ABCF->ab-angle angle

Average Error: 30.0 → 28.7

Time: 6.2s

Precision: binary64

Math

\[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\pi}\]

↓

\[\begin{array}{l} \mathbf{if}\;C \leq -3.2780186347505555 \cdot 10^{+32}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;C \leq -9.829888247783205 \cdot 10^{-172}:\\ \;\;\;\;180 \cdot \frac{1}{\frac{\pi}{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}}{B}\right)}}\\ \mathbf{elif}\;C \leq 5.241670643923542 \cdot 10^{-301}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;C \leq 2.405665199649213 \cdot 10^{-196}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\pi}\\ \mathbf{elif}\;C \leq 4.617632627264863 \cdot 10^{+151}:\\ \;\;\;\;\left(\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot 180\right) \cdot \frac{1}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0}{B}\right)}{\pi}\\ \end{array}\]

FPCore

(FPCore (A B C)
 :precision binary64
 (*
  180.0
  (/
   (atan (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
   PI)))

↓

(FPCore (A B C)
 :precision binary64
 (if (<= C -3.2780186347505555e+32)
   (* (atan (/ (- (- C A) B) B)) (/ 180.0 PI))
   (if (<= C -9.829888247783205e-172)
     (*
      180.0
      (/
       1.0
       (/ PI (atan (/ (- (- C A) (sqrt (+ (pow (- A C) 2.0) (* B B)))) B)))))
     (if (<= C 5.241670643923542e-301)
       (* (atan (/ (- (- C A) B) B)) (/ 180.0 PI))
       (if (<= C 2.405665199649213e-196)
         (*
          180.0
          (/
           (atan
            (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
           PI))
         (if (<= C 4.617632627264863e+151)
           (* (* (atan (/ (- (- C A) B) B)) 180.0) (/ 1.0 PI))
           (* 180.0 (/ (atan (/ 0.0 B)) PI))))))))

C

double code(double A, double B, double C) {
	return 180.0 * (atan((1.0 / B) * ((C - A) - sqrt(pow((A - C), 2.0) + pow(B, 2.0)))) / ((double) M_PI));
}

↓

double code(double A, double B, double C) {
	double tmp;
	if (C <= -3.2780186347505555e+32) {
		tmp = atan(((C - A) - B) / B) * (180.0 / ((double) M_PI));
	} else if (C <= -9.829888247783205e-172) {
		tmp = 180.0 * (1.0 / (((double) M_PI) / atan(((C - A) - sqrt(pow((A - C), 2.0) + (B * B))) / B)));
	} else if (C <= 5.241670643923542e-301) {
		tmp = atan(((C - A) - B) / B) * (180.0 / ((double) M_PI));
	} else if (C <= 2.405665199649213e-196) {
		tmp = 180.0 * (atan((1.0 / B) * ((C - A) - sqrt(pow((A - C), 2.0) + pow(B, 2.0)))) / ((double) M_PI));
	} else if (C <= 4.617632627264863e+151) {
		tmp = (atan(((C - A) - B) / B) * 180.0) * (1.0 / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(0.0 / B) / ((double) M_PI));
	}
	return tmp;
}

Error

Bits error versus A

Bits error versus B

Bits error versus C

Derivation

1. Split input into 5 regimes

2. if C < -3.2780186347505555e32 or -9.8298882477832054e-172 < C < 5.241670643923542e-301

   1. Initial program 18.8

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\pi}\]

   2. Simplified 18.8

      \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}}{B}\right)}{\pi}}\]

   3. Taylor expanded around 0 18.1

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{B}}{B}\right)}{\pi}\]
   4. Using strategy rm

   5. Applied div-inv_binary64_1780 18.1

      \[\leadsto 180 \cdot \color{blue}{\left(\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot \frac{1}{\pi}\right)}\]

   6. Applied associate-*r*_binary64_1723 18.1

      \[\leadsto \color{blue}{\left(180 \cdot \tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right)\right) \cdot \frac{1}{\pi}}\]

   7. Simplified 18.1

      \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot 180\right)} \cdot \frac{1}{\pi}\]
   8. Using strategy rm

   9. Applied associate-*l*_binary64_1724 18.1

      \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)}\]

   10. Simplified 18.1

       \[\leadsto \tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot \color{blue}{\frac{180}{\pi}}\]

   if -3.2780186347505555e32 < C < -9.8298882477832054e-172

   1. Initial program 22.5

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\pi}\]

   2. Simplified 22.5

      \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}}{B}\right)}{\pi}}\]
   3. Using strategy rm

   4. Applied clear-num_binary64_1782 22.5

      \[\leadsto 180 \cdot \color{blue}{\frac{1}{\frac{\pi}{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}}{B}\right)}}}\]

   if 5.241670643923542e-301 < C < 2.4056651996492128e-196

   1. Initial program 28.5

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\pi}\]

   if 2.4056651996492128e-196 < C < 4.617632627264863e151

   1. Initial program 36.1

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\pi}\]

   2. Simplified 36.1

      \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}}{B}\right)}{\pi}}\]

   3. Taylor expanded around 0 39.9

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \color{blue}{B}}{B}\right)}{\pi}\]
   4. Using strategy rm

   5. Applied div-inv_binary64_1780 39.9

      \[\leadsto 180 \cdot \color{blue}{\left(\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot \frac{1}{\pi}\right)}\]

   6. Applied associate-*r*_binary64_1723 39.9

      \[\leadsto \color{blue}{\left(180 \cdot \tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right)\right) \cdot \frac{1}{\pi}}\]

   7. Simplified 39.9

      \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot 180\right)} \cdot \frac{1}{\pi}\]

   if 4.617632627264863e151 < C

   1. Initial program 57.0

      \[180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\pi}\]

   2. Simplified 57.0

      \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}}{B}\right)}{\pi}}\]

   3. Taylor expanded around inf 40.2

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{0}}{B}\right)}{\pi}\]
3. Recombined 5 regimes into one program.

4. Final simplification 28.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -3.2780186347505555 \cdot 10^{+32}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;C \leq -9.829888247783205 \cdot 10^{-172}:\\ \;\;\;\;180 \cdot \frac{1}{\frac{\pi}{\tan^{-1} \left(\frac{\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + B \cdot B}}{B}\right)}}\\ \mathbf{elif}\;C \leq 5.241670643923542 \cdot 10^{-301}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;C \leq 2.405665199649213 \cdot 10^{-196}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}{\pi}\\ \mathbf{elif}\;C \leq 4.617632627264863 \cdot 10^{+151}:\\ \;\;\;\;\left(\tan^{-1} \left(\frac{\left(C - A\right) - B}{B}\right) \cdot 180\right) \cdot \frac{1}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{0}{B}\right)}{\pi}\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(FPCore (A B C)
  :name "ABCF->ab-angle angle"
  :precision binary64
  (* 180.0 (/ (atan (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))) PI)))