ABCF->ab-angle b

Percentage Accurate: 18.5% → 42.3%

Time: 25.9s

Alternatives: 19

Speedup: 5.5×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * 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]) / t$95$0), $MachinePrecision]]

Initial Program: 18.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * 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]) / t$95$0), $MachinePrecision]]

Alternative 1: 42.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{{\left(F \cdot \left(A \cdot C\right)\right)}^{0.5} \cdot \sqrt{C \cdot -16}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B\_m, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* 4.0 (* A C)) (* B_m B_m)))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0)))))
   (if (<= t_2 0.0)
     (/
      (*
       (sqrt (* 2.0 (+ (* B_m B_m) (* A (* C -4.0)))))
       (sqrt (* F (+ C (- A (hypot B_m (- A C)))))))
      t_0)
     (if (<= t_2 INFINITY)
       (/ (* (pow (* F (* A C)) 0.5) (sqrt (* C -16.0))) t_0)
       (/
        -1.0
        (* B_m (pow (/ (* 2.0 F) (/ -1.0 (- (hypot B_m C) C))) -0.5)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * (A * C)) - (B_m * B_m);
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (sqrt((2.0 * ((B_m * B_m) + (A * (C * -4.0))))) * sqrt((F * (C + (A - hypot(B_m, (A - C))))))) / t_0;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (pow((F * (A * C)), 0.5) * sqrt((C * -16.0))) / t_0;
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * (A * C)) - (B_m * B_m);
	double t_1 = (4.0 * A) * C;
	double t_2 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_1) * F)) * ((A + C) - Math.sqrt((Math.pow(B_m, 2.0) + Math.pow((A - C), 2.0)))))) / (t_1 - Math.pow(B_m, 2.0));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (Math.sqrt((2.0 * ((B_m * B_m) + (A * (C * -4.0))))) * Math.sqrt((F * (C + (A - Math.hypot(B_m, (A - C))))))) / t_0;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (Math.pow((F * (A * C)), 0.5) * Math.sqrt((C * -16.0))) / t_0;
	} else {
		tmp = -1.0 / (B_m * Math.pow(((2.0 * F) / (-1.0 / (Math.hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * (A * C)) - (B_m * B_m)
	t_1 = (4.0 * A) * C
	t_2 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_1) * F)) * ((A + C) - math.sqrt((math.pow(B_m, 2.0) + math.pow((A - C), 2.0)))))) / (t_1 - math.pow(B_m, 2.0))
	tmp = 0
	if t_2 <= 0.0:
		tmp = (math.sqrt((2.0 * ((B_m * B_m) + (A * (C * -4.0))))) * math.sqrt((F * (C + (A - math.hypot(B_m, (A - C))))))) / t_0
	elif t_2 <= math.inf:
		tmp = (math.pow((F * (A * C)), 0.5) * math.sqrt((C * -16.0))) / t_0
	else:
		tmp = -1.0 / (B_m * math.pow(((2.0 * F) / (-1.0 / (math.hypot(B_m, C) - C))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))))) * sqrt(Float64(F * Float64(C + Float64(A - hypot(B_m, Float64(A - C))))))) / t_0);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64((Float64(F * Float64(A * C)) ^ 0.5) * sqrt(Float64(C * -16.0))) / t_0);
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) / Float64(-1.0 / Float64(hypot(B_m, C) - C))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * (A * C)) - (B_m * B_m);
	t_1 = (4.0 * A) * C;
	t_2 = sqrt(((2.0 * (((B_m ^ 2.0) - t_1) * F)) * ((A + C) - sqrt(((B_m ^ 2.0) + ((A - C) ^ 2.0)))))) / (t_1 - (B_m ^ 2.0));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = (sqrt((2.0 * ((B_m * B_m) + (A * (C * -4.0))))) * sqrt((F * (C + (A - hypot(B_m, (A - C))))))) / t_0;
	elseif (t_2 <= Inf)
		tmp = (((F * (A * C)) ^ 0.5) * sqrt((C * -16.0))) / t_0;
	else
		tmp = -1.0 / (B_m * (((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(N[Sqrt[N[(2.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[(A - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Power[N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[N[(C * -16.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] / N[(-1.0 / N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0

   1. Initial program 33.4%

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

   3. Simplified 38.1%

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

   6. Applied egg-rr 52.3%

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

   if 0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 24.5%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 12.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 12.1%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(A \cdot C\right) \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot C\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(F \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 33.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 33.7%

      \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(\left(C \cdot A\right) \cdot \left(F \cdot C\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{-16 \cdot \left(\left(C \cdot A\right) \cdot \left(C \cdot F\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{-16 \cdot \left(\left(\left(C \cdot A\right) \cdot C\right) \cdot F\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{-16 \cdot \left(\left(C \cdot \left(C \cdot A\right)\right) \cdot F\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(C \cdot \left(C \cdot A\right)\right) \cdot F\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(F \cdot \left(C \cdot \left(C \cdot A\right)\right)\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(F \cdot C\right) \cdot \left(C \cdot A\right)\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(F \cdot C\right) \cdot \left(A \cdot C\right)\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(\left(F \cdot C\right) \cdot A\right) \cdot C\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(\left(A \cdot \left(F \cdot C\right)\right) \cdot C\right) \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(A \cdot \left(F \cdot C\right)\right) \cdot \left(C \cdot -16\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. sqrt-prod N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{A \cdot \left(F \cdot C\right)} \cdot \sqrt{C \cdot -16}\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. pow1/2 N/A

           \[\leadsto \mathsf{/.f64}\left(\left({\left(A \cdot \left(F \cdot C\right)\right)}^{\frac{1}{2}} \cdot \sqrt{C \cdot -16}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\left(A \cdot \left(F \cdot C\right)\right)}^{\frac{1}{2}}\right), \left(\sqrt{C \cdot -16}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right)}, \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 59.9%

       \[\leadsto \frac{\color{blue}{{\left(F \cdot \left(C \cdot A\right)\right)}^{0.5} \cdot \sqrt{C \cdot -16}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 17.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 17.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 17.6%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 17.7%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(F \cdot \left(2 \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       3. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}\right), \frac{-1}{2}\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{1}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       5. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\frac{F \cdot 2}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(F \cdot 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{\frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       9. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{C - \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \mathsf{/.f64}\left(1, \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   11. Applied egg-rr 17.7%

