ABCF->ab-angle b

Average Error: 52.3 → 31.5

Time: 36.1s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_0}\\ t_2 := \sqrt{-F}\\ \mathbf{if}\;t_1 \leq -2.52337341989472 \cdot 10^{+196}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{C}}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\\ \mathbf{if}\;t_1 \leq -5.110582906115813 \cdot 10^{-196}:\\ \;\;\;\;\begin{array}{l} t_4 := F \cdot \left(B \cdot B\right)\\ t_5 := C \cdot t_4\\ \frac{-\sqrt{\mathsf{fma}\left(2, \left(A \cdot t_5\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(A, A, B \cdot B\right)}}, \mathsf{fma}\left(2, t_5, \mathsf{fma}\left(2, A \cdot t_4, 8 \cdot \left(\left(A \cdot \left(C \cdot F\right)\right) \cdot \mathsf{hypot}\left(A, B\right)\right)\right)\right)\right) - \mathsf{fma}\left(8, \left(C \cdot F\right) \cdot \left(A \cdot A\right), 2 \cdot \left(t_4 \cdot \mathsf{hypot}\left(A, B\right)\right)\right)}}{t_3} \end{array}\\ \mathbf{elif}\;t_1 \leq 3.1080744412656145 \cdot 10^{+66}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A - 0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{t_3}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\begin{array}{l} t_6 := \sqrt{\left(C \cdot F\right) \cdot -8}\\ \frac{A \cdot \left(\sqrt{2} \cdot t_6\right) + \frac{F \cdot \left({B}^{2} \cdot \sqrt{2}\right)}{t_6}}{t_3} \end{array}\\ \mathbf{else}:\\ \;\;\;\;-\frac{t_2}{\sqrt{C}}\\ \end{array}\\ \end{array} \]

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 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_0))
        (t_2 (sqrt (- F))))
   (if (<= t_1 -2.52337341989472e+196)
     (/ -1.0 (/ (sqrt C) t_2))
     (let* ((t_3 (fma A (* C -4.0) (* B B))))
       (if (<= t_1 -5.110582906115813e-196)
         (let* ((t_4 (* F (* B B))) (t_5 (* C t_4)))
           (/
            (-
             (sqrt
              (-
               (fma
                2.0
                (* (* A t_5) (sqrt (/ 1.0 (fma A A (* B B)))))
                (fma
                 2.0
                 t_5
                 (fma 2.0 (* A t_4) (* 8.0 (* (* A (* C F)) (hypot A B))))))
               (fma 8.0 (* (* C F) (* A A)) (* 2.0 (* t_4 (hypot A B)))))))
            t_3))
         (if (<= t_1 3.1080744412656145e+66)
           (/
            (-
             (sqrt
              (* t_3 (* 2.0 (* F (- (* 2.0 A) (* 0.5 (/ (pow B 2.0) C))))))))
            t_3)
           (if (<= t_1 INFINITY)
             (let* ((t_6 (sqrt (* (* C F) -8.0))))
               (/
                (+
                 (* A (* (sqrt 2.0) t_6))
                 (/ (* F (* (pow B 2.0) (sqrt 2.0))) t_6))
                t_3))
             (- (/ 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 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt((2.0 * (t_0 * F)) * ((A + C) - sqrt(pow(B, 2.0) + pow((A - C), 2.0)))) / t_0;
	double t_2 = sqrt(-F);
	double tmp;
	if (t_1 <= -2.52337341989472e+196) {
		tmp = -1.0 / (sqrt(C) / t_2);
	} else {
		double t_3 = fma(A, (C * -4.0), (B * B));
		double tmp_1;
		if (t_1 <= -5.110582906115813e-196) {
			double t_4_2 = F * (B * B);
			double t_5_3 = C * t_4_2;
			tmp_1 = -sqrt(fma(2.0, ((A * t_5_3) * sqrt(1.0 / fma(A, A, (B * B)))), fma(2.0, t_5_3, fma(2.0, (A * t_4_2), (8.0 * ((A * (C * F)) * hypot(A, B)))))) - fma(8.0, ((C * F) * (A * A)), (2.0 * (t_4_2 * hypot(A, B))))) / t_3;
		} else if (t_1 <= 3.1080744412656145e+66) {
			tmp_1 = -sqrt(t_3 * (2.0 * (F * ((2.0 * A) - (0.5 * (pow(B, 2.0) / C)))))) / t_3;
		} else if (t_1 <= ((double) INFINITY)) {
			double t_6 = sqrt((C * F) * -8.0);
			tmp_1 = ((A * (sqrt(2.0) * t_6)) + ((F * (pow(B, 2.0) * sqrt(2.0))) / t_6)) / t_3;
		} else {
			tmp_1 = -(t_2 / sqrt(C));
		}
		tmp = tmp_1;
	}
	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))) < -2.52337341989472001e196

