10^{+18}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;A \leq -9.007271547567885 \cdot 10^{-287}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + A \cdot A}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;A \leq 1.1496526444247377 \cdot 10^{-282}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;A \leq 3.1969598146336614 \cdot 10^{-79}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + A \cdot A}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \end{array}\]

Alternative 3
Error	47.6
Cost	13633

\[\begin{array}{l} \mathbf{if}\;A \leq 1.3177812171524174 \cdot 10^{-282}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \end{array}\]

Alternative 4
Error	50.2
Cost	13633

\[\begin{array}{l} \mathbf{if}\;C \leq -3.5690494857006394 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \left(C - A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \end{array}\]

Alternative 5
Error	55.0
Cost	10370

\[\begin{array}{l} \mathbf{if}\;B \leq -1.0890342656946036 \cdot 10^{-35}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(B + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.1749683258543978 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(C + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(A + C\right) + \left(\frac{A \cdot C}{B} - \left(0.5 \cdot \frac{C \cdot C}{B} + \left(B + 0.5 \cdot \frac{A \cdot A}{B}\right)\right)\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 6
Error	54.9
Cost	8962

\[\begin{array}{l} \mathbf{if}\;B \leq -1.2261336475609622 \cdot 10^{-27}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(B + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1.072862699349279 \cdot 10^{-69}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(C + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(A + C\right) - B\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 7
Error	55.4
Cost	8641

\[\begin{array}{l} \mathbf{if}\;C \leq -1.0398432161930014 \cdot 10^{-100}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(C + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 8
Error	56.4
Cost	8641

\[\begin{array}{l} \mathbf{if}\;B \leq -6.137118346168398 \cdot 10^{-36}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(B + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(C + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \end{array}\]

Alternative 9
Error	58.3
Cost	8641

\[\begin{array}{l} \mathbf{if}\;B \leq 1.1749683258543978 \cdot 10^{-68}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(B + \left(A + C\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]

Alternative 10
Error	60.6
Cost	64

\[-1\]

Alternative 11
Error	61.6
Cost	64

\[0\]

Alternative 12
Error	62.1
Cost	64

\[1\]

Derivation

1. Split input into 3 regimes

2. if A < -7910126495996751900

   1. Initial program 52.8

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

   2. Simplified 52.8

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

   3. Taylor expanded around -inf 47.6

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

   4. Simplified 47.6

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

   5. Simplified 47.6

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

   if -7910126495996751900 < A < 1.84161992433098049e-58

   1. Initial program 46.9

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

   2. Simplified 46.9

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

   3. Simplified 46.9

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

   if 1.84161992433098049e-58 < A

   1. Initial program 60.0

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

   2. Simplified 60.0

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

   3. Taylor expanded around -inf 41.9

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

   4. Simplified 41.9

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

   5. Simplified 41.9

      \[\leadsto \color{blue}{-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 45.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -7.910126495996752 \cdot 10^{+18}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{C}}\\ \mathbf{elif}\;A \leq 1.8416199243309805 \cdot 10^{-58}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(B \cdot B - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + {\left(A - C\right)}^{2}}\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2} \cdot \sqrt{-0.5 \cdot \frac{F}{A}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))