ABCF->ab-angle b

Average Error: 52.8 → 40.2

Time: 30.9s

Precision: binary64

Cost: 14856

\[ \begin{array}{c}[A, C] = \mathsf{sort}([A, C])\\ \end{array} \]

Math

\[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

↓

\[\begin{array}{l} \mathbf{if}\;B \leq -2.7 \cdot 10^{+72}:\\ \;\;\;\;\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right) - F \cdot \left(C \cdot \left(A \cdot 4\right)\right)\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (/
  (-
   (sqrt
    (*
     (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F))
     (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
  (- (pow B 2.0) (* (* 4.0 A) C))))

↓

(FPCore (A B C F)
 :precision binary64
 (if (<= B -2.7e+72)
   (* (/ 2.0 B) (sqrt (* A F)))
   (if (<= B 1.7e+41)
     (/
      (- (sqrt (* 2.0 (* (- (* F (* B B)) (* F (* C (* A 4.0)))) (+ A A)))))
      (fma B B (* C (* A -4.0))))
     (/ (- (sqrt (* 2.0 (* F (- A (hypot B A)))))) B))))

C

double code(double A, double B, double C, double F) {
	return -sqrt(((2.0 * ((pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / (pow(B, 2.0) - ((4.0 * A) * C));
}

↓

double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= -2.7e+72) {
		tmp = (2.0 / B) * sqrt((A * F));
	} else if (B <= 1.7e+41) {
		tmp = -sqrt((2.0 * (((F * (B * B)) - (F * (C * (A * 4.0)))) * (A + A)))) / fma(B, B, (C * (A * -4.0)));
	} else {
		tmp = -sqrt((2.0 * (F * (A - hypot(B, A))))) / B;
	}
	return tmp;
}

Julia

function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)) * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)))
end

↓

function code(A, B, C, F)
	tmp = 0.0
	if (B <= -2.7e+72)
		tmp = Float64(Float64(2.0 / B) * sqrt(Float64(A * F)));
	elseif (B <= 1.7e+41)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(Float64(F * Float64(B * B)) - Float64(F * Float64(C * Float64(A * 4.0)))) * Float64(A + A))))) / fma(B, B, Float64(C * Float64(A * -4.0))));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B, A)))))) / B);
	end
	return tmp
end

Wolfram

code[A_, B_, C_, F_] := N[((-N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[A_, B_, C_, F_] := If[LessEqual[B, -2.7e+72], N[(N[(2.0 / B), $MachinePrecision] * N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 1.7e+41], N[((-N[Sqrt[N[(2.0 * N[(N[(N[(F * N[(B * B), $MachinePrecision]), $MachinePrecision] - N[(F * N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < -2.7000000000000001e72

   1. Initial program 58.8

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 57.9

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   3. Taylor expanded in C around inf 61.9

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \color{blue}{A}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Taylor expanded in B around 0 62.6

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(2 \cdot \left(A \cdot \left(F \cdot {B}^{2}\right)\right) + -8 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   5. Simplified 63.0

      \[\leadsto \frac{-\sqrt{2 \cdot \color{blue}{\left(F \cdot \left(-8 \cdot \left(C \cdot \left(A \cdot A\right)\right) + \left(2 \cdot A\right) \cdot \left(B \cdot B\right)\right)\right)}}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   6. Taylor expanded in B around -inf 56.8

      \[\leadsto \color{blue}{\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{2}\right)}^{2}}{B}} \]

   7. Simplified 56.7

      \[\leadsto \color{blue}{\frac{2}{B} \cdot \sqrt{A \cdot F}} \]

   if -2.7000000000000001e72 < B < 1.69999999999999999e41

   1. Initial program 49.1

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 42.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   3. Taylor expanded in C around inf 37.8

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \color{blue}{A}\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   4. Applied egg-rr 37.8

      \[\leadsto \frac{-\sqrt{2 \cdot \left(\color{blue}{\left(F \cdot \left(B \cdot B\right) + F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)} \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \]

   if 1.69999999999999999e41 < B

   1. Initial program 57.5

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   2. Simplified 56.0

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   3. Taylor expanded in C around 0 53.5

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

   4. Applied egg-rr 32.4

      \[\leadsto -1 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot F\right)}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq -2.7 \cdot 10^{+72}:\\ \;\;\;\;\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right) - F \cdot \left(C \cdot \left(A \cdot 4\right)\right)\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \]

Alternatives

Alternative 1
Error	40.1
Cost	14728

\[\begin{array}{l} \mathbf{if}\;B \leq -1 \cdot 10^{+110}:\\ \;\;\;\;\frac{2}{B} \cdot \sqrt{A \cdot F}\\ \mathbf{elif}\;B \leq 1.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 2
Error	41.3
Cost	14472

\[\begin{array}{l} \mathbf{if}\;B \leq -1.35 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\frac{B}{\sqrt{A \cdot F}}}\\ \mathbf{elif}\;B \leq 3.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(2 \cdot \left(\left(A \cdot F\right) \cdot \mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 3
Error	43.5
Cost	14344

\[\begin{array}{l} \mathbf{if}\;B \leq -8 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{B}{\sqrt{A \cdot F}}}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-{\left(\left(A \cdot -16\right) \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}^{0.5}}{C \cdot \left(A \cdot -4\right) + {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 4
Error	43.6
Cost	14280

\[\begin{array}{l} \mathbf{if}\;B \leq -8 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{B}{\sqrt{A \cdot F}}}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(A \cdot \left(A \cdot \left(F \cdot C\right)\right)\right)}}{C \cdot \left(A \cdot -4\right) + {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 5
Error	45.5
Cost	13768

\[\begin{array}{l} \mathbf{if}\;B \leq -8 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{B}{\sqrt{A \cdot F}}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}\\ \end{array} \]

Alternative 6
Error	46.7
Cost	13704

\[\begin{array}{l} \mathbf{if}\;B \leq -8 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{B}{\sqrt{A \cdot F}}}\\ \mathbf{elif}\;B \leq 3.35 \cdot 10^{-15}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{B \cdot B - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7
Error	52.4
Cost	7556

\[\begin{array}{l} \mathbf{if}\;A \leq -6.567434420042127 \cdot 10^{+148}:\\ \;\;\;\;\frac{2}{\frac{B}{\sqrt{A \cdot F}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{-16 \cdot \left(\left(F \cdot C\right) \cdot \left(A \cdot A\right)\right)}}{A \cdot \left(C \cdot -4\right)}\\ \end{array} \]

Alternative 8
Error	58.4
Cost	6980

\[\begin{array}{l} t_0 := \sqrt{A \cdot F}\\ \mathbf{if}\;B \leq -8.6 \cdot 10^{-281}:\\ \;\;\;\;\frac{2}{B} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \]

Alternative 9
Error	58.4
Cost	6980

\[\begin{array}{l} t_0 := \sqrt{A \cdot F}\\ \mathbf{if}\;B \leq -8.6 \cdot 10^{-281}:\\ \;\;\;\;\frac{2}{\frac{B}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{t_0}{B}\\ \end{array} \]

Alternative 10
Error	60.7
Cost	6848

\[\frac{-2}{\frac{B}{\sqrt{A \cdot F}}} \]

Alternative 11
Error	60.7
Cost	6848

\[-2 \cdot \frac{\sqrt{A \cdot F}}{B} \]

Alternative 12
Error	61.6
Cost	6656

\[-\sqrt{\frac{F}{A}} \]

Alternative 13
Error	63.1
Cost	6592

\[\sqrt{\frac{F}{A}} \]

Reproduce

herbie shell --seed 2022228 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))