ABCF->ab-angle b

Average Error: 52.1 → 33.5

Time: 37.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_0}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{C}}{\sqrt{-F}}}\\ \mathbf{elif}\;t_1 \leq -7.086899042058475 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;-{\left(\frac{-F}{C}\right)}^{0.5}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

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)))
   (if (<= t_1 (- INFINITY))
     (/ -1.0 (/ (sqrt C) (sqrt (- F))))
     (if (<= t_1 -7.086899042058475e-213)
       t_1
       (if (<= t_1 0.0)
         (- (pow (/ (- F) C) 0.5))
         (if (<= t_1 INFINITY)
           (/
            (* A (* (sqrt 2.0) (sqrt (* -8.0 (* C F)))))
            (fma A (* C -4.0) (* B B)))
           (- (* (sqrt (* F (- A (hypot A B)))) (/ (sqrt 2.0) B)))))))))

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 tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -1.0 / (sqrt(C) / sqrt(-F));
	} else if (t_1 <= -7.086899042058475e-213) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = -pow((-F / C), 0.5);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (A * (sqrt(2.0) * sqrt(-8.0 * (C * F)))) / fma(A, (C * -4.0), (B * B));
	} else {
		tmp = -(sqrt(F * (A - hypot(A, B))) * (sqrt(2.0) / B));
	}
	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. Simplified 53.8

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

   3. Taylor expanded in A around -inf 34.6

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

   4. Simplified 34.6

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

   5. Applied associate-*r/_binary64 34.6

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

   6. Applied sqrt-div_binary64 24.0

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

   7. Applied associate-*l/_binary64 24.0

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

   8. Applied sqrt-unprod_binary64 23.8

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

   9. Simplified 23.8

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

   10. Applied *-un-lft-identity_binary64 23.8

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

   11. Applied sqrt-prod_binary64 23.8

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

   12. Applied associate-/l*_binary64 23.9

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

   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))) < -7.0868990420584749e-213

   1. Initial program 1.4

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

   if -7.0868990420584749e-213 < (/.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 61.5

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

   2. Simplified 60.0

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

   3. Taylor expanded in A around -inf 31.9

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

   4. Simplified 31.9

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

   5. Applied pow1/2_binary64 31.9

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

   6. Applied pow1/2_binary64 31.9

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

   7. Applied pow-prod-down_binary64 31.7

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

   8. Simplified 31.7

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

   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 37.8

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

   2. Simplified 26.3

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

   3. Taylor expanded in A around -inf 14.6

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

   4. Simplified 14.6

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

   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 C around 0 63.4

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

   4. Simplified 53.3

      \[\leadsto \color{blue}{-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{\sqrt{2}}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{C}}{\sqrt{-F}}}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -7.086899042058475 \cdot 10^{-213}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;-{\left(\frac{-F}{C}\right)}^{0.5}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{A \cdot \left(\sqrt{2} \cdot \sqrt{-8 \cdot \left(C \cdot F\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{\sqrt{2}}{B}\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(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))))