ABCF->ab-angle b

Percentage Accurate: 19.7% → 50.3%

Time: 19.6s

Alternatives: 15

Speedup: 9.4×

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: 19.7% 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: 50.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(A + \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({2}^{0.25}\right)}^{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_1) F)))
        (t_3 (- t_1 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) t_3)
     (if (<= t_4 -5e-180)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (fma B_m (- B_m) (* A (* 4.0 C))))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_2 (+ A (fma (/ (* B_m B_m) C) -0.5 A)))) t_3)
         (/
          (* (pow (pow 2.0 0.25) 2.0) (sqrt (* F (- A (hypot B_m A)))))
          (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B_m, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) / t_3;
	} else if (t_4 <= -5e-180) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * (A + fma(((B_m * B_m) / C), -0.5, A)))) / t_3;
	} else {
		tmp = (pow(pow(2.0, 0.25), 2.0) * sqrt((F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) / t_3);
	elseif (t_4 <= -5e-180)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_2 * Float64(A + fma(Float64(Float64(B_m * B_m) / C), -0.5, A)))) / t_3);
	else
		tmp = Float64(Float64(((2.0 ^ 0.25) ^ 2.0) * sqrt(Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * 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] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -5e-180], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$2 * N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Power[N[Power[2.0, 0.25], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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))) < -inf.0

   1. Initial program 3.3%

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

   4. Applied rewrites 20.3%

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

   5. Taylor expanded in C around inf

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

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \color{blue}{\sqrt{F \cdot \left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\color{blue}{F \cdot \left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. mul-1-neg N/A

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

      5. lower-neg.f64 29.5

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

   7. Applied rewrites 29.5%

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

   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))) < -5.0000000000000001e-180

   1. Initial program 96.9%

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

   4. Applied rewrites 96.9%

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

   if -5.0000000000000001e-180 < (/.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 28.0%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 30.8

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 30.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   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 C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 2.1%

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

      1. lower-hypot.f64 17.1

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

   7. Applied rewrites 17.1%

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

      1. pow1/2 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 17.2

         \[\leadsto \frac{{\left({2}^{\color{blue}{0.25}}\right)}^{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B} \]

   9. Applied rewrites 17.2%

      \[\leadsto \frac{\color{blue}{{\left({2}^{0.25}\right)}^{2}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({2}^{0.25}\right)}^{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(A + \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_1) F)))
        (t_3 (- t_1 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/ (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) t_3)
     (if (<= t_4 -5e-180)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (fma B_m (- B_m) (* A (* 4.0 C))))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_2 (+ A (fma (/ (* B_m B_m) C) -0.5 A)))) t_3)
         (/ (* (sqrt (* F (- A (hypot B_m A)))) (sqrt 2.0)) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B_m, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) / t_3;
	} else if (t_4 <= -5e-180) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * (A + fma(((B_m * B_m) / C), -0.5, A)))) / t_3;
	} else {
		tmp = (sqrt((F * (A - hypot(B_m, A)))) * sqrt(2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) / t_3);
	elseif (t_4 <= -5e-180)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_2 * Float64(A + fma(Float64(Float64(B_m * B_m) / C), -0.5, A)))) / t_3);
	else
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * sqrt(2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * 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] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -5e-180], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$2 * N[(A + N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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))) < -inf.0

   1. Initial program 3.3%

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

   4. Applied rewrites 20.3%

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

   5. Taylor expanded in C around inf

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

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \color{blue}{\sqrt{F \cdot \left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\color{blue}{F \cdot \left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. mul-1-neg N/A

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

      5. lower-neg.f64 29.5

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

   7. Applied rewrites 29.5%

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

   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))) < -5.0000000000000001e-180

   1. Initial program 96.9%

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

   4. Applied rewrites 96.9%

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

   if -5.0000000000000001e-180 < (/.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 28.0%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 30.8

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 30.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   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 C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 2.1%

