ABCF->ab-angle a

Average Error: 52.4 → 42.1

Time: 44.6s

Precision: binary64

Cost: 217108

Math

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

↓

\[\begin{array}{l} t_0 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ t_1 := {\left(A - C\right)}^{2}\\ t_2 := \sqrt{{B}^{2} + t_1}\\ t_3 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{t_1 + {B}^{2}}\right)}}{t_3}\\ t_5 := {B}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;t_4 \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \left(A + t_2\right)\right) \cdot \left(t_5 \cdot \left(2 \cdot F\right)\right)}}{t_5}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(A + \left(C + t_2\right)\right)}}{t_0}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{-C \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot B}\\ \end{array} \]

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
 (let* ((t_0 (- (pow B 2.0) (* 4.0 (* A C))))
        (t_1 (pow (- A C) 2.0))
        (t_2 (sqrt (+ (pow B 2.0) t_1)))
        (t_3 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt (* (* 2.0 (* t_3 F)) (+ (+ A C) (sqrt (+ t_1 (pow B 2.0)))))))
          t_3))
        (t_5 (- (pow B 2.0) (* C (* A 4.0)))))
   (if (<= t_4 (- INFINITY))
     (- (sqrt (- (/ F C))))
     (if (<= t_4 -5e-205)
       (/ (- (sqrt (* (+ C (+ A t_2)) (* t_5 (* 2.0 F))))) t_5)
       (if (<= t_4 0.0)
         (- (sqrt (- (/ F A))))
         (if (<= t_4 2e+176)
           (/ (- (sqrt (* (* 2.0 (* t_0 F)) (+ A (+ C t_2))))) t_0)
           (if (<= t_4 INFINITY)
             (/ (- (* C (sqrt (* -16.0 (* A F))))) t_3)
             (- (* (/ (sqrt 2.0) B) (sqrt (* F 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 t_0 = pow(B, 2.0) - (4.0 * (A * C));
	double t_1 = pow((A - C), 2.0);
	double t_2 = sqrt((pow(B, 2.0) + t_1));
	double t_3 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((t_1 + pow(B, 2.0)))))) / t_3;
	double t_5 = pow(B, 2.0) - (C * (A * 4.0));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -sqrt(-(F / C));
	} else if (t_4 <= -5e-205) {
		tmp = -sqrt(((C + (A + t_2)) * (t_5 * (2.0 * F)))) / t_5;
	} else if (t_4 <= 0.0) {
		tmp = -sqrt(-(F / A));
	} else if (t_4 <= 2e+176) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * (A + (C + t_2)))) / t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = -(C * sqrt((-16.0 * (A * F)))) / t_3;
	} else {
		tmp = -((sqrt(2.0) / B) * sqrt((F * B)));
	}
	return tmp;
}

Java

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

↓

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - (4.0 * (A * C));
	double t_1 = Math.pow((A - C), 2.0);
	double t_2 = Math.sqrt((Math.pow(B, 2.0) + t_1));
	double t_3 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = -Math.sqrt(((2.0 * (t_3 * F)) * ((A + C) + Math.sqrt((t_1 + Math.pow(B, 2.0)))))) / t_3;
	double t_5 = Math.pow(B, 2.0) - (C * (A * 4.0));
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = -Math.sqrt(-(F / C));
	} else if (t_4 <= -5e-205) {
		tmp = -Math.sqrt(((C + (A + t_2)) * (t_5 * (2.0 * F)))) / t_5;
	} else if (t_4 <= 0.0) {
		tmp = -Math.sqrt(-(F / A));
	} else if (t_4 <= 2e+176) {
		tmp = -Math.sqrt(((2.0 * (t_0 * F)) * (A + (C + t_2)))) / t_0;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = -(C * Math.sqrt((-16.0 * (A * F)))) / t_3;
	} else {
		tmp = -((Math.sqrt(2.0) / B) * Math.sqrt((F * B)));
	}
	return tmp;
}