       \[\leadsto \frac{-1}{B \cdot {\color{blue}{\left(\frac{F \cdot 2}{\frac{1}{C - \mathsf{hypot}\left(B, C\right)}}\right)}}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(B \cdot B + A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{{\left(F \cdot \left(A \cdot C\right)\right)}^{0.5} \cdot \sqrt{C \cdot -16}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 40.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\\ t_1 := C + \left(A - \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ \mathbf{if}\;B\_m \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(t\_0 \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 2.15 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{t\_0} \cdot \frac{\sqrt{2 \cdot \left(F \cdot t\_1\right)}}{A \cdot \left(4 \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\frac{1}{F}}{C - \mathsf{hypot}\left(C, B\_m\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (* B_m B_m) (* A (* C -4.0))))
        (t_1 (+ C (- A (hypot B_m (- A C))))))
   (if (<= B_m 4e-54)
     (/ (sqrt (* t_1 (* t_0 (* 2.0 F)))) (- (* 4.0 (* A C)) (* B_m B_m)))
     (if (<= B_m 2.15e+137)
       (*
        (sqrt t_0)
        (/ (sqrt (* 2.0 (* F t_1))) (- (* A (* 4.0 C)) (* B_m B_m))))
       (/
        -1.0
        (* B_m (* (sqrt 0.5) (sqrt (/ (/ 1.0 F) (- C (hypot C B_m)))))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) + (A * (C * -4.0));
	double t_1 = C + (A - hypot(B_m, (A - C)));
	double tmp;
	if (B_m <= 4e-54) {
		tmp = sqrt((t_1 * (t_0 * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2.15e+137) {
		tmp = sqrt(t_0) * (sqrt((2.0 * (F * t_1))) / ((A * (4.0 * C)) - (B_m * B_m)));
	} else {
		tmp = -1.0 / (B_m * (sqrt(0.5) * sqrt(((1.0 / F) / (C - hypot(C, B_m))))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (B_m * B_m) + (A * (C * -4.0));
	double t_1 = C + (A - Math.hypot(B_m, (A - C)));
	double tmp;
	if (B_m <= 4e-54) {
		tmp = Math.sqrt((t_1 * (t_0 * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 2.15e+137) {
		tmp = Math.sqrt(t_0) * (Math.sqrt((2.0 * (F * t_1))) / ((A * (4.0 * C)) - (B_m * B_m)));
	} else {
		tmp = -1.0 / (B_m * (Math.sqrt(0.5) * Math.sqrt(((1.0 / F) / (C - Math.hypot(C, B_m))))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (B_m * B_m) + (A * (C * -4.0))
	t_1 = C + (A - math.hypot(B_m, (A - C)))
	tmp = 0
	if B_m <= 4e-54:
		tmp = math.sqrt((t_1 * (t_0 * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 2.15e+137:
		tmp = math.sqrt(t_0) * (math.sqrt((2.0 * (F * t_1))) / ((A * (4.0 * C)) - (B_m * B_m)))
	else:
		tmp = -1.0 / (B_m * (math.sqrt(0.5) * math.sqrt(((1.0 / F) / (C - math.hypot(C, B_m))))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0)))
	t_1 = Float64(C + Float64(A - hypot(B_m, Float64(A - C))))
	tmp = 0.0
	if (B_m <= 4e-54)
		tmp = Float64(sqrt(Float64(t_1 * Float64(t_0 * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 2.15e+137)
		tmp = Float64(sqrt(t_0) * Float64(sqrt(Float64(2.0 * Float64(F * t_1))) / Float64(Float64(A * Float64(4.0 * C)) - Float64(B_m * B_m))));
	else
		tmp = Float64(-1.0 / Float64(B_m * Float64(sqrt(0.5) * sqrt(Float64(Float64(1.0 / F) / Float64(C - hypot(C, B_m)))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m * B_m) + (A * (C * -4.0));
	t_1 = C + (A - hypot(B_m, (A - C)));
	tmp = 0.0;
	if (B_m <= 4e-54)
		tmp = sqrt((t_1 * (t_0 * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 2.15e+137)
		tmp = sqrt(t_0) * (sqrt((2.0 * (F * t_1))) / ((A * (4.0 * C)) - (B_m * B_m)));
	else
		tmp = -1.0 / (B_m * (sqrt(0.5) * sqrt(((1.0 / F) / (C - hypot(C, B_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(C + N[(A - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4e-54], N[(N[Sqrt[N[(t$95$1 * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.15e+137], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.0000000000000001e-54

   1. Initial program 17.9%

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

   3. Simplified 24.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + \left(A \cdot C\right) \cdot -4\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right), \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 24.7%

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

   if 4.0000000000000001e-54 < B < 2.14999999999999982e137

   1. Initial program 41.6%

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

   3. Simplified 45.0%

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

   6. Applied egg-rr 64.4%

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

   if 2.14999999999999982e137 < B

   1. Initial program 3.6%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 58.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 58.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 58.7%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in F around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \color{blue}{\left(\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \sqrt{\frac{1}{2}}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \left(\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)}\right)\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\color{blue}{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}}\right)\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right)\right)\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{F}}{C - \sqrt{{B}^{2} + {C}^{2}}}\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{F}\right), \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right)\right)\right) \]

       12. hypot-define N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. hypot-lowering-hypot.f64 58.1%

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 58.1%

       \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{\frac{\frac{1}{F}}{C - \mathsf{hypot}\left(C, B\right)}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{B \cdot B + A \cdot \left(C \cdot -4\right)} \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{A \cdot \left(4 \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\frac{1}{F}}{C - \mathsf{hypot}\left(C, B\right)}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 39.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(B\_m, A - C\right)\\ \mathbf{if}\;B\_m \leq 3.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - t\_0\right)\right) \cdot \left(\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;B\_m \leq 1.1 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - t\_0\right)}{B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\frac{1}{F}}{C - \mathsf{hypot}\left(C, B\_m\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot B_m (- A C))))
   (if (<= B_m 3.4e-52)
     (/
      (sqrt (* (+ C (- A t_0)) (* (+ (* B_m B_m) (* A (* C -4.0))) (* 2.0 F))))
      (- (* 4.0 (* A C)) (* B_m B_m)))
     (if (<= B_m 1.1e+137)
       (*
        (sqrt (/ (* F (- (+ A C) t_0)) (+ (* B_m B_m) (* -4.0 (* A C)))))
        (- 0.0 (sqrt 2.0)))
       (/
        -1.0
        (* B_m (* (sqrt 0.5) (sqrt (/ (/ 1.0 F) (- C (hypot C B_m)))))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 3.4e-52) {
		tmp = sqrt(((C + (A - t_0)) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 1.1e+137) {
		tmp = sqrt(((F * ((A + C) - t_0)) / ((B_m * B_m) + (-4.0 * (A * C))))) * (0.0 - sqrt(2.0));
	} else {
		tmp = -1.0 / (B_m * (sqrt(0.5) * sqrt(((1.0 / F) / (C - hypot(C, B_m))))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.hypot(B_m, (A - C));
	double tmp;
	if (B_m <= 3.4e-52) {
		tmp = Math.sqrt(((C + (A - t_0)) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else if (B_m <= 1.1e+137) {
		tmp = Math.sqrt(((F * ((A + C) - t_0)) / ((B_m * B_m) + (-4.0 * (A * C))))) * (0.0 - Math.sqrt(2.0));
	} else {
		tmp = -1.0 / (B_m * (Math.sqrt(0.5) * Math.sqrt(((1.0 / F) / (C - Math.hypot(C, B_m))))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.hypot(B_m, (A - C))
	tmp = 0
	if B_m <= 3.4e-52:
		tmp = math.sqrt(((C + (A - t_0)) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m))
	elif B_m <= 1.1e+137:
		tmp = math.sqrt(((F * ((A + C) - t_0)) / ((B_m * B_m) + (-4.0 * (A * C))))) * (0.0 - math.sqrt(2.0))
	else:
		tmp = -1.0 / (B_m * (math.sqrt(0.5) * math.sqrt(((1.0 / F) / (C - math.hypot(C, B_m))))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(B_m, Float64(A - C))
	tmp = 0.0
	if (B_m <= 3.4e-52)
		tmp = Float64(sqrt(Float64(Float64(C + Float64(A - t_0)) * Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	elseif (B_m <= 1.1e+137)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - t_0)) / Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))))) * Float64(0.0 - sqrt(2.0)));
	else
		tmp = Float64(-1.0 / Float64(B_m * Float64(sqrt(0.5) * sqrt(Float64(Float64(1.0 / F) / Float64(C - hypot(C, B_m)))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = hypot(B_m, (A - C));
	tmp = 0.0;
	if (B_m <= 3.4e-52)
		tmp = sqrt(((C + (A - t_0)) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	elseif (B_m <= 1.1e+137)
		tmp = sqrt(((F * ((A + C) - t_0)) / ((B_m * B_m) + (-4.0 * (A * C))))) * (0.0 - sqrt(2.0));
	else
		tmp = -1.0 / (B_m * (sqrt(0.5) * sqrt(((1.0 / F) / (C - hypot(C, B_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[B$95$m, 3.4e-52], N[(N[Sqrt[N[(N[(C + N[(A - t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.1e+137], N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.40000000000000017e-52

   1. Initial program 17.8%

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

   3. Simplified 24.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + \left(A \cdot C\right) \cdot -4\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right), \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 24.6%

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

   if 3.40000000000000017e-52 < B < 1.10000000000000008e137

   1. Initial program 42.7%

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

   3. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}}\right), \color{blue}{\left(\sqrt{2}\right)}\right)\right) \]

   5. Simplified 65.4%

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

   if 1.10000000000000008e137 < B

   1. Initial program 3.6%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 58.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 58.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 58.7%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in F around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \color{blue}{\left(\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \sqrt{\frac{1}{2}}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \left(\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)}\right)\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\color{blue}{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}}\right)\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right)\right)\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{F}}{C - \sqrt{{B}^{2} + {C}^{2}}}\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{F}\right), \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right)\right)\right) \]