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

      1. lower-hypot.f64 17.1

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

   7. Applied rewrites 17.1%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_0} \cdot \sqrt{F \cdot \left(A + A\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B\_m \cdot B\_m, \frac{-0.5}{C}, A\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (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_2)))
   (if (<= t_3 (- INFINITY))
     (/ (* (sqrt (* 2.0 t_0)) (sqrt (* F (+ A A)))) t_2)
     (if (<= t_3 -5e-180)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (fma B_m (- B_m) (* A (* 4.0 C))))
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (* F t_0) (* 2.0 (+ A (fma (* B_m B_m) (/ -0.5 C) A)))))
          (- t_0))
         (/ (* (sqrt (* F (- A (hypot B_m A)))) (sqrt 2.0)) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = 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_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt((2.0 * t_0)) * sqrt((F * (A + A)))) / t_2;
	} else if (t_3 <= -5e-180) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + fma((B_m * B_m), (-0.5 / C), A))))) / -t_0;
	} else {
		tmp = (sqrt((F * (A - hypot(B_m, A)))) * sqrt(2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = 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)))))) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * sqrt(Float64(F * Float64(A + A)))) / t_2);
	elseif (t_3 <= -5e-180)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + fma(Float64(B_m * B_m), Float64(-0.5 / C), A))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * sqrt(2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -5e-180], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(-0.5 / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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))) < -inf.0

   1. Initial program 3.3%

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

   4. Applied rewrites 20.3%

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

   5. Taylor expanded in C around inf

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

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \color{blue}{\sqrt{F \cdot \left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{\color{blue}{F \cdot \left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \color{blue}{\left(A - -1 \cdot A\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. mul-1-neg N/A

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

      5. lower-neg.f64 29.5

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

   7. Applied rewrites 29.5%

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

   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))) < -5.0000000000000001e-180

   1. Initial program 96.9%

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

   4. Applied rewrites 96.9%

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

   if -5.0000000000000001e-180 < (/.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 28.0%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 30.8

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 30.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   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 C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 2.1%

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

      1. lower-hypot.f64 17.1

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

   7. Applied rewrites 17.1%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := A + \mathsf{fma}\left(B\_m \cdot B\_m, \frac{-0.5}{C}, A\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)}{\sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{t\_1 \cdot \left(2 \cdot F\right)}}}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot t\_1\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{B\_m \cdot \left(-F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (t_1 (+ A (fma (* B_m B_m) (/ -0.5 C) A)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0)))))
   (if (<= t_3 (- INFINITY))
     (/
      -1.0
      (/
       (fma A (* C -4.0) (* B_m B_m))
       (* (sqrt (fma B_m B_m (* A (* C -4.0)))) (sqrt (* t_1 (* 2.0 F))))))
     (if (<= t_3 -5e-180)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (fma B_m (- B_m) (* A (* 4.0 C))))
       (if (<= t_3 INFINITY)
         (/ (sqrt (* (* F t_0) (* 2.0 t_1))) (- t_0))
         (/ (* (sqrt 2.0) (sqrt (* B_m (- F)))) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	double t_1 = A + fma((B_m * B_m), (-0.5 / C), A);
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = -1.0 / (fma(A, (C * -4.0), (B_m * B_m)) / (sqrt(fma(B_m, B_m, (A * (C * -4.0)))) * sqrt((t_1 * (2.0 * F)))));
	} else if (t_3 <= -5e-180) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * t_1))) / -t_0;
	} else {
		tmp = (sqrt(2.0) * sqrt((B_m * -F))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(A + fma(Float64(B_m * B_m), Float64(-0.5 / C), A))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(fma(A, Float64(C * -4.0), Float64(B_m * B_m)) / Float64(sqrt(fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * sqrt(Float64(t_1 * Float64(2.0 * F))))));
	elseif (t_3 <= -5e-180)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * t_1))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(B_m * Float64(-F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(-0.5 / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $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$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(-1.0 / N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-180], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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))) < -inf.0