Python

def code(A, B, C, F):
	return -math.sqrt(((2.0 * ((math.pow(B, 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / (math.pow(B, 2.0) - ((4.0 * A) * C))

↓

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - (4.0 * (A * C))
	t_1 = math.pow((A - C), 2.0)
	t_2 = math.sqrt((math.pow(B, 2.0) + t_1))
	t_3 = math.pow(B, 2.0) - ((4.0 * A) * C)
	t_4 = -math.sqrt(((2.0 * (t_3 * F)) * ((A + C) + math.sqrt((t_1 + math.pow(B, 2.0)))))) / t_3
	t_5 = math.pow(B, 2.0) - (C * (A * 4.0))
	tmp = 0
	if t_4 <= -math.inf:
		tmp = -math.sqrt(-(F / C))
	elif t_4 <= -5e-205:
		tmp = -math.sqrt(((C + (A + t_2)) * (t_5 * (2.0 * F)))) / t_5
	elif t_4 <= 0.0:
		tmp = -math.sqrt(-(F / A))
	elif t_4 <= 2e+176:
		tmp = -math.sqrt(((2.0 * (t_0 * F)) * (A + (C + t_2)))) / t_0
	elif t_4 <= math.inf:
		tmp = -(C * math.sqrt((-16.0 * (A * F)))) / t_3
	else:
		tmp = -((math.sqrt(2.0) / B) * math.sqrt((F * 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)
	t_0 = Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C)))
	t_1 = Float64(A - C) ^ 2.0
	t_2 = sqrt(Float64((B ^ 2.0) + t_1))
	t_3 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64(t_1 + (B ^ 2.0))))))) / t_3)
	t_5 = Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(-Float64(F / C))));
	elseif (t_4 <= -5e-205)
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + Float64(A + t_2)) * Float64(t_5 * Float64(2.0 * F))))) / t_5);
	elseif (t_4 <= 0.0)
		tmp = Float64(-sqrt(Float64(-Float64(F / A))));
	elseif (t_4 <= 2e+176)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(A + Float64(C + t_2))))) / t_0);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(-Float64(C * sqrt(Float64(-16.0 * Float64(A * F))))) / t_3);
	else
		tmp = Float64(-Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * B))));
	end
	return tmp
end

MATLAB

function tmp = code(A, B, C, F)
	tmp = -sqrt(((2.0 * (((B ^ 2.0) - ((4.0 * A) * C)) * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / ((B ^ 2.0) - ((4.0 * A) * C));
end

↓

function tmp_2 = code(A, B, C, F)
	t_0 = (B ^ 2.0) - (4.0 * (A * C));
	t_1 = (A - C) ^ 2.0;
	t_2 = sqrt(((B ^ 2.0) + t_1));
	t_3 = (B ^ 2.0) - ((4.0 * A) * C);
	t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((t_1 + (B ^ 2.0)))))) / t_3;
	t_5 = (B ^ 2.0) - (C * (A * 4.0));
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = -sqrt(-(F / C));
	elseif (t_4 <= -5e-205)
		tmp = -sqrt(((C + (A + t_2)) * (t_5 * (2.0 * F)))) / t_5;
	elseif (t_4 <= 0.0)
		tmp = -sqrt(-(F / A));
	elseif (t_4 <= 2e+176)
		tmp = -sqrt(((2.0 * (t_0 * F)) * (A + (C + t_2)))) / t_0;
	elseif (t_4 <= Inf)
		tmp = -(C * sqrt((-16.0 * (A * F)))) / t_3;
	else
		tmp = -((sqrt(2.0) / B) * sqrt((F * B)));
	end
	tmp_2 = 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_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(t$95$1 + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], (-N[Sqrt[(-N[(F / C), $MachinePrecision])], $MachinePrecision]), If[LessEqual[t$95$4, -5e-205], N[((-N[Sqrt[N[(N[(C + N[(A + t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$5 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision], If[LessEqual[t$95$4, 0.0], (-N[Sqrt[(-N[(F / A), $MachinePrecision])], $MachinePrecision]), If[LessEqual[t$95$4, 2e+176], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[((-N[(C * N[Sqrt[N[(-16.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$3), $MachinePrecision], (-N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in B around 0 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right)} \]

