ABCF->ab-angle b

Percentage Accurate: 18.3% → 49.5%

Time: 19.3s

Alternatives: 15

Speedup: 10.2×

• could not determine a ground truth

  1. A = 6.175696926593882e-97
  2. B = -4.178281027786984e-201
  3. C = 8.461694592006135e-306
  4. F = -146627679315152.56

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.3% 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: 49.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -8, 2 \cdot \left(B\_m \cdot B\_m\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{-\mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-212}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}\right)}}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(A + \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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 (fma (* A C) -8.0 (* 2.0 (* B_m B_m)))) (sqrt (* F (+ A A))))
      (- (fma B_m B_m (* (* A C) -4.0))))
     (if (<= t_4 -2e-212)
       (/
        -1.0
        (/
         t_0
         (sqrt
          (*
           (* t_0 (* 2.0 F))
           (- (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m))))))))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_2 (+ A (fma (/ (* B_m B_m) C) -0.5 A)))) t_3)
         (* (sqrt 2.0) (* (sqrt (* F (- C (hypot C B_m)))) (/ -1.0 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(fma((A * C), -8.0, (2.0 * (B_m * B_m)))) * sqrt((F * (A + A)))) / -fma(B_m, B_m, ((A * C) * -4.0));
	} else if (t_4 <= -2e-212) {
		tmp = -1.0 / (t_0 / sqrt(((t_0 * (2.0 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B_m * B_m)))))));
	} 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(2.0) * (sqrt((F * (C - hypot(C, B_m)))) * (-1.0 / 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(fma(Float64(A * C), -8.0, Float64(2.0 * Float64(B_m * B_m)))) * sqrt(Float64(F * Float64(A + A)))) / Float64(-fma(B_m, B_m, Float64(Float64(A * C) * -4.0))));
	elseif (t_4 <= -2e-212)
		tmp = Float64(-1.0 / Float64(t_0 / 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))))))));
	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(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(C - hypot(C, B_m)))) * Float64(-1.0 / 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[(N[(A * C), $MachinePrecision] * -8.0 + N[(2.0 * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -2e-212], N[(-1.0 / N[(t$95$0 / 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]), $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[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $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 2.8%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 22.1

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

   5. Simplified 22.1%

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

   7. Applied egg-rr 22.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

   9. Applied egg-rr 26.3%

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

   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))) < -1.99999999999999991e-212

   1. Initial program 99.3%

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

   4. Applied egg-rr 99.5%

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

   if -1.99999999999999991e-212 < (/.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 25.6%

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

   3. Taylor expanded in C around 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 26.2

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

   5. Simplified 26.2%

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

   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

   4. Applied egg-rr 0.0%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 1.9

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

   7. Simplified 1.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-hypot.f64 14.4

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

   9. Applied egg-rr 14.4%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A \cdot C, -8, 2 \cdot \left(B \cdot B\right)\right)} \cdot \sqrt{F \cdot \left(A + A\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-212}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}{\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 44.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\\ \mathbf{if}\;B\_m \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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 (* (* A C) -4.0))))
   (if (<= B_m 1.65e+38)
     (/ (sqrt (* (+ A A) (* 2.0 (* F t_0)))) (- t_0))
     (* (sqrt 2.0) (* (sqrt (* F (- C (hypot C B_m)))) (/ -1.0 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, ((A * C) * -4.0));
	double tmp;
	if (B_m <= 1.65e+38) {
		tmp = sqrt(((A + A) * (2.0 * (F * t_0)))) / -t_0;
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (C - hypot(C, B_m)))) * (-1.0 / 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(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B_m <= 1.65e+38)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(2.0 * Float64(F * t_0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(C - hypot(C, B_m)))) * Float64(-1.0 / 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[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.65e+38], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.65e38

   1. Initial program 23.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.3

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   if 1.65e38 < B

   1. Initial program 18.2%

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

   4. Applied egg-rr 18.1%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 23.3

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

   7. Simplified 23.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-hypot.f64 46.2

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

   9. Applied egg-rr 46.2%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 27.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\\ t_1 := -t\_0\\ \mathbf{if}\;A \leq -3.2 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{t\_0}}\\ \mathbf{elif}\;A \leq -1.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{t\_1}\\ \mathbf{elif}\;A \leq 8.5 \cdot 10^{-151}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{t\_1}\\ \end{array} \end{array} \]