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 14.3

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 14.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 15.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 15.8%

      \[\leadsto \color{blue}{\frac{1}{-\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right) \cdot 2\right)}}}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\left(A \cdot \color{blue}{\left(C \cdot -4\right)} + B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \left(\left(B \cdot B\right) \cdot \frac{\frac{-1}{2}}{C} + A\right)\right)\right) \cdot 2\right)}}\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\left(A \cdot \left(C \cdot -4\right) + \color{blue}{B \cdot B}\right) \cdot \left(\left(F \cdot \left(A + \left(\left(B \cdot B\right) \cdot \frac{\frac{-1}{2}}{C} + A\right)\right)\right) \cdot 2\right)}}\right)} \]

       3. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\color{blue}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \left(\left(F \cdot \left(A + \left(\left(B \cdot B\right) \cdot \frac{\frac{-1}{2}}{C} + A\right)\right)\right) \cdot 2\right)}}\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \left(\color{blue}{\left(B \cdot B\right)} \cdot \frac{\frac{-1}{2}}{C} + A\right)\right)\right) \cdot 2\right)}}\right)} \]

       5. lift-/.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \left(\left(B \cdot B\right) \cdot \color{blue}{\frac{\frac{-1}{2}}{C}} + A\right)\right)\right) \cdot 2\right)}}\right)} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \color{blue}{\mathsf{fma}\left(B \cdot B, \frac{\frac{-1}{2}}{C}, A\right)}\right)\right) \cdot 2\right)}}\right)} \]

       7. lift-+.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \color{blue}{\left(A + \mathsf{fma}\left(B \cdot B, \frac{\frac{-1}{2}}{C}, A\right)\right)}\right) \cdot 2\right)}}\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\color{blue}{\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{\frac{-1}{2}}{C}, A\right)\right)\right)} \cdot 2\right)}}\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{\frac{-1}{2}}{C}, A\right)\right)\right) \cdot 2\right)}}}\right)} \]

       10. sqrt-prod N/A

           \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{\frac{-1}{2}}{C}, A\right)\right)\right) \cdot 2}}}\right)} \]

       11. pow1/2 N/A

           \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\color{blue}{{\left(\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{\frac{-1}{2}}{C}, A\right)\right)\right) \cdot 2}}\right)} \]

   11. Applied rewrites 23.4%

       \[\leadsto \frac{1}{-\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{\left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right) \cdot \left(2 \cdot F\right)}}}} \]

   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))) < -5.0000000000000001e-180

   1. Initial program 96.9%

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

   4. Applied rewrites 96.9%

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

   if -5.0000000000000001e-180 < (/.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 28.0%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 30.8

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 30.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   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 C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 2.1%

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

      1. lower-hypot.f64 17.1

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

   7. Applied rewrites 17.1%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 15.9

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

   10. Applied rewrites 15.9%

       \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{-B \cdot F}}}{-B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{\left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{B \cdot \left(-F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B\_m \cdot B\_m\right)}{F \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, 2 \cdot A\right)}}}{\sqrt{2}}}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B\_m \cdot B\_m, \frac{-0.5}{C}, A\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{B\_m \cdot \left(-F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* C (* A -4.0))))
        (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 (- INFINITY))
     (/
      -1.0
      (/
       (sqrt
        (/
         (fma -4.0 (* A C) (* B_m B_m))
         (* F (fma -0.5 (/ (* B_m B_m) C) (* 2.0 A)))))
       (sqrt 2.0)))
     (if (<= t_2 -5e-180)
       (/
        (sqrt
         (*
          (* t_0 (* 2.0 F))
          (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))
        (fma B_m (- B_m) (* A (* 4.0 C))))
       (if (<= t_2 INFINITY)
         (/
          (sqrt (* (* F t_0) (* 2.0 (+ A (fma (* B_m B_m) (/ -0.5 C) A)))))
          (- t_0))
         (/ (* (sqrt 2.0) (sqrt (* B_m (- F)))) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (C * (A * -4.0)));
	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 <= -((double) INFINITY)) {
		tmp = -1.0 / (sqrt((fma(-4.0, (A * C), (B_m * B_m)) / (F * fma(-0.5, ((B_m * B_m) / C), (2.0 * A))))) / sqrt(2.0));
	} else if (t_2 <= -5e-180) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + fma((B_m * B_m), (-0.5 / C), A))))) / -t_0;
	} else {
		tmp = (sqrt(2.0) * sqrt((B_m * -F))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	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 <= Float64(-Inf))
		tmp = Float64(-1.0 / Float64(sqrt(Float64(fma(-4.0, Float64(A * C), Float64(B_m * B_m)) / Float64(F * fma(-0.5, Float64(Float64(B_m * B_m) / C), Float64(2.0 * A))))) / sqrt(2.0)));
	elseif (t_2 <= -5e-180)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + fma(Float64(B_m * B_m), Float64(-0.5 / C), A))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(B_m * Float64(-F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $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, (-Infinity)], N[(-1.0 / N[(N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-180], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(-0.5 / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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))) < -inf.0