   4. Simplified 48.1

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{C}}} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right) \]
      rational_best-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}\right) \cdot -1} \]
      rational_best-simplify-12 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{C}}} \]
      exponential-simplify-19 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{C}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{C}} \]
      exponential-simplify-19 [=>] 48.1	\[ -\color{blue}{\sqrt{\frac{F}{C} \cdot -1}} \]
      rational_best-simplify-12 [=>] 48.1	\[ -\sqrt{\color{blue}{-\frac{F}{C}}} \]

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -5.00000000000000001e-205

   1. Initial program 1.4

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

   2. Simplified 1.4

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

      [Start] 1.4	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Applied egg-rr 1.4

      \[\leadsto \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} + 0} \]

   4. Simplified 1.4

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

      [Start] 1.4	\[ \frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)} + 0 \]
      rational_best-simplify-4 [=>] 1.4	\[ \color{blue}{\frac{-\sqrt{2 \cdot \left(\left(\left({B}^{2} - A \cdot \left(C \cdot 4\right)\right) \cdot F\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)\right)}}{{B}^{2} - A \cdot \left(C \cdot 4\right)}} \]

   if -5.00000000000000001e-205 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -0.0

   1. Initial program 60.9

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

   2. Simplified 58.7

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

      [Start] 60.9	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in C around inf 64.0

      \[\leadsto \color{blue}{-1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right)} \]

   4. Simplified 47.8

      \[\leadsto \color{blue}{-\sqrt{-\frac{F}{A}}} \]
      Proof

      [Start] 64.0	\[ -1 \cdot \left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \]
      rational_best-simplify-2 [=>] 64.0	\[ \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}\right) \cdot -1} \]
      rational_best-simplify-12 [=>] 64.0	\[ \color{blue}{-\left(\sqrt{2} \cdot \sqrt{-0.5}\right) \cdot \sqrt{\frac{F}{A}}} \]
      exponential-simplify-19 [=>] 64.0	\[ -\color{blue}{\sqrt{-0.5 \cdot 2}} \cdot \sqrt{\frac{F}{A}} \]
      metadata-eval [=>] 64.0	\[ -\sqrt{\color{blue}{-1}} \cdot \sqrt{\frac{F}{A}} \]
      exponential-simplify-19 [=>] 47.8	\[ -\color{blue}{\sqrt{\frac{F}{A} \cdot -1}} \]
      rational_best-simplify-12 [=>] 47.8	\[ -\sqrt{\color{blue}{-\frac{F}{A}}} \]

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 2e176

   1. Initial program 1.6

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

   2. Simplified 1.6

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

      [Start] 1.6	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if 2e176 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 60.0

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

   2. Taylor expanded in C around inf 64.0

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

   3. Simplified 39.0

      \[\leadsto \frac{-\color{blue}{C \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      Proof

      [Start] 64.0	\[ \frac{-\sqrt{A \cdot F} \cdot \left(\sqrt{2} \cdot \left(C \cdot \sqrt{-8}\right)\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best-simplify-44 [=>] 64.0	\[ \frac{-\sqrt{A \cdot F} \cdot \color{blue}{\left(C \cdot \left(\sqrt{2} \cdot \sqrt{-8}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      rational_best-simplify-44 [=>] 64.0	\[ \frac{-\color{blue}{C \cdot \left(\sqrt{A \cdot F} \cdot \left(\sqrt{2} \cdot \sqrt{-8}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      exponential-simplify-19 [=>] 64.0	\[ \frac{-C \cdot \left(\sqrt{A \cdot F} \cdot \color{blue}{\sqrt{-8 \cdot 2}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      exponential-simplify-19 [=>] 39.0	\[ \frac{-C \cdot \color{blue}{\sqrt{\left(-8 \cdot 2\right) \cdot \left(A \cdot F\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
      metadata-eval [=>] 39.0	\[ \frac{-C \cdot \sqrt{\color{blue}{-16} \cdot \left(A \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   3. Taylor expanded in A around 0 63.6

