ABCF->ab-angle b

Average Error: 52.7 → 31.1

Time: 23.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt{2}}\\ t_1 := t_0 \cdot t_0\\ t_2 := \sqrt{F \cdot -0.5}\\ t_3 := {\left(A - C\right)}^{2}\\ t_4 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{-\sqrt{\left(2 \cdot \left(t_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + t_3}\right)}}{t_4}\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;t_1 \cdot \left(t_2 \cdot \left(t_0 \cdot \frac{-1}{\sqrt{C}}\right)\right)\\ \mathbf{elif}\;t_5 \leq -1.2953816470925393 \cdot 10^{-207}:\\ \;\;\;\;\begin{array}{l} t_6 := 2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\\ \frac{-\sqrt{A \cdot t_6 + t_6 \cdot \left(C - \sqrt{t_3 + B \cdot B}\right)}}{t_4} \end{array}\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;\begin{array}{l} t_7 := \sqrt{\sqrt{2}}\\ -t_7 \cdot \left(t_7 \cdot \frac{t_2}{\sqrt{C}}\right) \end{array}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\frac{A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{t_4}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_1 \cdot \left(t_0 \cdot t_2\right)}{\sqrt{C}}\\ \end{array} \]

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
 (let* ((t_0 (cbrt (sqrt 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (sqrt (* F -0.5)))
        (t_3 (pow (- A C) 2.0))
        (t_4 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (-
           (sqrt (* (* 2.0 (* t_4 F)) (- (+ A C) (sqrt (+ (pow B 2.0) t_3))))))
          t_4)))
   (if (<= t_5 (- INFINITY))
     (* t_1 (* t_2 (* t_0 (/ -1.0 (sqrt C)))))
     (if (<= t_5 -1.2953816470925393e-207)
       (let* ((t_6 (* 2.0 (* F (- (* B B) (* 4.0 (* A C)))))))
         (/ (- (sqrt (+ (* A t_6) (* t_6 (- C (sqrt (+ t_3 (* B B)))))))) t_4))
       (if (<= t_5 0.0)
         (let* ((t_7 (sqrt (sqrt 2.0)))) (- (* t_7 (* t_7 (/ t_2 (sqrt C))))))
         (if (<= t_5 INFINITY)
           (/ (* A (* (sqrt 2.0) (sqrt (* -8.0 (* C F))))) t_4)
           (- (/ (* t_1 (* t_0 t_2)) (sqrt 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 t_0 = cbrt(sqrt(2.0));
	double t_1 = t_0 * t_0;
	double t_2 = sqrt(F * -0.5);
	double t_3 = pow((A - C), 2.0);
	double t_4 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_5 = -sqrt((2.0 * (t_4 * F)) * ((A + C) - sqrt(pow(B, 2.0) + t_3))) / t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_1 * (t_2 * (t_0 * (-1.0 / sqrt(C))));
	} else if (t_5 <= -1.2953816470925393e-207) {
		double t_6 = 2.0 * (F * ((B * B) - (4.0 * (A * C))));
		tmp = -sqrt((A * t_6) + (t_6 * (C - sqrt(t_3 + (B * B))))) / t_4;
	} else if (t_5 <= 0.0) {
		double t_7 = sqrt(sqrt(2.0));
		tmp = -(t_7 * (t_7 * (t_2 / sqrt(C))));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (A * (sqrt(2.0) * sqrt(-8.0 * (C * F)))) / t_4;
	} else {
		tmp = -((t_1 * (t_0 * t_2)) / sqrt(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))) < -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 34.8

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

   3. Simplified 34.8

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

   5. Applied associate-*r/_binary64 34.7

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

   6. Applied sqrt-div_binary64 24.3

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

   7. Simplified 24.3

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

   9. Applied add-cube-cbrt_binary64 24.3

      \[\leadsto -\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}} \]

   10. Applied associate-*l*_binary64 24.3

       \[\leadsto -\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\right)} \]

   11. Simplified 24.3

       \[\leadsto -\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \color{blue}{\left(\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}} \cdot \sqrt[3]{\sqrt{2}}\right)} \]
   12. Using strategy rm

   13. Applied div-inv_binary64 24.3

       \[\leadsto -\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\color{blue}{\left(\sqrt{F \cdot -0.5} \cdot \frac{1}{\sqrt{C}}\right)} \cdot \sqrt[3]{\sqrt{2}}\right) \]

   14. Applied associate-*l*_binary64 24.3

       \[\leadsto -\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \color{blue}{\left(\sqrt{F \cdot -0.5} \cdot \left(\frac{1}{\sqrt{C}} \cdot \sqrt[3]{\sqrt{2}}\right)\right)} \]

   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))) < -1.29538164709253926e-207

   1. Initial program 1.4

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

   3. Applied associate--l+_binary64 1.4

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

   4. Applied distribute-rgt-in_binary64 1.4

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

   5. Simplified 1.4

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

   6. Simplified 1.4

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

   if -1.29538164709253926e-207 < (/.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 60.9

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

   2. Taylor expanded around -inf 33.1

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

   3. Simplified 33.1

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

   5. Applied associate-*r/_binary64 33.1

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

   6. Applied sqrt-div_binary64 29.5

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

   7. Simplified 29.5

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

   9. Applied add-sqr-sqrt_binary64 29.6

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

   10. Applied associate-*l*_binary64 29.6

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

   11. Simplified 29.6

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

   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 39.5

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

   2. Taylor expanded around -inf 14.5

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

   3. Simplified 14.5

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

   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 52.4

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

   3. Simplified 52.4

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

   5. Applied associate-*r/_binary64 52.4

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

   6. Applied sqrt-div_binary64 48.1

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

   7. Simplified 48.1

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

   9. Applied add-cube-cbrt_binary64 48.1

      \[\leadsto -\color{blue}{\left(\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}\right)} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}} \]

   10. Applied associate-*l*_binary64 48.1

       \[\leadsto -\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\right)} \]

   11. Simplified 48.1

       \[\leadsto -\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \color{blue}{\left(\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}} \cdot \sqrt[3]{\sqrt{2}}\right)} \]
   12. Using strategy rm

   13. Applied associate-*l/_binary64 48.1

       \[\leadsto -\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \color{blue}{\frac{\sqrt{F \cdot -0.5} \cdot \sqrt[3]{\sqrt{2}}}{\sqrt{C}}} \]

   14. Applied associate-*r/_binary64 48.1

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

4. Final simplification 31.1

   \[\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:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt{F \cdot -0.5} \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \frac{-1}{\sqrt{C}}\right)\right)\\ \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 -1.2953816470925393 \cdot 10^{-207}:\\ \;\;\;\;\frac{-\sqrt{A \cdot \left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) + \left(2 \cdot \left(F \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right) \cdot \left(C - \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{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}\right)\\ \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{A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt{F \cdot -0.5}\right)}{\sqrt{C}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))))