ABCF->ab-angle angle

Percentage Accurate: 53.3% → 72.7%

Time: 12.1s

Alternatives: 12

Speedup: 0.6×

Specification

Math

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

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)))

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));
}

Java

public static double code(double A, double B, double C) {
	return 180.0 * (Math.atan(((1.0 / B) * ((C - A) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / Math.PI);
}

Python

def code(A, B, C):
	return 180.0 * (math.atan(((1.0 / B) * ((C - A) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / math.pi)

Julia

function code(A, B, C)
	return Float64(180.0 * Float64(atan(Float64(Float64(1.0 / B) * Float64(Float64(C - A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))))) / pi))
end

MATLAB

function tmp = code(A, B, C)
	tmp = 180.0 * (atan(((1.0 / B) * ((C - A) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / pi);
end

Wolfram

code[A_, B_, C_] := N[(180.0 * N[(N[ArcTan[N[(N[(1.0 / B), $MachinePrecision] * N[(N[(C - A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Initial Program: 53.3% accurate, 1.0× speedup

Math

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

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)))

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));
}

Java

public static double code(double A, double B, double C) {
	return 180.0 * (Math.atan(((1.0 / B) * ((C - A) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / Math.PI);
}

Python

def code(A, B, C):
	return 180.0 * (math.atan(((1.0 / B) * ((C - A) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / math.pi)

Julia

function code(A, B, C)
	return Float64(180.0 * Float64(atan(Float64(Float64(1.0 / B) * Float64(Float64(C - A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))))) / pi))
end

MATLAB

function tmp = code(A, B, C)
	tmp = 180.0 * (atan(((1.0 / B) * ((C - A) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / pi);
end

Wolfram

code[A_, B_, C_] := N[(180.0 * N[(N[ArcTan[N[(N[(1.0 / B), $MachinePrecision] * N[(N[(C - A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Alternative 1: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(t\_1 + -1\right)}{\pi}\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(B, \frac{-0.5}{C}, 0\right)\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 + t\_1\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -0.5)
     (/ (* 180.0 (atan (+ t_1 -1.0))) PI)
     (if (<= t_0 1e-7)
       (* (atan (fma B (/ -0.5 C) 0.0)) (* 180.0 (/ 1.0 PI)))
       (* 180.0 (/ (atan (+ 1.0 t_1)) PI))))))

C

double code(double A, double B, double C) {
	double t_0 = (1.0 / B) * ((C - A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0))));
	double t_1 = (C - A) / B;
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (180.0 * atan((t_1 + -1.0))) / ((double) M_PI);
	} else if (t_0 <= 1e-7) {
		tmp = atan(fma(B, (-0.5 / C), 0.0)) * (180.0 * (1.0 / ((double) M_PI)));
	} else {
		tmp = 180.0 * (atan((1.0 + t_1)) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(A, B, C)
	t_0 = Float64(Float64(1.0 / B) * Float64(Float64(C - A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))))
	t_1 = Float64(Float64(C - A) / B)
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(180.0 * atan(Float64(t_1 + -1.0))) / pi);
	elseif (t_0 <= 1e-7)
		tmp = Float64(atan(fma(B, Float64(-0.5 / C), 0.0)) * Float64(180.0 * Float64(1.0 / pi)));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(1.0 + t_1)) / pi));
	end
	return tmp
end

Wolfram

code[A_, B_, C_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] * N[(N[(C - A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C - A), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(180.0 * N[ArcTan[N[(t$95$1 + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[(N[ArcTan[N[(B * N[(-0.5 / C), $MachinePrecision] + 0.0), $MachinePrecision]], $MachinePrecision] * N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(1.0 + t$95$1), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -0.5

   1. Initial program 57.3%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 76.7

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

   5. Applied rewrites 76.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + -1\right)}}{\pi} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 76.7

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

   7. Applied rewrites 76.7%

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

   if -0.5 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 9.9999999999999995e-8