       12. hypot-define N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. hypot-lowering-hypot.f64 58.1%

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 58.1%

       \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{\frac{\frac{1}{F}}{C - \mathsf{hypot}\left(C, B\right)}}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B + -4 \cdot \left(A \cdot C\right)}} \cdot \left(0 - \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\frac{1}{F}}{C - \mathsf{hypot}\left(C, B\right)}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.65 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\frac{1}{F}}{C - \mathsf{hypot}\left(C, B\_m\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.65e+58)
   (/
    (sqrt
     (*
      (+ C (- A (hypot B_m (- A C))))
      (* (+ (* B_m B_m) (* A (* C -4.0))) (* 2.0 F))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (/ -1.0 (* B_m (* (sqrt 0.5) (sqrt (/ (/ 1.0 F) (- C (hypot C B_m)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.65e+58) {
		tmp = sqrt(((C + (A - hypot(B_m, (A - C)))) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * (sqrt(0.5) * sqrt(((1.0 / F) / (C - hypot(C, B_m))))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.65e+58) {
		tmp = Math.sqrt(((C + (A - Math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * (Math.sqrt(0.5) * Math.sqrt(((1.0 / F) / (C - Math.hypot(C, B_m))))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.65e+58:
		tmp = math.sqrt(((C + (A - math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m))
	else:
		tmp = -1.0 / (B_m * (math.sqrt(0.5) * math.sqrt(((1.0 / F) / (C - math.hypot(C, B_m))))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.65e+58)
		tmp = Float64(sqrt(Float64(Float64(C + Float64(A - hypot(B_m, Float64(A - C)))) * Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(-1.0 / Float64(B_m * Float64(sqrt(0.5) * sqrt(Float64(Float64(1.0 / F) / Float64(C - hypot(C, B_m)))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.65e+58)
		tmp = sqrt(((C + (A - hypot(B_m, (A - C)))) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = -1.0 / (B_m * (sqrt(0.5) * sqrt(((1.0 / F) / (C - hypot(C, B_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.65e+58], N[(N[Sqrt[N[(N[(C + N[(A - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.64999999999999991e58

   1. Initial program 20.2%

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

   3. Simplified 25.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + \left(A \cdot C\right) \cdot -4\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right), \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 26.6%

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

   if 1.64999999999999991e58 < B

   1. Initial program 16.5%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 61.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 61.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 61.2%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in F around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \color{blue}{\left(\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \cdot \sqrt{\frac{1}{2}}\right)}\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \left(\sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)}\right)\right)\right) \]

       3. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(\sqrt{\color{blue}{\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}}\right)\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right)\right)\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{F}}{C - \sqrt{{B}^{2} + {C}^{2}}}\right)\right)\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{F}\right), \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       9. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{{C}^{2} + {B}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + {B}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right)\right)\right)\right)\right)\right) \]

       12. hypot-define N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right)\right)\right)\right)\right)\right) \]

       13. hypot-lowering-hypot.f64 60.5%

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(1, F\right), \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right)\right)\right)\right)\right)\right) \]

   12. Simplified 60.5%

       \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(\sqrt{0.5} \cdot \sqrt{\frac{\frac{1}{F}}{C - \mathsf{hypot}\left(C, B\right)}}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 39.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B\_m, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6e+58)
   (/
    (sqrt
     (*
      (+ C (- A (hypot B_m (- A C))))
      (* (+ (* B_m B_m) (* A (* C -4.0))) (* 2.0 F))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (/ -1.0 (* B_m (pow (/ (* 2.0 F) (/ -1.0 (- (hypot B_m C) C))) -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6e+58) {
		tmp = sqrt(((C + (A - hypot(B_m, (A - C)))) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6e+58) {
		tmp = Math.sqrt(((C + (A - Math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * Math.pow(((2.0 * F) / (-1.0 / (Math.hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 6e+58:
		tmp = math.sqrt(((C + (A - math.hypot(B_m, (A - C)))) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m))
	else:
		tmp = -1.0 / (B_m * math.pow(((2.0 * F) / (-1.0 / (math.hypot(B_m, C) - C))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6e+58)
		tmp = Float64(sqrt(Float64(Float64(C + Float64(A - hypot(B_m, Float64(A - C)))) * Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(2.0 * F)))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) / Float64(-1.0 / Float64(hypot(B_m, C) - C))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 6e+58)
		tmp = sqrt(((C + (A - hypot(B_m, (A - C)))) * (((B_m * B_m) + (A * (C * -4.0))) * (2.0 * F)))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = -1.0 / (B_m * (((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6e+58], N[(N[Sqrt[N[(N[(C + N[(A - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] / N[(-1.0 / N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.0000000000000005e58

   1. Initial program 20.2%

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

   3. Simplified 25.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + \left(A \cdot C\right) \cdot -4\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(B \cdot B - 4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right) \cdot \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right), \left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 26.6%

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

   if 6.0000000000000005e58 < B

   1. Initial program 16.5%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 61.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 61.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 61.2%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(F \cdot \left(2 \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       3. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}\right), \frac{-1}{2}\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{1}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       5. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\frac{F \cdot 2}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(F \cdot 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{\frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       9. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{C - \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \mathsf{/.f64}\left(1, \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   11. Applied egg-rr 61.2%

       \[\leadsto \frac{-1}{B \cdot {\color{blue}{\left(\frac{F \cdot 2}{\frac{1}{C - \mathsf{hypot}\left(B, C\right)}}\right)}}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left(B\_m \cdot B\_m + A \cdot \left(C \cdot -4\right)\right) \cdot \left(C + \left(A - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B\_m, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.2e+46)
   (/
    (sqrt
     (*
      (* 2.0 F)
      (* (+ (* B_m B_m) (* A (* C -4.0))) (+ C (- A (hypot B_m (- A C)))))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (/ -1.0 (* B_m (pow (/ (* 2.0 F) (/ -1.0 (- (hypot B_m C) C))) -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.2e+46) {
		tmp = sqrt(((2.0 * F) * (((B_m * B_m) + (A * (C * -4.0))) * (C + (A - hypot(B_m, (A - C))))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.2e+46) {
		tmp = Math.sqrt(((2.0 * F) * (((B_m * B_m) + (A * (C * -4.0))) * (C + (A - Math.hypot(B_m, (A - C))))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * Math.pow(((2.0 * F) / (-1.0 / (Math.hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.2e+46:
		tmp = math.sqrt(((2.0 * F) * (((B_m * B_m) + (A * (C * -4.0))) * (C + (A - math.hypot(B_m, (A - C))))))) / ((4.0 * (A * C)) - (B_m * B_m))
	else:
		tmp = -1.0 / (B_m * math.pow(((2.0 * F) / (-1.0 / (math.hypot(B_m, C) - C))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.2e+46)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(Float64(Float64(B_m * B_m) + Float64(A * Float64(C * -4.0))) * Float64(C + Float64(A - hypot(B_m, Float64(A - C))))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) / Float64(-1.0 / Float64(hypot(B_m, C) - C))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.2e+46)
		tmp = sqrt(((2.0 * F) * (((B_m * B_m) + (A * (C * -4.0))) * (C + (A - hypot(B_m, (A - C))))))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = -1.0 / (B_m * (((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.2e+46], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C + N[(A - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] / N[(-1.0 / N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.20000000000000004e46

   1. Initial program 19.9%

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

   3. Simplified 25.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(\left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right) \cdot \left(2 \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right)\right) \cdot \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(B \cdot B + \left(A \cdot C\right) \cdot -4\right) \cdot \left(A - \left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)} - C\right)\right)\right), \left(2 \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 23.7%