   1. Initial program 3.3%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 14.3

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 14.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 15.7%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 15.8%

      \[\leadsto \color{blue}{\frac{1}{-\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right) \cdot 2\right)}}}} \]

   10. Taylor expanded in F around 0

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

       1. mul-1-neg N/A

          \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + 2 \cdot A\right)}} \cdot \frac{1}{\sqrt{2}}\right)}} \]

       2. associate-*r/ N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\frac{\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + 2 \cdot A\right)}} \cdot 1}{\sqrt{2}}}\right)} \]

       3. *-rgt-identity N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\color{blue}{\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + 2 \cdot A\right)}}}}{\sqrt{2}}\right)} \]

       4. distribute-neg-frac2 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + 2 \cdot A\right)}}}{\mathsf{neg}\left(\sqrt{2}\right)}}} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{-4 \cdot \left(A \cdot C\right) + {B}^{2}}{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + 2 \cdot A\right)}}}{\mathsf{neg}\left(\sqrt{2}\right)}}} \]

   12. Applied rewrites 22.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A \cdot 2\right)}}}{-\sqrt{2}}}} \]

   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))) < -5.0000000000000001e-180

   1. Initial program 96.9%

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

   4. Applied rewrites 96.9%

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

   if -5.0000000000000001e-180 < (/.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 28.0%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 30.8

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 30.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   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 C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 2.1%

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

      1. lower-hypot.f64 17.1

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

   7. Applied rewrites 17.1%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 15.9

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

   10. Applied rewrites 15.9%

       \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{-B \cdot F}}}{-B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{-1}{\frac{\sqrt{\frac{\mathsf{fma}\left(-4, A \cdot C, B \cdot B\right)}{F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, 2 \cdot A\right)}}}{\sqrt{2}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{B \cdot \left(-F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+262}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{F \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, 2 \cdot A\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)}}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_3\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B\_m \cdot B\_m, \frac{-0.5}{C}, A\right)\right)\right)}}{-t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{B\_m \cdot \left(-F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0)))
        (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))))
        (t_3 (fma B_m B_m (* C (* A -4.0)))))
   (if (<= t_2 -5e+262)
     (*
      t_0
      (sqrt
       (/
        (* F (fma -0.5 (/ (* B_m B_m) C) (* 2.0 A)))
        (fma B_m B_m (* -4.0 (* A C))))))
     (if (<= t_2 -5e-180)
       (*
        t_0
        (sqrt
         (/
          (* F (- (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C))))))
          (fma (* A C) -4.0 (* B_m B_m)))))
       (if (<= t_2 INFINITY)
         (/
          (sqrt (* (* F t_3) (* 2.0 (+ A (fma (* B_m B_m) (/ -0.5 C) A)))))
          (- t_3))
         (/ (* (sqrt 2.0) (sqrt (* B_m (- F)))) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	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 t_3 = fma(B_m, B_m, (C * (A * -4.0)));
	double tmp;
	if (t_2 <= -5e+262) {
		tmp = t_0 * sqrt(((F * fma(-0.5, ((B_m * B_m) / C), (2.0 * A))) / fma(B_m, B_m, (-4.0 * (A * C)))));
	} else if (t_2 <= -5e-180) {
		tmp = t_0 * sqrt(((F * ((A + C) - sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / fma((A * C), -4.0, (B_m * B_m))));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_3) * (2.0 * (A + fma((B_m * B_m), (-0.5 / C), A))))) / -t_3;
	} else {
		tmp = (sqrt(2.0) * sqrt((B_m * -F))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	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)))
	t_3 = fma(B_m, B_m, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (t_2 <= -5e+262)
		tmp = Float64(t_0 * sqrt(Float64(Float64(F * fma(-0.5, Float64(Float64(B_m * B_m) / C), Float64(2.0 * A))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))));
	elseif (t_2 <= -5e-180)
		tmp = Float64(t_0 * sqrt(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / fma(Float64(A * C), -4.0, Float64(B_m * B_m)))));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_3) * Float64(2.0 * Float64(A + fma(Float64(B_m * B_m), Float64(-0.5 / C), A))))) / Float64(-t_3));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(B_m * Float64(-F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $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]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+262], N[(t$95$0 * N[Sqrt[N[(N[(F * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-180], N[(t$95$0 * N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$3), $MachinePrecision] * N[(2.0 * N[(A + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(-0.5 / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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))) < -5.00000000000000008e262

