ABCF->ab-angle angle

Average Error: 29.9 → 10.5

Time: 10.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\\ t_1 := \left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)\\ t_2 := \mathsf{fma}\left(A, A, C \cdot \left(C - A \cdot 2\right)\right)\\ \mathbf{if}\;t_0 \leq -0.5:\\ \;\;\;\;\frac{1}{\frac{\pi}{\tan^{-1} \left({\left(\frac{B}{t_1}\right)}^{-1}\right) \cdot 180}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\frac{C - \left(A + \sqrt{t_2}\right)}{B} - B \cdot \left(0.5 \cdot \sqrt{\frac{1}{t_2}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{t_1}{B}\right)}{\pi \cdot 0.005555555555555556}\\ \end{array} \]

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
         (* (/ 1.0 B) (- (- C A) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))
        (t_1 (- (- C A) (hypot B (- A C))))
        (t_2 (fma A A (* C (- C (* A 2.0))))))
   (if (<= t_0 -0.5)
     (/ 1.0 (/ PI (* (atan (pow (/ B t_1) -1.0)) 180.0)))
     (if (<= t_0 0.0)
       (/
        1.0
        (/
         PI
         (*
          180.0
          (atan
           (-
            (/ (- C (+ A (sqrt t_2))) B)
            (* B (* 0.5 (sqrt (/ 1.0 t_2)))))))))
       (/ (atan (/ t_1 B)) (* PI 0.005555555555555556))))))

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 = (1.0 / B) * ((C - A) - sqrt((pow((A - C), 2.0) + pow(B, 2.0))));
	double t_1 = (C - A) - hypot(B, (A - C));
	double t_2 = fma(A, A, (C * (C - (A * 2.0))));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = 1.0 / (((double) M_PI) / (atan(pow((B / t_1), -1.0)) * 180.0));
	} else if (t_0 <= 0.0) {
		tmp = 1.0 / (((double) M_PI) / (180.0 * atan((((C - (A + sqrt(t_2))) / B) - (B * (0.5 * sqrt((1.0 / t_2))))))));
	} else {
		tmp = atan((t_1 / B)) / (((double) M_PI) * 0.005555555555555556);
	}
	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(Float64(1.0 / B) * Float64(Float64(C - A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0)))))
	t_1 = Float64(Float64(C - A) - hypot(B, Float64(A - C)))
	t_2 = fma(A, A, Float64(C * Float64(C - Float64(A * 2.0))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(1.0 / Float64(pi / Float64(atan((Float64(B / t_1) ^ -1.0)) * 180.0)));
	elseif (t_0 <= 0.0)
		tmp = Float64(1.0 / Float64(pi / Float64(180.0 * atan(Float64(Float64(Float64(C - Float64(A + sqrt(t_2))) / B) - Float64(B * Float64(0.5 * sqrt(Float64(1.0 / t_2)))))))));
	else
		tmp = Float64(atan(Float64(t_1 / B)) / Float64(pi * 0.005555555555555556));
	end
	return 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[(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] - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(A * A + N[(C * N[(C - N[(A * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(1.0 / N[(Pi / N[(N[ArcTan[N[Power[N[(B / t$95$1), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(1.0 / N[(Pi / N[(180.0 * N[ArcTan[N[(N[(N[(C - N[(A + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(B * N[(0.5 * N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(t$95$1 / B), $MachinePrecision]], $MachinePrecision] / N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]]]]]]

Error

Bits error versus A

Bits error versus B

Bits error versus C

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.6

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

   2. Simplified 8.7

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

   3. Applied egg-rr 11.6

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

   4. Applied egg-rr 8.7

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

   if -0.5 < (*.f64 (/.f64 1 B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))) < 0.0

   1. Initial program 51.3

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

   2. Simplified 50.1

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

   3. Applied egg-rr 55.0

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

   4. Taylor expanded in B around 0 61.2

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

   5. Simplified 24.7

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

   if 0.0 < (*.f64 (/.f64 1 B) (-.f64 (-.f64 C A) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2)))))

   1. Initial program 26.6

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

   2. Simplified 8.0

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

   3. Applied egg-rr 11.0

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

   4. Applied egg-rr 8.0

      \[\leadsto \color{blue}{\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\pi \cdot 0.005555555555555556}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq -0.5:\\ \;\;\;\;\frac{1}{\frac{\pi}{\tan^{-1} \left({\left(\frac{B}{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}\right)}^{-1}\right) \cdot 180}}\\ \mathbf{elif}\;\frac{1}{B} \cdot \left(\left(C - A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \leq 0:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\frac{C - \left(A + \sqrt{\mathsf{fma}\left(A, A, C \cdot \left(C - A \cdot 2\right)\right)}\right)}{B} - B \cdot \left(0.5 \cdot \sqrt{\frac{1}{\mathsf{fma}\left(A, A, C \cdot \left(C - A \cdot 2\right)\right)}}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\left(C - A\right) - \mathsf{hypot}\left(B, A - C\right)}{B}\right)}{\pi \cdot 0.005555555555555556}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))