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

   if 1.20000000000000004e46 < B

   1. Initial program 18.0%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 60.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 60.5%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 60.6%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(F \cdot \left(2 \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       3. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}\right), \frac{-1}{2}\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{1}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       5. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\frac{F \cdot 2}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(F \cdot 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{\frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       9. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{C - \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \mathsf{/.f64}\left(1, \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   11. Applied egg-rr 60.7%

       \[\leadsto \frac{-1}{B \cdot {\color{blue}{\left(\frac{F \cdot 2}{\frac{1}{C - \mathsf{hypot}\left(B, C\right)}}\right)}}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(\left(B \cdot B + A \cdot \left(C \cdot -4\right)\right) \cdot \left(C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B\_m, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.5e-46)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* -4.0 (* A C)))
      (* (* 2.0 F) (- A (- (hypot B_m (- A C)) C)))))
    (* 4.0 (* A C)))
   (/ -1.0 (* B_m (pow (/ (* 2.0 F) (/ -1.0 (- (hypot B_m C) C))) -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.5e-46) {
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (A - (hypot(B_m, (A - C)) - C))))) / (4.0 * (A * C));
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.5e-46) {
		tmp = Math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (A - (Math.hypot(B_m, (A - C)) - C))))) / (4.0 * (A * C));
	} else {
		tmp = -1.0 / (B_m * Math.pow(((2.0 * F) / (-1.0 / (Math.hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 8.5e-46:
		tmp = math.sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (A - (math.hypot(B_m, (A - C)) - C))))) / (4.0 * (A * C))
	else:
		tmp = -1.0 / (B_m * math.pow(((2.0 * F) / (-1.0 / (math.hypot(B_m, C) - C))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 8.5e-46)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C))) * Float64(Float64(2.0 * F) * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C))))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) / Float64(-1.0 / Float64(hypot(B_m, C) - C))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 8.5e-46)
		tmp = sqrt((((B_m * B_m) + (-4.0 * (A * C))) * ((2.0 * F) * (A - (hypot(B_m, (A - C)) - C))))) / (4.0 * (A * C));
	else
		tmp = -1.0 / (B_m * (((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8.5e-46], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] / N[(-1.0 / N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.5000000000000001e-46

   1. Initial program 17.8%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in A around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), -4\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), -4\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left(A \cdot C\right)}\right)\right) \]

      2. *-lowering-*.f64 17.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, C\right), -4\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, \color{blue}{C}\right)\right)\right) \]

   7. Simplified 17.3%

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

   if 8.5000000000000001e-46 < B

   1. Initial program 25.1%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(F \cdot \left(2 \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       3. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}\right), \frac{-1}{2}\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{1}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       5. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\frac{F \cdot 2}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(F \cdot 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{\frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       9. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{C - \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \mathsf{/.f64}\left(1, \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   11. Applied egg-rr 57.1%

       \[\leadsto \frac{-1}{B \cdot {\color{blue}{\left(\frac{F \cdot 2}{\frac{1}{C - \mathsf{hypot}\left(B, C\right)}}\right)}}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.5 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B\_m, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.8e-46)
   (/
    (sqrt (* (* -4.0 (* A C)) (* (* 2.0 F) (- A (- (hypot B_m (- A C)) C)))))
    (- (* 4.0 (* A C)) (* B_m B_m)))
   (/ -1.0 (* B_m (pow (/ (* 2.0 F) (/ -1.0 (- (hypot B_m C) C))) -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.8e-46) {
		tmp = sqrt(((-4.0 * (A * C)) * ((2.0 * F) * (A - (hypot(B_m, (A - C)) - C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.8e-46) {
		tmp = Math.sqrt(((-4.0 * (A * C)) * ((2.0 * F) * (A - (Math.hypot(B_m, (A - C)) - C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * Math.pow(((2.0 * F) / (-1.0 / (Math.hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.8e-46:
		tmp = math.sqrt(((-4.0 * (A * C)) * ((2.0 * F) * (A - (math.hypot(B_m, (A - C)) - C))))) / ((4.0 * (A * C)) - (B_m * B_m))
	else:
		tmp = -1.0 / (B_m * math.pow(((2.0 * F) / (-1.0 / (math.hypot(B_m, C) - C))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.8e-46)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(A * C)) * Float64(Float64(2.0 * F) * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C))))) / Float64(Float64(4.0 * Float64(A * C)) - Float64(B_m * B_m)));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) / Float64(-1.0 / Float64(hypot(B_m, C) - C))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.8e-46)
		tmp = sqrt(((-4.0 * (A * C)) * ((2.0 * F) * (A - (hypot(B_m, (A - C)) - C))))) / ((4.0 * (A * C)) - (B_m * B_m));
	else
		tmp = -1.0 / (B_m * (((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.8e-46], N[(N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] / N[(-1.0 / N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.80000000000000027e-46

   1. Initial program 17.8%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right)}, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 17.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(A, \mathsf{\_.f64}\left(\mathsf{hypot.f64}\left(B, \mathsf{\_.f64}\left(A, C\right)\right), C\right)\right), \mathsf{*.f64}\left(2, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 17.6%

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

   if 4.80000000000000027e-46 < B

   1. Initial program 25.1%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(F \cdot \left(2 \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       3. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}\right), \frac{-1}{2}\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{1}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       5. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\frac{F \cdot 2}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(F \cdot 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{\frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       9. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{C - \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \mathsf{/.f64}\left(1, \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   11. Applied egg-rr 57.1%

       \[\leadsto \frac{-1}{B \cdot {\color{blue}{\left(\frac{F \cdot 2}{\frac{1}{C - \mathsf{hypot}\left(B, C\right)}}\right)}}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 4 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 3.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B\_m, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))))
   (if (<= B_m 4e-266)
     (/ (sqrt (* -16.0 (* (* A C) (* C F)))) t_0)
     (if (<= B_m 3.15e-51)
       (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (- t_0 (* B_m B_m)))
       (/
        -1.0
        (* B_m (pow (/ (* 2.0 F) (/ -1.0 (- (hypot B_m C) C))) -0.5)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (B_m <= 4e-266) {
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	} else if (B_m <= 3.15e-51) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (B_m <= 4e-266) {
		tmp = Math.sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	} else if (B_m <= 3.15e-51) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * Math.pow(((2.0 * F) / (-1.0 / (Math.hypot(B_m, C) - C))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 4.0 * (A * C)
	tmp = 0
	if B_m <= 4e-266:
		tmp = math.sqrt((-16.0 * ((A * C) * (C * F)))) / t_0
	elif B_m <= 3.15e-51:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m))
	else:
		tmp = -1.0 / (B_m * math.pow(((2.0 * F) / (-1.0 / (math.hypot(B_m, C) - C))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	tmp = 0.0
	if (B_m <= 4e-266)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(C * F)))) / t_0);
	elseif (B_m <= 3.15e-51)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) / Float64(-1.0 / Float64(hypot(B_m, C) - C))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 4.0 * (A * C);
	tmp = 0.0;
	if (B_m <= 4e-266)
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	elseif (B_m <= 3.15e-51)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	else
		tmp = -1.0 / (B_m * (((2.0 * F) / (-1.0 / (hypot(B_m, C) - C))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4e-266], N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 3.15e-51], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] / N[(-1.0 / N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.9999999999999999e-266

   1. Initial program 17.2%

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

   3. Simplified 24.3%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(A \cdot C\right) \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot C\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(F \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 17.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.6%

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

   10. Taylor expanded in A around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left(A \cdot C\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot \color{blue}{A}\right)\right)\right) \]

       3. *-lowering-*.f64 18.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, \color{blue}{A}\right)\right)\right) \]

   12. Simplified 18.1%

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

   if 3.9999999999999999e-266 < B < 3.1499999999999999e-51

   1. Initial program 20.6%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-8 \cdot A\right), \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(1 \cdot A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 18.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{*.f64}\left(1, A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A + 1 \cdot A\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   if 3.1499999999999999e-51 < B