   1. Initial program 28.3%

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

   3. Taylor expanded in A around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{\frac{-1}{2} \cdot B + \frac{-1}{2} \cdot \frac{B \cdot C}{A}}{A}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-out N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(B, \frac{C}{A}, B\right)}}{A}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-/.f64 28.5

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{0.5 \cdot \mathsf{fma}\left(B, \color{blue}{\frac{C}{A}}, B\right)}{A}\right)}{\pi} \]

   5. Applied rewrites 28.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{0.5 \cdot \mathsf{fma}\left(B, \frac{C}{A}, B\right)}{A}\right)}}{\pi} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{180 \cdot \frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \mathsf{fma}\left(B, \frac{C}{A}, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \mathsf{fma}\left(B, \frac{C}{A}, B\right)}{A}\right)}{\mathsf{PI}\left(\right)} \cdot 180} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \mathsf{fma}\left(B, \frac{C}{A}, B\right)}{A}\right)}{\mathsf{PI}\left(\right)}} \cdot 180 \]

      4. div-inv N/A

         \[\leadsto \color{blue}{\left(\tan^{-1} \left(\frac{\frac{1}{2} \cdot \mathsf{fma}\left(B, \frac{C}{A}, B\right)}{A}\right) \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)} \cdot 180 \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \mathsf{fma}\left(B, \frac{C}{A}, B\right)}{A}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\tan^{-1} \left(\frac{\frac{1}{2} \cdot \mathsf{fma}\left(B, \frac{C}{A}, B\right)}{A}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right)} \]

   7. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\tan^{-1} \left(0.5 \cdot \frac{\mathsf{fma}\left(B, \frac{C}{A}, B\right)}{A}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   8. Taylor expanded in C around inf

      \[\leadsto \tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B} + \frac{-1}{2} \cdot \frac{B}{C}\right)} \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(B \cdot \frac{\frac{-1}{2}}{C} + \frac{-1 \cdot \left(\color{blue}{0} \cdot A\right)}{B}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

      8. mul0-lft N/A

         \[\leadsto \tan^{-1} \left(B \cdot \frac{\frac{-1}{2}}{C} + \frac{-1 \cdot \color{blue}{0}}{B}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

      9. metadata-eval N/A

         \[\leadsto \tan^{-1} \left(B \cdot \frac{\frac{-1}{2}}{C} + \frac{\color{blue}{0}}{B}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

      10. div0 N/A

          \[\leadsto \tan^{-1} \left(B \cdot \frac{\frac{-1}{2}}{C} + \color{blue}{0}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(B, \frac{\frac{-1}{2}}{C}, 0\right)\right)} \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

      12. lower-/.f64 69.9

          \[\leadsto \tan^{-1} \left(\mathsf{fma}\left(B, \color{blue}{\frac{-0.5}{C}}, 0\right)\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   10. Applied rewrites 69.9%

       \[\leadsto \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(B, \frac{-0.5}{C}, 0\right)\right)} \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 9.9999999999999995e-8 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))

   1. Initial program 60.6%

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

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 75.4

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

   5. Applied rewrites 75.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -0.5:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\frac{C - A}{B} + -1\right)}{\pi}\\ \mathbf{elif}\;\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 10^{-7}:\\ \;\;\;\;\tan^{-1} \left(\mathsf{fma}\left(B, \frac{-0.5}{C}, 0\right)\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 + \frac{C - A}{B}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \frac{C - A}{B}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(t\_1 + -1\right)}{\pi}\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(\mathsf{fma}\left(B, \frac{-0.5}{C}, 0\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(1 + t\_1\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

(FPCore (A B C)
 :precision binary64
 (let* ((t_0
         (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (/ (- C A) B)))
   (if (<= t_0 -0.5)
     (/ (* 180.0 (atan (+ t_1 -1.0))) PI)
     (if (<= t_0 1e-7)
       (/ (* 180.0 (atan (fma B (/ -0.5 C) 0.0))) PI)
       (* 180.0 (/ (atan (+ 1.0 t_1)) PI))))))