   1. Initial program 4.8%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 14.0

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 14.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + 2 \cdot A\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + 2 \cdot A\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{\frac{F \cdot \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + 2 \cdot A\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}}\right) \]

   8. Applied rewrites 21.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, 2 \cdot A\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \sqrt{2}} \]

   if -5.00000000000000008e262 < (/.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))) < -5.0000000000000001e-180

   1. Initial program 96.8%

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

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

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

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 87.9%

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

   if -5.0000000000000001e-180 < (/.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 28.0%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 30.8

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 30.8%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}} \]

   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 C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 2.1%

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

      1. lower-hypot.f64 17.1

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

   7. Applied rewrites 17.1%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 15.9

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

   10. Applied rewrites 15.9%

       \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{-B \cdot F}}}{-B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{+262}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, 2 \cdot A\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{B \cdot \left(-F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_0 \cdot \left(\left(4 \cdot A\right) \cdot F\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+90}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B\_m \cdot B\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B\_m, -F, A \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 4.8e-105)
     (/ -1.0 (/ t_0 (sqrt (* t_0 (* (* 4.0 A) F)))))
     (if (<= (pow B_m 2.0) 1e+90)
       (*
        (- (sqrt 2.0))
        (sqrt
         (/
          (* F (- (+ A C) (sqrt (fma B_m B_m (* (- A C) (- A C))))))
          (fma (* A C) -4.0 (* B_m B_m)))))
       (/ (* (sqrt 2.0) (sqrt (fma B_m (- F) (* A F)))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 4.8e-105) {
		tmp = -1.0 / (t_0 / sqrt((t_0 * ((4.0 * A) * F))));
	} else if (pow(B_m, 2.0) <= 1e+90) {
		tmp = -sqrt(2.0) * sqrt(((F * ((A + C) - sqrt(fma(B_m, B_m, ((A - C) * (A - C)))))) / fma((A * C), -4.0, (B_m * B_m))));
	} else {
		tmp = (sqrt(2.0) * sqrt(fma(B_m, -F, (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4.8e-105)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(t_0 * Float64(Float64(4.0 * A) * F)))));
	elseif ((B_m ^ 2.0) <= 1e+90)
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C)))))) / fma(Float64(A * C), -4.0, Float64(B_m * B_m)))));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(fma(B_m, Float64(-F), Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4.8e-105], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(t$95$0 * N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+90], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F) + N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.8000000000000003e-105

   1. Initial program 21.8%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 25.3

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 25.3%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 24.3%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 23.5%

      \[\leadsto \color{blue}{\frac{1}{-\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right) \cdot 2\right)}}}} \]

   10. Taylor expanded in A around inf

       \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(\left(4 \cdot A\right) \cdot F\right)}}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(\left(4 \cdot A\right) \cdot F\right)}}}\right)} \]

       3. lower-*.f64 22.7

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

   12. Applied rewrites 22.7%

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

   if 4.8000000000000003e-105 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999966e89

   1. Initial program 44.4%

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

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

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

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 45.3%

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

   if 9.99999999999999966e89 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.1%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 8.5%