   1. Initial program 25.1%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(F \cdot \left(2 \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       3. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}\right), \frac{-1}{2}\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(F \cdot 2\right) \cdot \frac{1}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       5. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\frac{F \cdot 2}{\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}}\right), \frac{-1}{2}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(F \cdot 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{C + \sqrt{C \cdot C + B \cdot B}}{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{\frac{C \cdot C - \sqrt{C \cdot C + B \cdot B} \cdot \sqrt{C \cdot C + B \cdot B}}{C + \sqrt{C \cdot C + B \cdot B}}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       9. flip-- N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \left(\frac{1}{C - \sqrt{C \cdot C + B \cdot B}}\right)\right), \frac{-1}{2}\right)\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(F, 2\right), \mathsf{/.f64}\left(1, \left(C - \sqrt{C \cdot C + B \cdot B}\right)\right)\right), \frac{-1}{2}\right)\right)\right) \]

   11. Applied egg-rr 57.1%

       \[\leadsto \frac{-1}{B \cdot {\color{blue}{\left(\frac{F \cdot 2}{\frac{1}{C - \mathsf{hypot}\left(B, C\right)}}\right)}}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.15 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\frac{2 \cdot F}{\frac{-1}{\mathsf{hypot}\left(B, C\right) - C}}\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 3 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))))
   (if (<= B_m 3e-266)
     (/ (sqrt (* -16.0 (* (* A C) (* C F)))) t_0)
     (if (<= B_m 1.2e-51)
       (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (- t_0 (* B_m B_m)))
       (/ (pow (* 2.0 (* F (- C (hypot B_m C)))) 0.5) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (B_m <= 3e-266) {
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	} else if (B_m <= 1.2e-51) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = pow((2.0 * (F * (C - hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (B_m <= 3e-266) {
		tmp = Math.sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	} else if (B_m <= 1.2e-51) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = Math.pow((2.0 * (F * (C - Math.hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 4.0 * (A * C)
	tmp = 0
	if B_m <= 3e-266:
		tmp = math.sqrt((-16.0 * ((A * C) * (C * F)))) / t_0
	elif B_m <= 1.2e-51:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m))
	else:
		tmp = math.pow((2.0 * (F * (C - math.hypot(B_m, C)))), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	tmp = 0.0
	if (B_m <= 3e-266)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(C * F)))) / t_0);
	elseif (B_m <= 1.2e-51)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64((Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C)))) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 4.0 * (A * C);
	tmp = 0.0;
	if (B_m <= 3e-266)
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	elseif (B_m <= 1.2e-51)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	else
		tmp = ((2.0 * (F * (C - hypot(B_m, C)))) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3e-266], N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.2e-51], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3e-266

   1. Initial program 17.2%

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

   3. Simplified 24.3%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(A \cdot C\right) \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot C\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(F \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 17.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.6%

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

   10. Taylor expanded in A around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left(A \cdot C\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot \color{blue}{A}\right)\right)\right) \]

       3. *-lowering-*.f64 18.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, \color{blue}{A}\right)\right)\right) \]

   12. Simplified 18.1%

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

   if 3e-266 < B < 1.2e-51

   1. Initial program 20.6%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-8 \cdot A\right), \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(1 \cdot A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 18.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{*.f64}\left(1, A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A + 1 \cdot A\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   if 1.2e-51 < B

   1. Initial program 25.1%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 3.1 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 2.06 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))))
   (if (<= B_m 3.1e-266)
     (/ (sqrt (* -16.0 (* (* A C) (* C F)))) t_0)
     (if (<= B_m 2.06e-50)
       (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (- t_0 (* B_m B_m)))
       (/ (sqrt (* F (* 2.0 (- C (hypot C B_m))))) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (B_m <= 3.1e-266) {
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	} else if (B_m <= 2.06e-50) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = sqrt((F * (2.0 * (C - hypot(C, B_m))))) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (B_m <= 3.1e-266) {
		tmp = Math.sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	} else if (B_m <= 2.06e-50) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = Math.sqrt((F * (2.0 * (C - Math.hypot(C, B_m))))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 4.0 * (A * C)
	tmp = 0
	if B_m <= 3.1e-266:
		tmp = math.sqrt((-16.0 * ((A * C) * (C * F)))) / t_0
	elif B_m <= 2.06e-50:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m))
	else:
		tmp = math.sqrt((F * (2.0 * (C - math.hypot(C, B_m))))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	tmp = 0.0
	if (B_m <= 3.1e-266)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(C * F)))) / t_0);
	elseif (B_m <= 2.06e-50)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * Float64(C - hypot(C, B_m))))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 4.0 * (A * C);
	tmp = 0.0;
	if (B_m <= 3.1e-266)
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	elseif (B_m <= 2.06e-50)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	else
		tmp = sqrt((F * (2.0 * (C - hypot(C, B_m))))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.1e-266], N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2.06e-50], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(2.0 * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.09999999999999995e-266

   1. Initial program 17.2%

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

   3. Simplified 24.3%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(A \cdot C\right) \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot C\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(F \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 17.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.6%

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

   10. Taylor expanded in A around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left(A \cdot C\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot \color{blue}{A}\right)\right)\right) \]

       3. *-lowering-*.f64 18.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, \color{blue}{A}\right)\right)\right) \]

   12. Simplified 18.1%

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

   if 3.09999999999999995e-266 < B < 2.06e-50

   1. Initial program 20.6%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-8 \cdot A\right), \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(1 \cdot A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 18.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{*.f64}\left(1, A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A + 1 \cdot A\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   if 2.06e-50 < B

   1. Initial program 25.1%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right) \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right), B\right) \]

      3. sqrt-prod N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)\right), B\right) \]

      4. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}\right)\right), B\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left({\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}\right)\right), B\right) \]

      6. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)\right), B\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right), B\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right) \cdot 2\right)\right)\right), B\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot \left(\left(C - \sqrt{B \cdot B + C \cdot C}\right) \cdot 2\right)\right)\right)\right), B\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(\left(C - \sqrt{B \cdot B + C \cdot C}\right) \cdot 2\right)\right)\right)\right), B\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\left(C - \sqrt{B \cdot B + C \cdot C}\right), 2\right)\right)\right)\right), B\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right), 2\right)\right)\right)\right), B\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right), 2\right)\right)\right)\right), B\right) \]

      14. hypot-define N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right), 2\right)\right)\right)\right), B\right) \]

      15. hypot-lowering-hypot.f64 57.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right)\right)\right), B\right) \]

   9. Applied egg-rr 57.2%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.06 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;B\_m \leq 2.85 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 4.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(F \cdot \left(2 \cdot \left(C - B\_m\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* 4.0 (* A C))))
   (if (<= B_m 2.85e-266)
     (/ (sqrt (* -16.0 (* (* A C) (* C F)))) t_0)
     (if (<= B_m 4.6e-49)
       (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (- t_0 (* B_m B_m)))
       (/ -1.0 (* B_m (pow (* F (* 2.0 (- C B_m))) -0.5)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (B_m <= 2.85e-266) {
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	} else if (B_m <= 4.6e-49) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * pow((F * (2.0 * (C - B_m))), -0.5));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 4.0d0 * (a * c)
    if (b_m <= 2.85d-266) then
        tmp = sqrt(((-16.0d0) * ((a * c) * (c * f)))) / t_0
    else if (b_m <= 4.6d-49) then
        tmp = sqrt(((a * (-8.0d0)) * (c * (f * (a + a))))) / (t_0 - (b_m * b_m))
    else
        tmp = (-1.0d0) / (b_m * ((f * (2.0d0 * (c - b_m))) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = 4.0 * (A * C);
	double tmp;
	if (B_m <= 2.85e-266) {
		tmp = Math.sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	} else if (B_m <= 4.6e-49) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	} else {
		tmp = -1.0 / (B_m * Math.pow((F * (2.0 * (C - B_m))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = 4.0 * (A * C)
	tmp = 0
	if B_m <= 2.85e-266:
		tmp = math.sqrt((-16.0 * ((A * C) * (C * F)))) / t_0
	elif B_m <= 4.6e-49:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m))
	else:
		tmp = -1.0 / (B_m * math.pow((F * (2.0 * (C - B_m))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(4.0 * Float64(A * C))
	tmp = 0.0
	if (B_m <= 2.85e-266)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(C * F)))) / t_0);
	elseif (B_m <= 4.6e-49)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(F * Float64(2.0 * Float64(C - B_m))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = 4.0 * (A * C);
	tmp = 0.0;
	if (B_m <= 2.85e-266)
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / t_0;
	elseif (B_m <= 4.6e-49)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (t_0 - (B_m * B_m));
	else
		tmp = -1.0 / (B_m * ((F * (2.0 * (C - B_m))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.85e-266], N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 4.6e-49], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(F * N[(2.0 * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.8500000000000001e-266

