ABCF->ab-angle angle

Average Error: 28.9 → 12.0

Time: 11.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 180 \cdot \frac{\tan^{-1} \left(\frac{B}{2 \cdot \left(A - C\right)}\right)}{\pi}\\ \mathbf{if}\;A \leq -6.955844732265596 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;A \leq -6.382237549682215 \cdot 10^{-43}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{\frac{B}{C - \mathsf{hypot}\left(B, C\right)}}\right)}{\pi}\\ \mathbf{elif}\;A \leq -3.4226817542894903 \cdot 10^{-77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{\frac{B}{\left(C - A\right) - \mathsf{hypot}\left(B, C - A\right)}}\right)}{\pi}\\ \end{array} \]

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)))

↓

(FPCore (A B C)
 :precision binary64
 (let* ((t_0 (* 180.0 (/ (atan (/ B (* 2.0 (- A C)))) PI))))
   (if (<= A -6.955844732265596e-7)
     t_0
     (if (<= A -6.382237549682215e-43)
       (* 180.0 (/ (atan (/ 1.0 (/ B (- C (hypot B C))))) PI))
       (if (<= A -3.4226817542894903e-77)
         t_0
         (*
          180.0
          (/ (atan (/ 1.0 (/ B (- (- C A) (hypot B (- C A)))))) 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 t_0 = 180.0 * (atan((B / (2.0 * (A - C)))) / ((double) M_PI));
	double tmp;
	if (A <= -6.955844732265596e-7) {
		tmp = t_0;
	} else if (A <= -6.382237549682215e-43) {
		tmp = 180.0 * (atan((1.0 / (B / (C - hypot(B, C))))) / ((double) M_PI));
	} else if (A <= -3.4226817542894903e-77) {
		tmp = t_0;
	} else {
		tmp = 180.0 * (atan((1.0 / (B / ((C - A) - hypot(B, (C - A)))))) / ((double) M_PI));
	}
	return tmp;
}

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);
}

↓

public static double code(double A, double B, double C) {
	double t_0 = 180.0 * (Math.atan((B / (2.0 * (A - C)))) / Math.PI);
	double tmp;
	if (A <= -6.955844732265596e-7) {
		tmp = t_0;
	} else if (A <= -6.382237549682215e-43) {
		tmp = 180.0 * (Math.atan((1.0 / (B / (C - Math.hypot(B, C))))) / Math.PI);
	} else if (A <= -3.4226817542894903e-77) {
		tmp = t_0;
	} else {
		tmp = 180.0 * (Math.atan((1.0 / (B / ((C - A) - Math.hypot(B, (C - A)))))) / Math.PI);
	}
	return tmp;
}

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)

↓

def code(A, B, C):
	t_0 = 180.0 * (math.atan((B / (2.0 * (A - C)))) / math.pi)
	tmp = 0
	if A <= -6.955844732265596e-7:
		tmp = t_0
	elif A <= -6.382237549682215e-43:
		tmp = 180.0 * (math.atan((1.0 / (B / (C - math.hypot(B, C))))) / math.pi)
	elif A <= -3.4226817542894903e-77:
		tmp = t_0
	else:
		tmp = 180.0 * (math.atan((1.0 / (B / ((C - A) - math.hypot(B, (C - A)))))) / math.pi)
	return tmp

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

↓

function code(A, B, C)
	t_0 = Float64(180.0 * Float64(atan(Float64(B / Float64(2.0 * Float64(A - C)))) / pi))
	tmp = 0.0
	if (A <= -6.955844732265596e-7)
		tmp = t_0;
	elseif (A <= -6.382237549682215e-43)
		tmp = Float64(180.0 * Float64(atan(Float64(1.0 / Float64(B / Float64(C - hypot(B, C))))) / pi));
	elseif (A <= -3.4226817542894903e-77)
		tmp = t_0;
	else
		tmp = Float64(180.0 * Float64(atan(Float64(1.0 / Float64(B / Float64(Float64(C - A) - hypot(B, Float64(C - A)))))) / pi));
	end
	return tmp
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

↓

function tmp_2 = code(A, B, C)
	t_0 = 180.0 * (atan((B / (2.0 * (A - C)))) / pi);
	tmp = 0.0;
	if (A <= -6.955844732265596e-7)
		tmp = t_0;
	elseif (A <= -6.382237549682215e-43)
		tmp = 180.0 * (atan((1.0 / (B / (C - hypot(B, C))))) / pi);
	elseif (A <= -3.4226817542894903e-77)
		tmp = t_0;
	else
		tmp = 180.0 * (atan((1.0 / (B / ((C - A) - hypot(B, (C - A)))))) / pi);
	end
	tmp_2 = tmp;
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]

↓

code[A_, B_, C_] := Block[{t$95$0 = N[(180.0 * N[(N[ArcTan[N[(B / N[(2.0 * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[A, -6.955844732265596e-7], t$95$0, If[LessEqual[A, -6.382237549682215e-43], N[(180.0 * N[(N[ArcTan[N[(1.0 / N[(B / N[(C - N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -3.4226817542894903e-77], t$95$0, N[(180.0 * N[(N[ArcTan[N[(1.0 / N[(B / N[(N[(C - A), $MachinePrecision] - N[Sqrt[B ^ 2 + N[(C - A), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus A

Bits error versus B

Bits error versus C

Derivation

1. Split input into 3 regimes

2. if A < -6.9558447322655957e-7 or -6.3822375496822152e-43 < A < -3.4226817542894903e-77

   1. Initial program 46.0

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

   2. Simplified 25.2

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

   3. Applied clear-num_binary64 25.2

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

   4. Taylor expanded in A around -inf 24.4

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{\color{blue}{\left(0.5 \cdot \frac{B}{A} + 2 \cdot \frac{A}{B}\right) - 2 \cdot \frac{C}{B}}}\right)}{\pi} \]

   5. Simplified 24.4

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{B}{A}, 2 \cdot \left(\frac{A}{B} - \frac{C}{B}\right)\right)}}\right)}{\pi} \]

   6. Taylor expanded in B around 0 20.2

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

   7. Simplified 20.2

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

   if -6.9558447322655957e-7 < A < -6.3822375496822152e-43

   1. Initial program 32.3

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

   2. Simplified 21.2

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

   3. Applied clear-num_binary64 21.2

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

   4. Taylor expanded in A around 0 32.7

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

   5. Simplified 21.7

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

   if -3.4226817542894903e-77 < A

   1. Initial program 22.0

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

   2. Simplified 8.4

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

   3. Applied clear-num_binary64 8.4

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -6.955844732265596 \cdot 10^{-7}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{2 \cdot \left(A - C\right)}\right)}{\pi}\\ \mathbf{elif}\;A \leq -6.382237549682215 \cdot 10^{-43}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{\frac{B}{C - \mathsf{hypot}\left(B, C\right)}}\right)}{\pi}\\ \mathbf{elif}\;A \leq -3.4226817542894903 \cdot 10^{-77}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{B}{2 \cdot \left(A - C\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{1}{\frac{B}{\left(C - A\right) - \mathsf{hypot}\left(B, C - A\right)}}\right)}{\pi}\\ \end{array} \]

Reproduce

herbie shell --seed 2022132 
(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)))