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

      1. lower-hypot.f64 24.2

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

   7. Applied rewrites 24.2%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(B, -1 \cdot F, A \cdot F\right)}}}{\mathsf{neg}\left(B\right)} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 21.0

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

   10. Applied rewrites 21.0%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(4 \cdot A\right) \cdot F\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+90}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, -F, A \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+38}:\\ \;\;\;\;\sqrt{t\_0 \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B\_m \cdot B\_m, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \frac{-\sqrt{2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e+38)
     (*
      (sqrt (* t_0 (* F (+ A (fma (* B_m B_m) (/ -0.5 C) A)))))
      (/ (- (sqrt 2.0)) t_0))
     (/ (* (sqrt (* F (- A (hypot B_m A)))) (sqrt 2.0)) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+38) {
		tmp = sqrt((t_0 * (F * (A + fma((B_m * B_m), (-0.5 / C), A))))) * (-sqrt(2.0) / t_0);
	} else {
		tmp = (sqrt((F * (A - hypot(B_m, A)))) * sqrt(2.0)) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+38)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * Float64(A + fma(Float64(B_m * B_m), Float64(-0.5 / C), A))))) * Float64(Float64(-sqrt(2.0)) / t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * sqrt(2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+38], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(A + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(-0.5 / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999977e37

   1. Initial program 26.7%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 23.0

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 23.0%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 22.3%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 22.2%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \frac{-\sqrt{2}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   if 9.99999999999999977e37 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.5%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 11.4%

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

      1. lower-hypot.f64 25.8

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

   7. Applied rewrites 25.8%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+38}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \frac{-\sqrt{2}}{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(2 \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B\_m \cdot B\_m, \frac{-0.5}{C}, A\right)\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B\_m, -F, A \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e+61)
     (/
      (sqrt (* t_0 (* 2.0 (* F (+ A (fma (* B_m B_m) (/ -0.5 C) A))))))
      (- t_0))
     (/ (* (sqrt 2.0) (sqrt (fma B_m (- F) (* A F)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+61) {
		tmp = sqrt((t_0 * (2.0 * (F * (A + fma((B_m * B_m), (-0.5 / C), A)))))) / -t_0;
	} else {
		tmp = (sqrt(2.0) * sqrt(fma(B_m, -F, (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+61)
		tmp = Float64(sqrt(Float64(t_0 * Float64(2.0 * Float64(F * Float64(A + fma(Float64(B_m * B_m), Float64(-0.5 / C), A)))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(fma(B_m, Float64(-F), Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+61], N[(N[Sqrt[N[(t$95$0 * N[(2.0 * N[(F * N[(A + N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(-0.5 / C), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F) + N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000018e61

   1. Initial program 27.1%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 22.7

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 22.7%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 21.9%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 21.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right) \cdot 2\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}} \]

   if 5.00000000000000018e61 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.8%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 11.5%

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

      1. lower-hypot.f64 26.1

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(B, -1 \cdot F, A \cdot F\right)}}}{\mathsf{neg}\left(B\right)} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 21.7

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

   10. Applied rewrites 21.7%

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

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)\right)}}{-\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, -F, A \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{t\_0 \cdot \left(\left(4 \cdot A\right) \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B\_m, -F, A \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e+58)
     (/ -1.0 (/ t_0 (sqrt (* t_0 (* (* 4.0 A) F)))))
     (/ (* (sqrt 2.0) (sqrt (fma B_m (- F) (* A F)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+58) {
		tmp = -1.0 / (t_0 / sqrt((t_0 * ((4.0 * A) * F))));
	} else {
		tmp = (sqrt(2.0) * sqrt(fma(B_m, -F, (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+58)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(t_0 * Float64(Float64(4.0 * A) * F)))));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(fma(B_m, Float64(-F), Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+58], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(t$95$0 * N[(N[(4.0 * A), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F) + N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999944e57

   1. Initial program 27.2%

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

   3. Taylor expanded in C around inf

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

      1. associate--l+ N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} - -1 \cdot A\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