   1. Initial program 17.2%

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

   3. Simplified 24.3%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(A \cdot C\right) \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot C\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(F \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 17.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.6%

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

   10. Taylor expanded in A around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left(A \cdot C\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot \color{blue}{A}\right)\right)\right) \]

       3. *-lowering-*.f64 18.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, \color{blue}{A}\right)\right)\right) \]

   12. Simplified 18.1%

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

   if 2.8500000000000001e-266 < B < 4.5999999999999998e-49

   1. Initial program 20.6%

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

   3. Simplified 22.5%

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

   5. Taylor expanded in C around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(-8 \cdot A\right), \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(A - -1 \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \left(A + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \left(1 \cdot A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 18.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-8, A\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(F, \mathsf{+.f64}\left(A, \mathsf{*.f64}\left(1, A\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 18.5%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A + 1 \cdot A\right)\right)\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]

   if 4.5999999999999998e-49 < B

   1. Initial program 25.1%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 57.0%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 57.2%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\color{blue}{\left(C - B\right)}, 2\right)\right), \frac{-1}{2}\right)\right)\right) \]
   11. Step-by-step derivation

       1. --lowering--.f64 51.0%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, B\right), 2\right)\right), \frac{-1}{2}\right)\right)\right) \]

   12. Simplified 51.0%

       \[\leadsto \frac{-1}{B \cdot {\left(F \cdot \left(\color{blue}{\left(C - B\right)} \cdot 2\right)\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.85 \cdot 10^{-266}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 4.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(F \cdot \left(2 \cdot \left(C - B\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(F \cdot \left(2 \cdot \left(C - B\_m\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.6e-25)
   (/ (sqrt (* -16.0 (* (* A C) (* C F)))) (* 4.0 (* A C)))
   (/ -1.0 (* B_m (pow (* F (* 2.0 (- C B_m))) -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.6e-25) {
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / (4.0 * (A * C));
	} else {
		tmp = -1.0 / (B_m * pow((F * (2.0 * (C - B_m))), -0.5));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.6d-25) then
        tmp = sqrt(((-16.0d0) * ((a * c) * (c * f)))) / (4.0d0 * (a * c))
    else
        tmp = (-1.0d0) / (b_m * ((f * (2.0d0 * (c - b_m))) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.6e-25) {
		tmp = Math.sqrt((-16.0 * ((A * C) * (C * F)))) / (4.0 * (A * C));
	} else {
		tmp = -1.0 / (B_m * Math.pow((F * (2.0 * (C - B_m))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.6e-25:
		tmp = math.sqrt((-16.0 * ((A * C) * (C * F)))) / (4.0 * (A * C))
	else:
		tmp = -1.0 / (B_m * math.pow((F * (2.0 * (C - B_m))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.6e-25)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(Float64(A * C) * Float64(C * F)))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(F * Float64(2.0 * Float64(C - B_m))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.6e-25)
		tmp = sqrt((-16.0 * ((A * C) * (C * F)))) / (4.0 * (A * C));
	else
		tmp = -1.0 / (B_m * ((F * (2.0 * (C - B_m))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.6e-25], N[(N[Sqrt[N[(-16.0 * N[(N[(A * C), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(F * N[(2.0 * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.6000000000000001e-25

   1. Initial program 18.3%

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

   3. Simplified 24.3%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.9%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(\left(A \cdot C\right) \cdot C\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot C\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(C \cdot A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \left(F \cdot C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 17.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Applied egg-rr 17.9%

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

   10. Taylor expanded in A around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot C\right)\right)}\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \color{blue}{\left(A \cdot C\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot \color{blue}{A}\right)\right)\right) \]

       3. *-lowering-*.f64 18.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(C, A\right), \mathsf{*.f64}\left(F, C\right)\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, \color{blue}{A}\right)\right)\right) \]

   12. Simplified 18.6%

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

   if 1.6000000000000001e-25 < B

   1. Initial program 24.0%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 59.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 59.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\color{blue}{\left(C - B\right)}, 2\right)\right), \frac{-1}{2}\right)\right)\right) \]
   11. Step-by-step derivation

       1. --lowering--.f64 52.5%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, B\right), 2\right)\right), \frac{-1}{2}\right)\right)\right) \]

   12. Simplified 52.5%

       \[\leadsto \frac{-1}{B \cdot {\left(F \cdot \left(\color{blue}{\left(C - B\right)} \cdot 2\right)\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(\left(A \cdot C\right) \cdot \left(C \cdot F\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(F \cdot \left(2 \cdot \left(C - B\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{C} \cdot \sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(F \cdot \left(2 \cdot \left(C - B\_m\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.5e-25)
   (* (/ (/ 0.25 A) C) (sqrt (* (* C (* A C)) (* F -16.0))))
   (/ -1.0 (* B_m (pow (* F (* 2.0 (- C B_m))) -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.5e-25) {
		tmp = ((0.25 / A) / C) * sqrt(((C * (A * C)) * (F * -16.0)));
	} else {
		tmp = -1.0 / (B_m * pow((F * (2.0 * (C - B_m))), -0.5));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 3.5d-25) then
        tmp = ((0.25d0 / a) / c) * sqrt(((c * (a * c)) * (f * (-16.0d0))))
    else
        tmp = (-1.0d0) / (b_m * ((f * (2.0d0 * (c - b_m))) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.5e-25) {
		tmp = ((0.25 / A) / C) * Math.sqrt(((C * (A * C)) * (F * -16.0)));
	} else {
		tmp = -1.0 / (B_m * Math.pow((F * (2.0 * (C - B_m))), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.5e-25:
		tmp = ((0.25 / A) / C) * math.sqrt(((C * (A * C)) * (F * -16.0)))
	else:
		tmp = -1.0 / (B_m * math.pow((F * (2.0 * (C - B_m))), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.5e-25)
		tmp = Float64(Float64(Float64(0.25 / A) / C) * sqrt(Float64(Float64(C * Float64(A * C)) * Float64(F * -16.0))));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(F * Float64(2.0 * Float64(C - B_m))) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.5e-25)
		tmp = ((0.25 / A) / C) * sqrt(((C * (A * C)) * (F * -16.0)));
	else
		tmp = -1.0 / (B_m * ((F * (2.0 * (C - B_m))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.5e-25], N[(N[(N[(0.25 / A), $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(N[(C * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(F * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(F * N[(2.0 * N[(C - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.5000000000000002e-25

   1. Initial program 18.3%

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

   3. Simplified 24.3%

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

   5. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(A \cdot \left({C}^{2} \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \left(\left(A \cdot {C}^{2}\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\left(A \cdot {C}^{2}\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left({C}^{2}\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \left(C \cdot C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-lowering-*.f64 11.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(\mathsf{*.f64}\left(A, \mathsf{*.f64}\left(C, C\right)\right), F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, C\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 11.9%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}{4 \cdot \left(A \cdot C\right) - B \cdot B} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{4 \cdot \left(A \cdot C\right) - B \cdot B}{\sqrt{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}}} \]

      2. associate-/r/ N/A

         \[\leadsto \frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B} \cdot \color{blue}{\sqrt{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\left(A \cdot C\right) \cdot 4 - B \cdot B} \cdot \sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{A \cdot \left(C \cdot 4\right) - B \cdot B} \cdot \sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{A \cdot \left(C \cdot 4\right) - B \cdot B}\right), \color{blue}{\left(\sqrt{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}\right)}\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\left(A \cdot C\right) \cdot 4 - B \cdot B}\right), \left(\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4 \cdot \left(A \cdot C\right) - B \cdot B}\right), \left(\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(4 \cdot \left(A \cdot C\right) - B \cdot B\right)\right), \left(\sqrt{\color{blue}{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(4 \cdot \left(A \cdot C\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \color{blue}{\left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)}}\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(\left(4 \cdot A\right) \cdot C\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\left(C \cdot \left(4 \cdot A\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \left(4 \cdot A\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\color{blue}{\left(A \cdot \left(C \cdot C\right)\right)} \cdot F\right)}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \left(A \cdot 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \left(B \cdot B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\left(A \cdot \color{blue}{\left(C \cdot C\right)}\right) \cdot F\right)}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \left(\sqrt{-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot \color{blue}{F}\right)}\right)\right) \]