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

   4. Simplified 63.6

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

      [Start] 63.6	\[ -1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \]
      rational_best-simplify-2 [=>] 63.6	\[ \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}\right) \cdot -1} \]
      rational_best-simplify-12 [=>] 63.6	\[ \color{blue}{-\frac{\sqrt{2}}{B} \cdot \sqrt{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]
      rational_best-simplify-2 [=>] 63.6	\[ -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   5. Taylor expanded in C around 0 54.5

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot B}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 42.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\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} \leq -\infty:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;\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} \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \left(A + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right) \cdot \left(\left({B}^{2} - C \cdot \left(A \cdot 4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;\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} \leq 0:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;\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} \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;\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} \leq \infty:\\ \;\;\;\;\frac{-C \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot B}\\ \end{array} \]

Alternatives

Alternative 1
Error	42.1
Cost	217108

\[\begin{array}{l} t_0 := {\left(A - C\right)}^{2}\\ t_1 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(A + \left(C + \sqrt{{B}^{2} + t_0}\right)\right)}}{t_1}\\ t_3 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{t_0 + {B}^{2}}\right)}}{t_3}\\ \mathbf{if}\;t_4 \leq -\infty:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;t_4 \leq -5 \cdot 10^{-205}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \mathbf{elif}\;t_4 \leq 2 \cdot 10^{+176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{-C \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot B}\\ \end{array} \]

Alternative 2
Error	45.6
Cost	41104

\[\begin{array}{l} t_0 := \frac{-\sqrt{2 \cdot \left(\left({B}^{2} - C \cdot \left(A \cdot 4\right)\right) \cdot \left(F \cdot \left(C + \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} + A\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -4.8 \cdot 10^{-125}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -2.55 \cdot 10^{-254}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 7.6 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	46.3
Cost	14680

\[\begin{array}{l} t_0 := \sqrt{F \cdot B}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -1.4 \cdot 10^{-122}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -4.2 \cdot 10^{-265}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot t_0\\ \mathbf{elif}\;C \leq 7 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 3.9 \cdot 10^{-44}:\\ \;\;\;\;\frac{1}{\sqrt{2} \cdot B} \cdot \left(-2 \cdot t_0\right)\\ \mathbf{elif}\;C \leq 1.65 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 3.3 \cdot 10^{+233}:\\ \;\;\;\;\frac{-C \cdot \sqrt{-16 \cdot \left(A \cdot F\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	47.4
Cost	14032

\[\begin{array}{l} t_0 := \sqrt{F \cdot B}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -6.5 \cdot 10^{-123}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -6.8 \cdot 10^{-265}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot t_0\\ \mathbf{elif}\;C \leq 8 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 5.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{1}{\sqrt{2} \cdot B} \cdot \left(-2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	47.4
Cost	13840

\[\begin{array}{l} t_0 := -\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot B}\\ t_1 := -\sqrt{-\frac{F}{A}}\\ \mathbf{if}\;C \leq -1.8 \cdot 10^{-121}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{elif}\;C \leq -4.8 \cdot 10^{-264}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;C \leq 2.8 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;C \leq 7.8 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	47.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;C \leq -2.45 \cdot 10^{-253}:\\ \;\;\;\;-\sqrt{-\frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-\frac{F}{A}}\\ \end{array} \]

Alternative 7
Error	54.8
Cost	6720

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

Reproduce

herbie shell --seed 2023092 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))