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

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \left(\mathsf{neg}\left(-1\right)\right) \cdot A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{1} \cdot A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + \color{blue}{A}\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \left(A + \frac{-1}{2} \cdot \frac{{B}^{2}}{C}\right)\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\left(\frac{-1}{2} \cdot \frac{{B}^{2}}{C} + A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(\color{blue}{\frac{{B}^{2}}{C} \cdot \frac{-1}{2}} + A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \color{blue}{\mathsf{fma}\left(\frac{{B}^{2}}{C}, \frac{-1}{2}, A\right)}\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\color{blue}{\frac{{B}^{2}}{C}}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, \frac{-1}{2}, A\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. lower-*.f64 22.9

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{\color{blue}{B \cdot B}}{C}, -0.5, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Applied rewrites 22.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 22.1%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right)} \cdot \sqrt{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   9. Applied rewrites 21.5%

      \[\leadsto \color{blue}{\frac{1}{-\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \left(\left(F \cdot \left(A + \mathsf{fma}\left(B \cdot B, \frac{-0.5}{C}, A\right)\right)\right) \cdot 2\right)}}}} \]

   10. Taylor expanded in A around inf

       \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(\left(4 \cdot A\right) \cdot F\right)}}}\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\mathsf{neg}\left(\frac{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right)}{\sqrt{\mathsf{fma}\left(A, C \cdot -4, B \cdot B\right) \cdot \color{blue}{\left(\left(4 \cdot A\right) \cdot F\right)}}}\right)} \]

       3. lower-*.f64 20.7

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

   12. Applied rewrites 20.7%

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

   if 9.99999999999999944e57 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 11.4%

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

      1. lower-hypot.f64 25.9

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

   7. Applied rewrites 25.9%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(B, -1 \cdot F, A \cdot F\right)}}}{\mathsf{neg}\left(B\right)} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 21.5

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

   10. Applied rewrites 21.5%

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

4. Final simplification 21.1%

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

Alternative 11: 34.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-184}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B\_m, -F, A \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-184)
   (/ (sqrt (* (* C F) (* -16.0 (* A A)))) (fma B_m (- B_m) (* A (* 4.0 C))))
   (/ (* (sqrt 2.0) (sqrt (fma B_m (- F) (* A F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-184) {
		tmp = sqrt(((C * F) * (-16.0 * (A * A)))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else {
		tmp = (sqrt(2.0) * sqrt(fma(B_m, -F, (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-184)
		tmp = Float64(sqrt(Float64(Float64(C * F) * Float64(-16.0 * Float64(A * A)))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(fma(B_m, Float64(-F), Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-184], N[(N[Sqrt[N[(N[(C * F), $MachinePrecision] * N[(-16.0 * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F) + N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e-184

   1. Initial program 22.8%

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

   4. Applied rewrites 22.8%

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

   5. Taylor expanded in A around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 15.8

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

   7. Applied rewrites 15.8%

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

   if 1.0000000000000001e-184 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 21.0%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 9.9%

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

      1. lower-hypot.f64 20.0

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

   7. Applied rewrites 20.0%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(B, -1 \cdot F, A \cdot F\right)}}}{\mathsf{neg}\left(B\right)} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 16.6

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

   10. Applied rewrites 16.6%

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

4. Final simplification 16.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-184}:\\ \;\;\;\;\frac{\sqrt{\left(C \cdot F\right) \cdot \left(-16 \cdot \left(A \cdot A\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, -F, A \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B\_m, -B\_m, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B\_m, -F, A \cdot F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.5e-90)
   (/
    (sqrt (* (* A -8.0) (* (+ A A) (* C F))))
    (fma B_m (- B_m) (* A (* 4.0 C))))
   (/ (* (sqrt 2.0) (sqrt (fma B_m (- F) (* A F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.5e-90) {
		tmp = sqrt(((A * -8.0) * ((A + A) * (C * F)))) / fma(B_m, -B_m, (A * (4.0 * C)));
	} else {
		tmp = (sqrt(2.0) * sqrt(fma(B_m, -F, (A * F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.5e-90)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(Float64(A + A) * Float64(C * F)))) / fma(B_m, Float64(-B_m), Float64(A * Float64(4.0 * C))));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(fma(B_m, Float64(-F), Float64(A * F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 6.5e-90], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(B$95$m * (-B$95$m) + N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F) + N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.4999999999999996e-90