      16. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, 4\right)\right), \mathsf{*.f64}\left(B, B\right)\right)\right), \mathsf{sqrt.f64}\left(\left(-16 \cdot \left(\left(A \cdot \left(C \cdot C\right)\right) \cdot F\right)\right)\right)\right) \]

   9. Applied egg-rr 15.7%

      \[\leadsto \color{blue}{\frac{1}{C \cdot \left(A \cdot 4\right) - B \cdot B} \cdot \sqrt{\left(C \cdot \left(C \cdot A\right)\right) \cdot \left(F \cdot -16\right)}} \]

   10. Taylor expanded in C around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{\frac{1}{4}}{A \cdot C}\right)}, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{*.f64}\left(F, -16\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{1}{4}}{A}}{C}\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{*.f64}\left(F, -16\right)\right)}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\frac{1}{4} \cdot 1}{A}}{C}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{C}, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{*.f64}\left(F, -16\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{1}{4} \cdot \frac{1}{A}}{C}\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right)}, \mathsf{*.f64}\left(F, -16\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \frac{1}{A}\right), C\right), \mathsf{sqrt.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{*.f64}\left(F, -16\right)\right)}\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{4} \cdot 1}{A}\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right)}, \mathsf{*.f64}\left(F, -16\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{A}\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{C}, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{*.f64}\left(F, -16\right)\right)\right)\right) \]

       7. /-lowering-/.f64 16.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, A\right), C\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(C, \mathsf{*.f64}\left(C, A\right)\right)}, \mathsf{*.f64}\left(F, -16\right)\right)\right)\right) \]

   12. Simplified 16.3%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{A}}{C}} \cdot \sqrt{\left(C \cdot \left(C \cdot A\right)\right) \cdot \left(F \cdot -16\right)} \]

   if 3.5000000000000002e-25 < B

   1. Initial program 24.0%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 59.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 59.1%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\color{blue}{\left(C - B\right)}, 2\right)\right), \frac{-1}{2}\right)\right)\right) \]
   11. Step-by-step derivation

       1. --lowering--.f64 52.5%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, B\right), 2\right)\right), \frac{-1}{2}\right)\right)\right) \]

   12. Simplified 52.5%

       \[\leadsto \frac{-1}{B \cdot {\left(F \cdot \left(\color{blue}{\left(C - B\right)} \cdot 2\right)\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{0.25}{A}}{C} \cdot \sqrt{\left(C \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot -16\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(F \cdot \left(2 \cdot \left(C - B\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -9.9 \cdot 10^{+253}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(F \cdot \left(4 \cdot C\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\left(B\_m \cdot F\right) \cdot -2\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -9.9e+253)
   (/ -1.0 (* B_m (pow (* F (* 4.0 C)) -0.5)))
   (/ -1.0 (* B_m (pow (* (* B_m F) -2.0) -0.5)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -9.9e+253) {
		tmp = -1.0 / (B_m * pow((F * (4.0 * C)), -0.5));
	} else {
		tmp = -1.0 / (B_m * pow(((B_m * F) * -2.0), -0.5));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-9.9d+253)) then
        tmp = (-1.0d0) / (b_m * ((f * (4.0d0 * c)) ** (-0.5d0)))
    else
        tmp = (-1.0d0) / (b_m * (((b_m * f) * (-2.0d0)) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -9.9e+253) {
		tmp = -1.0 / (B_m * Math.pow((F * (4.0 * C)), -0.5));
	} else {
		tmp = -1.0 / (B_m * Math.pow(((B_m * F) * -2.0), -0.5));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -9.9e+253:
		tmp = -1.0 / (B_m * math.pow((F * (4.0 * C)), -0.5))
	else:
		tmp = -1.0 / (B_m * math.pow(((B_m * F) * -2.0), -0.5))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -9.9e+253)
		tmp = Float64(-1.0 / Float64(B_m * (Float64(F * Float64(4.0 * C)) ^ -0.5)));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(B_m * F) * -2.0) ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -9.9e+253)
		tmp = -1.0 / (B_m * ((F * (4.0 * C)) ^ -0.5));
	else
		tmp = -1.0 / (B_m * (((B_m * F) * -2.0) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -9.9e+253], N[(-1.0 / N[(B$95$m * N[Power[N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(B$95$m * F), $MachinePrecision] * -2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -9.8999999999999998e253

   1. Initial program 1.5%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 21.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 21.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 22.4%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in C around -inf

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(4 \cdot C\right)}\right), \frac{-1}{2}\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(C \cdot 4\right)\right), \frac{-1}{2}\right)\right)\right) \]

       2. *-lowering-*.f64 22.4%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, 4\right)\right), \frac{-1}{2}\right)\right)\right) \]

   12. Simplified 22.4%

       \[\leadsto \frac{-1}{B \cdot {\left(F \cdot \color{blue}{\left(C \cdot 4\right)}\right)}^{-0.5}} \]

   if -9.8999999999999998e253 < C

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 15.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 15.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 15.7%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}, \frac{-1}{2}\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\left(\left(B \cdot F\right) \cdot -2\right), \frac{-1}{2}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(B \cdot F\right), -2\right), \frac{-1}{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(F \cdot B\right), -2\right), \frac{-1}{2}\right)\right)\right) \]

       4. *-lowering-*.f64 14.6%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(F, B\right), -2\right), \frac{-1}{2}\right)\right)\right) \]

   12. Simplified 14.6%

       \[\leadsto \frac{-1}{B \cdot {\color{blue}{\left(\left(F \cdot B\right) \cdot -2\right)}}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9.9 \cdot 10^{+253}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(F \cdot \left(4 \cdot C\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\left(B \cdot F\right) \cdot -2\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -2.15 \cdot 10^{+250}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(F \cdot \left(4 \cdot C\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(B\_m \cdot F\right) \cdot -2\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -2.15e+250)
   (/ -1.0 (* B_m (pow (* F (* 4.0 C)) -0.5)))
   (/ (pow (* (* B_m F) -2.0) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -2.15e+250) {
		tmp = -1.0 / (B_m * pow((F * (4.0 * C)), -0.5));
	} else {
		tmp = pow(((B_m * F) * -2.0), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-2.15d+250)) then
        tmp = (-1.0d0) / (b_m * ((f * (4.0d0 * c)) ** (-0.5d0)))
    else
        tmp = (((b_m * f) * (-2.0d0)) ** 0.5d0) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -2.15e+250) {
		tmp = -1.0 / (B_m * Math.pow((F * (4.0 * C)), -0.5));
	} else {
		tmp = Math.pow(((B_m * F) * -2.0), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -2.15e+250:
		tmp = -1.0 / (B_m * math.pow((F * (4.0 * C)), -0.5))
	else:
		tmp = math.pow(((B_m * F) * -2.0), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -2.15e+250)
		tmp = Float64(-1.0 / Float64(B_m * (Float64(F * Float64(4.0 * C)) ^ -0.5)));
	else
		tmp = Float64((Float64(Float64(B_m * F) * -2.0) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -2.15e+250)
		tmp = -1.0 / (B_m * ((F * (4.0 * C)) ^ -0.5));
	else
		tmp = (((B_m * F) * -2.0) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -2.15e+250], N[(-1.0 / N[(B$95$m * N[Power[N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(B$95$m * F), $MachinePrecision] * -2.0), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -2.15e250

   1. Initial program 1.5%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 21.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 21.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B} \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)}{B} \]

      3. sqrt-prod N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)}\right)}{B} \]

      4. unpow1/2 N/A

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

      5. distribute-neg-frac N/A

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

      6. clear-num N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{B}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}\right)}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \left(B \cdot \color{blue}{\frac{1}{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}}\right)\right) \]