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 (* (* A C) -4.0))) (t_1 (- t_0)))
   (if (<= A -3.2e+142)
     (* -2.0 (sqrt (/ (* A F) t_0)))
     (if (<= A -1.2e-125)
       (/ (sqrt (* -16.0 (* F (* C (* A A))))) t_1)
       (if (<= A 8.5e-151)
         (- (/ (sqrt (* 2.0 (* F (- C (sqrt (fma B_m B_m (* C C))))))) B_m))
         (/ (sqrt (* (+ A A) (* -8.0 (* A (* C F))))) t_1))))))

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, ((A * C) * -4.0));
	double t_1 = -t_0;
	double tmp;
	if (A <= -3.2e+142) {
		tmp = -2.0 * sqrt(((A * F) / t_0));
	} else if (A <= -1.2e-125) {
		tmp = sqrt((-16.0 * (F * (C * (A * A))))) / t_1;
	} else if (A <= 8.5e-151) {
		tmp = -(sqrt((2.0 * (F * (C - sqrt(fma(B_m, B_m, (C * C))))))) / B_m);
	} else {
		tmp = sqrt(((A + A) * (-8.0 * (A * (C * F))))) / t_1;
	}
	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(Float64(A * C) * -4.0))
	t_1 = Float64(-t_0)
	tmp = 0.0
	if (A <= -3.2e+142)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / t_0)));
	elseif (A <= -1.2e-125)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(C * Float64(A * A))))) / t_1);
	elseif (A <= 8.5e-151)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - sqrt(fma(B_m, B_m, Float64(C * C))))))) / B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(-8.0 * Float64(A * Float64(C * F))))) / t_1);
	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[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[A, -3.2e+142], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -1.2e-125], N[(N[Sqrt[N[(-16.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[A, 8.5e-151], (-N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if A < -3.20000000000000005e142

   1. Initial program 7.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 24.5

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

   5. Simplified 24.5%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 22.5

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

   8. Simplified 22.5%

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

   if -3.20000000000000005e142 < A < -1.2000000000000001e-125

   1. Initial program 40.2%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 37.8

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

   5. Simplified 37.8%

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

   7. Applied egg-rr 37.8%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 30.9

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

   10. Simplified 30.9%

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

   if -1.2000000000000001e-125 < A < 8.49999999999999999e-151

   1. Initial program 29.1%

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

   4. Applied egg-rr 27.7%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 24.7

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

   7. Simplified 24.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. neg-mul-1 N/A

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

      11. lower-/.f64 N/A

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

   9. Applied egg-rr 24.8%

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

   if 8.49999999999999999e-151 < A

   1. Initial program 11.0%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 10.4

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

   5. Simplified 10.4%

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

   7. Applied egg-rr 10.4%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 10.6

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

   10. Simplified 10.6%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(A + A\right)}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.2 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\\ \mathbf{elif}\;A \leq -1.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{elif}\;A \leq 8.5 \cdot 10^{-151}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\\ \mathbf{if}\;B\_m \leq 8.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, \frac{C \cdot C}{B\_m}, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

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 (* (* A C) -4.0))))
   (if (<= B_m 8.6e+37)
     (/ (sqrt (* (+ A A) (* 2.0 (* F t_0)))) (- t_0))
     (*
      (sqrt 2.0)
      (* (sqrt (* F (- C (fma 0.5 (/ (* C C) B_m) B_m)))) (/ -1.0 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, ((A * C) * -4.0));
	double tmp;
	if (B_m <= 8.6e+37) {
		tmp = sqrt(((A + A) * (2.0 * (F * t_0)))) / -t_0;
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (C - fma(0.5, ((C * C) / B_m), B_m)))) * (-1.0 / 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(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B_m <= 8.6e+37)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(2.0 * Float64(F * t_0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(C - fma(0.5, Float64(Float64(C * C) / B_m), B_m)))) * Float64(-1.0 / 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[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.6e+37], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(C - N[(0.5 * N[(N[(C * C), $MachinePrecision] / B$95$m), $MachinePrecision] + B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 8.5999999999999994e37