   1. Initial program 22.1%

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

   4. Applied rewrites 22.1%

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

   5. Taylor expanded in C around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 13.4

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

   7. Applied rewrites 13.4%

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

   if 6.4999999999999996e-90 < B

   1. Initial program 20.3%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 19.5%

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

      1. lower-hypot.f64 40.1

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

   7. Applied rewrites 40.1%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(B, -1 \cdot F, A \cdot F\right)}}}{\mathsf{neg}\left(B\right)} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 35.5

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

   10. Applied rewrites 35.5%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(B, -B, A \cdot \left(4 \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{\mathsf{fma}\left(B, -F, A \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2.2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{B\_m \cdot \left(-F\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -2.2e+148)
   (* (sqrt (* A F)) (/ -2.0 B_m))
   (/ (* (sqrt 2.0) (sqrt (* B_m (- F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -2.2e+148) {
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = (sqrt(2.0) * sqrt((B_m * -F))) / -B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 (a <= (-2.2d+148)) then
        tmp = sqrt((a * f)) * ((-2.0d0) / b_m)
    else
        tmp = (sqrt(2.0d0) * sqrt((b_m * -f))) / -b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -2.2e+148) {
		tmp = Math.sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = (Math.sqrt(2.0) * Math.sqrt((B_m * -F))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -2.2e+148:
		tmp = math.sqrt((A * F)) * (-2.0 / B_m)
	else:
		tmp = (math.sqrt(2.0) * math.sqrt((B_m * -F))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -2.2e+148)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(B_m * Float64(-F)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -2.2e+148)
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	else
		tmp = (sqrt(2.0) * sqrt((B_m * -F))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -2.2e+148], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -2.1999999999999999e148

   1. Initial program 4.9%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 1.4%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

         \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot \color{blue}{2}}{B} \]

      8. metadata-eval N/A

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

      9. lower-/.f64 7.0

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

   8. Applied rewrites 7.0%

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

   if -2.1999999999999999e148 < A

   1. Initial program 24.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. associate-*l/ N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 8.4%

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

      1. lower-hypot.f64 15.4

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

   7. Applied rewrites 15.4%

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

   8. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 14.1

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

   10. Applied rewrites 14.1%

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

4. Final simplification 13.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.2 \cdot 10^{+148}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot \sqrt{B \cdot \left(-F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 9.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{A \cdot F} \cdot \frac{-2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (sqrt (* A F)) (/ -2.0 B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((A * F)) * (-2.0 / B_m);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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((a * f)) * ((-2.0d0) / b_m)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((A * F)) * (-2.0 / B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((A * F)) * (-2.0 / B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((A * F)) * (-2.0 / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.6%

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

3. Taylor expanded in C around 0

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

   1. mul-1-neg N/A

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

   2. associate-*l/ N/A

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

   3. distribute-neg-frac2 N/A

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

   4. mul-1-neg N/A

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

   5. lower-/.f64 N/A

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

5. Applied rewrites 7.3%

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

6. Taylor expanded in A around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. rem-square-sqrt N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

      \[\leadsto \sqrt{A \cdot F} \cdot \frac{-1 \cdot \color{blue}{2}}{B} \]

   8. metadata-eval N/A

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

   9. lower-/.f64 3.5

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

8. Applied rewrites 3.5%

   \[\leadsto \color{blue}{\sqrt{A \cdot F} \cdot \frac{-2}{B}} \]
9. Add Preprocessing

Alternative 15: 1.5% accurate, 18.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (* F (/ 2.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(F * Float64(2.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 21.6%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 2.0

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

5. Applied rewrites 2.0%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. mul-1-neg N/A

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

   4. remove-double-neg N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-unprod N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-*.f64 2.0

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

7. Applied rewrites 2.0%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 2.0

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

9. Applied rewrites 2.0%

   \[\leadsto \sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))