   9. Applied egg-rr 22.4%

      \[\leadsto \color{blue}{\frac{-1}{B \cdot {\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{-0.5}}} \]

   10. Taylor expanded in C around -inf

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(4 \cdot C\right)}\right), \frac{-1}{2}\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(C \cdot 4\right)\right), \frac{-1}{2}\right)\right)\right) \]

       2. *-lowering-*.f64 22.4%

          \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(B, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(C, 4\right)\right), \frac{-1}{2}\right)\right)\right) \]

   12. Simplified 22.4%

       \[\leadsto \frac{-1}{B \cdot {\left(F \cdot \color{blue}{\left(C \cdot 4\right)}\right)}^{-0.5}} \]

   if -2.15e250 < C

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 15.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 15.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \frac{1}{2}\right), B\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(F \cdot B\right)\right), \frac{1}{2}\right), B\right)\right) \]

      3. *-lowering-*.f64 14.6%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(F, B\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Simplified 14.6%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(F \cdot B\right)\right)}}^{0.5}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -2.15 \cdot 10^{+250}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(F \cdot \left(4 \cdot C\right)\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(B \cdot F\right) \cdot -2\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -1.24 \cdot 10^{+250}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(B\_m \cdot F\right) \cdot -2\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -1.24e+250)
   (* -2.0 (/ (sqrt (* C F)) B_m))
   (/ (pow (* (* B_m F) -2.0) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -1.24e+250) {
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	} else {
		tmp = pow(((B_m * F) * -2.0), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-1.24d+250)) then
        tmp = (-2.0d0) * (sqrt((c * f)) / b_m)
    else
        tmp = (((b_m * f) * (-2.0d0)) ** 0.5d0) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -1.24e+250) {
		tmp = -2.0 * (Math.sqrt((C * F)) / B_m);
	} else {
		tmp = Math.pow(((B_m * F) * -2.0), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -1.24e+250:
		tmp = -2.0 * (math.sqrt((C * F)) / B_m)
	else:
		tmp = math.pow(((B_m * F) * -2.0), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -1.24e+250)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * F)) / B_m));
	else
		tmp = Float64((Float64(Float64(B_m * F) * -2.0) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -1.24e+250)
		tmp = -2.0 * (sqrt((C * F)) / B_m);
	else
		tmp = (((B_m * F) * -2.0) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -1.24e+250], N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(B$95$m * F), $MachinePrecision] * -2.0), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -1.24000000000000006e250

   1. Initial program 1.5%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 21.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 21.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

   7. Applied egg-rr 21.7%

      \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]

   8. Taylor expanded in C around -inf

      \[\leadsto \color{blue}{\frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{C \cdot F}\right), \color{blue}{\left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)}\right) \]

      3. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}}{B}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot C\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}{B}\right)\right) \]

      8. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-1 \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}{B}\right)\right) \]

      9. rem-square-sqrt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-1 \cdot 2}{B}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-2}{B}\right)\right) \]

      11. /-lowering-/.f64 21.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \mathsf{/.f64}\left(-2, \color{blue}{B}\right)\right) \]

   10. Simplified 21.8%

       \[\leadsto \color{blue}{\sqrt{F \cdot C} \cdot \frac{-2}{B}} \]
   11. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \sqrt{F \cdot C} \cdot \frac{1}{\color{blue}{\frac{B}{-2}}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{\sqrt{F \cdot C} \cdot 1}{\color{blue}{\frac{B}{-2}}} \]

       3. div-inv N/A

          \[\leadsto \frac{\sqrt{F \cdot C} \cdot 1}{B \cdot \color{blue}{\frac{1}{-2}}} \]

       4. metadata-eval N/A

          \[\leadsto \frac{\sqrt{F \cdot C} \cdot 1}{B \cdot \frac{-1}{2}} \]

       5. times-frac N/A

          \[\leadsto \frac{\sqrt{F \cdot C}}{B} \cdot \color{blue}{\frac{1}{\frac{-1}{2}}} \]

       6. metadata-eval N/A

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

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{F \cdot C}}{B}\right), \color{blue}{-2}\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{F \cdot C}\right), B\right), -2\right) \]

       9. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot C\right)\right), B\right), -2\right) \]

       10. *-lowering-*.f64 21.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), B\right), -2\right) \]

   12. Applied egg-rr 21.8%

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

   if -1.24000000000000006e250 < C

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 15.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 15.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

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

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right), \frac{1}{2}\right), B\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \left(F \cdot B\right)\right), \frac{1}{2}\right), B\right)\right) \]

      3. *-lowering-*.f64 14.6%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(F, B\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Simplified 14.6%

       \[\leadsto -\frac{{\color{blue}{\left(-2 \cdot \left(F \cdot B\right)\right)}}^{0.5}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.24 \cdot 10^{+250}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\left(B \cdot F\right) \cdot -2\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 5.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ {\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (* (pow (* C F) 0.5) (/ -2.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return pow((C * F), 0.5) * (-2.0 / B_m);
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = ((c * f) ** 0.5d0) * ((-2.0d0) / b_m)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.pow((C * F), 0.5) * (-2.0 / B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.pow((C * F), 0.5) * (-2.0 / B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64((Float64(C * F) ^ 0.5) * Float64(-2.0 / B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = ((C * F) ^ 0.5) * (-2.0 / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.6%

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

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

   11. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

   12. hypot-lowering-hypot.f64 16.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

5. Simplified 16.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

7. Applied egg-rr 16.1%

   \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]

8. Taylor expanded in C around -inf

   \[\leadsto \color{blue}{\frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}} \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{C \cdot F}\right), \color{blue}{\left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)}\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}}{B}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot C\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}{B}\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-1 \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}{B}\right)\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-1 \cdot 2}{B}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-2}{B}\right)\right) \]

   11. /-lowering-/.f64 3.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \mathsf{/.f64}\left(-2, \color{blue}{B}\right)\right) \]

10. Simplified 3.3%

    \[\leadsto \color{blue}{\sqrt{F \cdot C} \cdot \frac{-2}{B}} \]
11. Step-by-step derivation

    1. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(F \cdot C\right)}^{\frac{1}{2}}\right), \mathsf{/.f64}\left(\color{blue}{-2}, B\right)\right) \]

    2. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(F \cdot C\right), \frac{1}{2}\right), \mathsf{/.f64}\left(\color{blue}{-2}, B\right)\right) \]

    3. *-lowering-*.f64 3.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, C\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-2, B\right)\right) \]

12. Applied egg-rr 3.4%

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

13. Final simplification 3.4%

    \[\leadsto {\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B} \]
14. Add Preprocessing

Alternative 19: 5.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{C \cdot F} \cdot \frac{-2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (* (sqrt (* C F)) (/ -2.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((C * F)) * (-2.0 / B_m);
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((c * f)) * ((-2.0d0) / b_m)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((C * F)) * (-2.0 / B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((C * F)) * (-2.0 / B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(C * F)) * Float64(-2.0 / B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((C * F)) * (-2.0 / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.6%

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

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

   11. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

   12. hypot-lowering-hypot.f64 16.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

5. Simplified 16.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. distribute-neg-frac N/A

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

   3. associate-*l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), \color{blue}{B}\right) \]

7. Applied egg-rr 16.1%

   \[\leadsto \color{blue}{\frac{\left(0 - \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}}{B}} \]

8. Taylor expanded in C around -inf

   \[\leadsto \color{blue}{\frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B} \cdot \sqrt{C \cdot F}} \]
9. Step-by-step derivation

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{C \cdot F}\right), \color{blue}{\left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)}\right) \]

   3. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}}{B}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot C\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}{B}\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-1 \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}{B}\right)\right) \]

   9. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-1 \cdot 2}{B}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \left(\frac{-2}{B}\right)\right) \]

   11. /-lowering-/.f64 3.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right), \mathsf{/.f64}\left(-2, \color{blue}{B}\right)\right) \]

10. Simplified 3.3%

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

11. Final simplification 3.3%

    \[\leadsto \sqrt{C \cdot F} \cdot \frac{-2}{B} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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))))