   1. Initial program 61.9

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

   2. Simplified 53.6

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

   3. Taylor expanded in A around -inf 33.7

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

   4. Simplified 33.7

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

   5. Applied associate-*r/_binary64 33.7

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

   6. Applied sqrt-div_binary64 22.1

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

   7. Applied associate-*l/_binary64 22.1

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

   8. Applied sqrt-unprod_binary64 21.9

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

   9. Simplified 21.9

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

   10. Applied *-un-lft-identity_binary64 21.9

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

   11. Applied sqrt-prod_binary64 21.9

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

   12. Applied associate-/l*_binary64 22.0

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

   if -2.52337341989472001e196 < (/.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))) < -5.1105829061158132e-196

   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. Simplified 1.5

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

   3. Taylor expanded in C around 0 3.5

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

   4. Simplified 3.5

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

   if -5.1105829061158132e-196 < (/.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))) < 3.10807444126561446e66

   1. Initial program 53.6

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

   2. Simplified 52.4

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

   3. Taylor expanded in C around inf 26.6

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \color{blue}{\left(2 \cdot A - 0.5 \cdot \frac{{B}^{2}}{C}\right)}\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \]

   if 3.10807444126561446e66 < (/.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 50.2

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

   2. Simplified 35.3

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

   3. Taylor expanded in A around -inf 15.5

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

   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. Simplified 63.5

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

   3. Taylor expanded in A around -inf 53.3

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

   4. Simplified 53.3

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

   5. Applied associate-*r/_binary64 53.3

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

   6. Applied sqrt-div_binary64 49.7

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

   7. Applied associate-*l/_binary64 49.7

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

   8. Applied sqrt-unprod_binary64 49.7

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

   9. Simplified 49.7

      \[\leadsto -\frac{\sqrt{\color{blue}{-F}}}{\sqrt{C}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 31.5

   \[\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 -2.52337341989472 \cdot 10^{+196}:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{C}}{\sqrt{-F}}}\\ \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 -5.110582906115813 \cdot 10^{-196}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(2, \left(A \cdot \left(C \cdot \left(F \cdot \left(B \cdot B\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(A, A, B \cdot B\right)}}, \mathsf{fma}\left(2, C \cdot \left(F \cdot \left(B \cdot B\right)\right), \mathsf{fma}\left(2, A \cdot \left(F \cdot \left(B \cdot B\right)\right), 8 \cdot \left(\left(A \cdot \left(C \cdot F\right)\right) \cdot \mathsf{hypot}\left(A, B\right)\right)\right)\right)\right) - \mathsf{fma}\left(8, \left(C \cdot F\right) \cdot \left(A \cdot A\right), 2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \mathsf{hypot}\left(A, B\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\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 3.1080744412656145 \cdot 10^{+66}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(2 \cdot A - 0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\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{\left(C \cdot F\right) \cdot -8}\right) + \frac{F \cdot \left({B}^{2} \cdot \sqrt{2}\right)}{\sqrt{\left(C \cdot F\right) \cdot -8}}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{-F}}{\sqrt{C}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))