C

double code(double A, double B, double C) {
	double t_0 = (1.0 / B) * ((C - A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0))));
	double t_1 = (C - A) / B;
	double tmp;
	if (t_0 <= -0.5) {
		tmp = (180.0 * atan((t_1 + -1.0))) / ((double) M_PI);
	} else if (t_0 <= 1e-7) {
		tmp = (180.0 * atan(fma(B, (-0.5 / C), 0.0))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan((1.0 + t_1)) / ((double) M_PI));
	}
	return tmp;
}

Julia

function code(A, B, C)
	t_0 = Float64(Float64(1.0 / B) * Float64(Float64(C - A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))))
	t_1 = Float64(Float64(C - A) / B)
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(180.0 * atan(Float64(t_1 + -1.0))) / pi);
	elseif (t_0 <= 1e-7)
		tmp = Float64(Float64(180.0 * atan(fma(B, Float64(-0.5 / C), 0.0))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(1.0 + t_1)) / pi));
	end
	return tmp
end

Wolfram

code[A_, B_, C_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] * N[(N[(C - A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C - A), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(180.0 * N[ArcTan[N[(t$95$1 + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[(N[(180.0 * N[ArcTan[N[(B * N[(-0.5 / C), $MachinePrecision] + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(1.0 + t$95$1), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < -0.5

   1. Initial program 57.9%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + -1\right)}}{\pi} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 75.5

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

   7. Applied rewrites 75.5%

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

   if -0.5 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))) < 9.9999999999999995e-8

   1. Initial program 20.0%

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

   3. Taylor expanded in B around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C}{B} - \left(1 + \frac{A}{B}\right)\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 3.7

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

   5. Applied rewrites 3.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{C - A}{B} + -1\right)}}{\pi} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 3.7

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in C around inf

      \[\leadsto \frac{180 \cdot \tan^{-1} \color{blue}{\left(-1 \cdot \frac{A + -1 \cdot A}{B} + \frac{-1}{2} \cdot \frac{B}{C}\right)}}{\mathsf{PI}\left(\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r/ N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(B \cdot \frac{\frac{-1}{2}}{C} + \frac{-1 \cdot \left(\color{blue}{0} \cdot A\right)}{B}\right)}{\mathsf{PI}\left(\right)} \]

      8. mul0-lft N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(B \cdot \frac{\frac{-1}{2}}{C} + \frac{-1 \cdot \color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{180 \cdot \tan^{-1} \left(B \cdot \frac{\frac{-1}{2}}{C} + \frac{\color{blue}{0}}{B}\right)}{\mathsf{PI}\left(\right)} \]

      10. div0 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(B \cdot \frac{\frac{-1}{2}}{C} + \color{blue}{0}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{180 \cdot \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(B, \frac{\frac{-1}{2}}{C}, 0\right)\right)}}{\mathsf{PI}\left(\right)} \]

      12. lower-/.f64 52.2

          \[\leadsto \frac{180 \cdot \tan^{-1} \left(\mathsf{fma}\left(B, \color{blue}{\frac{-0.5}{C}}, 0\right)\right)}{\pi} \]

   10. Applied rewrites 52.2%

       \[\leadsto \frac{180 \cdot \tan^{-1} \color{blue}{\left(\mathsf{fma}\left(B, \frac{-0.5}{C}, 0\right)\right)}}{\pi} \]

   if 9.9999999999999995e-8 < (*.f64 (/.f64 #s(literal 1 binary64) B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64))))))

   1. Initial program 59.5%

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

   3. Taylor expanded in B around -inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\left(1 + \frac{C}{B}\right) - \frac{A}{B}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(1 + \frac{C - A}{B}\right)}}{\pi} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))