ABCF->ab-angle angle

Average Error: 29.4 → 13.0

Time: 7.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\tan^{-1} \left(\left(\frac{B}{C} \cdot \left(1 + \frac{A}{C}\right)\right) \cdot -0.5\right)}{\pi \cdot 0.005555555555555556}\\ t_1 := \tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)\\ \mathbf{if}\;C \leq 8.995628272771583 \cdot 10^{-100}:\\ \;\;\;\;t_1 \cdot \frac{180}{\pi}\\ \mathbf{elif}\;C \leq 0.23170752377600393:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 1.7991825338144448 \cdot 10^{+115}:\\ \;\;\;\;\frac{t_1 \cdot 180}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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
         (/
          (atan (* (* (/ B C) (+ 1.0 (/ A C))) -0.5))
          (* PI 0.005555555555555556)))
        (t_1 (atan (/ (- (- C A) (hypot B (- A C))) B))))
   (if (<= C 8.995628272771583e-100)
     (* t_1 (/ 180.0 PI))
     (if (<= C 0.23170752377600393)
       t_0
       (if (<= C 1.7991825338144448e+115) (/ (* t_1 180.0) PI) t_0)))))

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 = atan((((B / C) * (1.0 + (A / C))) * -0.5)) / (((double) M_PI) * 0.005555555555555556);
	double t_1 = atan((((C - A) - hypot(B, (A - C))) / B));
	double tmp;
	if (C <= 8.995628272771583e-100) {
		tmp = t_1 * (180.0 / ((double) M_PI));
	} else if (C <= 0.23170752377600393) {
		tmp = t_0;
	} else if (C <= 1.7991825338144448e+115) {
		tmp = (t_1 * 180.0) / ((double) M_PI);
	} else {
		tmp = t_0;
	}
	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 = Math.atan((((B / C) * (1.0 + (A / C))) * -0.5)) / (Math.PI * 0.005555555555555556);
	double t_1 = Math.atan((((C - A) - Math.hypot(B, (A - C))) / B));
	double tmp;
	if (C <= 8.995628272771583e-100) {
		tmp = t_1 * (180.0 / Math.PI);
	} else if (C <= 0.23170752377600393) {
		tmp = t_0;
	} else if (C <= 1.7991825338144448e+115) {
		tmp = (t_1 * 180.0) / Math.PI;
	} else {
		tmp = t_0;
	}
	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 = math.atan((((B / C) * (1.0 + (A / C))) * -0.5)) / (math.pi * 0.005555555555555556)
	t_1 = math.atan((((C - A) - math.hypot(B, (A - C))) / B))
	tmp = 0
	if C <= 8.995628272771583e-100:
		tmp = t_1 * (180.0 / math.pi)
	elif C <= 0.23170752377600393:
		tmp = t_0
	elif C <= 1.7991825338144448e+115:
		tmp = (t_1 * 180.0) / math.pi
	else:
		tmp = t_0
	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(atan(Float64(Float64(Float64(B / C) * Float64(1.0 + Float64(A / C))) * -0.5)) / Float64(pi * 0.005555555555555556))
	t_1 = atan(Float64(Float64(Float64(C - A) - hypot(B, Float64(A - C))) / B))
	tmp = 0.0
	if (C <= 8.995628272771583e-100)
		tmp = Float64(t_1 * Float64(180.0 / pi));
	elseif (C <= 0.23170752377600393)
		tmp = t_0;
	elseif (C <= 1.7991825338144448e+115)
		tmp = Float64(Float64(t_1 * 180.0) / pi);
	else
		tmp = t_0;
	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 = atan((((B / C) * (1.0 + (A / C))) * -0.5)) / (pi * 0.005555555555555556);
	t_1 = atan((((C - A) - hypot(B, (A - C))) / B));
	tmp = 0.0;
	if (C <= 8.995628272771583e-100)
		tmp = t_1 * (180.0 / pi);
	elseif (C <= 0.23170752377600393)
		tmp = t_0;
	elseif (C <= 1.7991825338144448e+115)
		tmp = (t_1 * 180.0) / pi;
	else
		tmp = t_0;
	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[(N[ArcTan[N[(N[(N[(B / C), $MachinePrecision] * N[(1.0 + N[(A / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[(N[(C - A), $MachinePrecision] - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[C, 8.995628272771583e-100], N[(t$95$1 * N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[C, 0.23170752377600393], t$95$0, If[LessEqual[C, 1.7991825338144448e+115], N[(N[(t$95$1 * 180.0), $MachinePrecision] / Pi), $MachinePrecision], t$95$0]]]]]

Error

Bits error versus A

Bits error versus B

Bits error versus C

Derivation

1. Split input into 3 regimes

2. if C < 8.995628272771583e-100

   1. Initial program 22.3

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

   2. Simplified 8.3

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

   if 8.995628272771583e-100 < C < 0.231707523776003926 or 1.79918253381444479e115 < C

   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 25.1

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

   3. Taylor expanded in C around inf 24.7

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

   4. Simplified 22.4

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

   5. Applied egg-rr 22.3

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\left(\frac{B}{C} \cdot \left(1 + \frac{A}{C}\right)\right) \cdot -0.5\right)}{\pi \cdot 0.005555555555555556}} \]

   if 0.231707523776003926 < C < 1.79918253381444479e115

   1. Initial program 38.5

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

   2. Simplified 23.4

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

   3. Taylor expanded in C around 0 27.0

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

   4. Simplified 23.4

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 8.995628272771583 \cdot 10^{-100}:\\ \;\;\;\;\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot \frac{180}{\pi}\\ \mathbf{elif}\;C \leq 0.23170752377600393:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{B}{C} \cdot \left(1 + \frac{A}{C}\right)\right) \cdot -0.5\right)}{\pi \cdot 0.005555555555555556}\\ \mathbf{elif}\;C \leq 1.7991825338144448 \cdot 10^{+115}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right) \cdot 180}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{B}{C} \cdot \left(1 + \frac{A}{C}\right)\right) \cdot -0.5\right)}{\pi \cdot 0.005555555555555556}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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)))