   1. Initial program 23.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.3

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   if 8.5999999999999994e37 < B

   1. Initial program 18.2%

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

   4. Applied egg-rr 18.1%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 23.3

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

   7. Simplified 23.3%

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

   8. Taylor expanded in C around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 37.8

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

   10. Simplified 37.8%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C - \mathsf{fma}\left(0.5, \frac{C \cdot C}{B}, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\\ \mathbf{if}\;B\_m \leq 7.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B\_m \cdot B\_m\right)}\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 (* (* A C) -4.0))))
   (if (<= B_m 7.6e+37)
     (/ (sqrt (* (+ A A) (* 2.0 (* F t_0)))) (- t_0))
     (*
      (sqrt 2.0)
      (/ (sqrt (* F (- A (sqrt (fma A A (* B_m B_m)))))) (- 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, ((A * C) * -4.0));
	double tmp;
	if (B_m <= 7.6e+37) {
		tmp = sqrt(((A + A) * (2.0 * (F * t_0)))) / -t_0;
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - sqrt(fma(A, A, (B_m * B_m)))))) / -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(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B_m <= 7.6e+37)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(2.0 * Float64(F * t_0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B_m * B_m)))))) / 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[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 7.6e+37], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 7.59999999999999979e37

   1. Initial program 23.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.3

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   if 7.59999999999999979e37 < B

   1. Initial program 18.2%

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

   4. Applied egg-rr 18.1%

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

   5. Taylor expanded in C around 0

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 24.2

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

   7. Simplified 24.2%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\\ \mathbf{if}\;B\_m \leq 5.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot t\_0\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \end{array} \end{array} \]

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 (* (* A C) -4.0))))
   (if (<= B_m 5.1e+38)
     (/ (sqrt (* (+ A A) (* 2.0 (* F t_0)))) (- t_0))
     (*
      (sqrt (* F (- A (sqrt (fma B_m B_m (* A 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, ((A * C) * -4.0));
	double tmp;
	if (B_m <= 5.1e+38) {
		tmp = sqrt(((A + A) * (2.0 * (F * t_0)))) / -t_0;
	} else {
		tmp = sqrt((F * (A - sqrt(fma(B_m, B_m, (A * 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(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B_m <= 5.1e+38)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(2.0 * Float64(F * t_0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A)))))) * Float64(-Float64(sqrt(2.0) / 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[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 5.1e+38], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 5.1000000000000001e38

   1. Initial program 23.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.3

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   if 5.1000000000000001e38 < B

   1. Initial program 18.2%

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

   3. Taylor expanded in C around 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. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Simplified 24.2%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\\ t_1 := \frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-t\_0}\\ \mathbf{if}\;A \leq -5.8 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{t\_0}}\\ \mathbf{elif}\;A \leq -4.6 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;A \leq 1.1 \cdot 10^{-150}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 (* (* A C) -4.0)))
        (t_1 (/ (sqrt (* -16.0 (* F (* C (* A A))))) (- t_0))))
   (if (<= A -5.8e+142)
     (* -2.0 (sqrt (/ (* A F) t_0)))
     (if (<= A -4.6e-126)
       t_1
       (if (<= A 1.1e-150)
         (- (/ (sqrt (* 2.0 (* F (- C (sqrt (fma B_m B_m (* C C))))))) B_m))
         t_1)))))

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, ((A * C) * -4.0));
	double t_1 = sqrt((-16.0 * (F * (C * (A * A))))) / -t_0;
	double tmp;
	if (A <= -5.8e+142) {
		tmp = -2.0 * sqrt(((A * F) / t_0));
	} else if (A <= -4.6e-126) {
		tmp = t_1;
	} else if (A <= 1.1e-150) {
		tmp = -(sqrt((2.0 * (F * (C - sqrt(fma(B_m, B_m, (C * C))))))) / B_m);
	} else {
		tmp = t_1;
	}
	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(Float64(A * C) * -4.0))
	t_1 = Float64(sqrt(Float64(-16.0 * Float64(F * Float64(C * Float64(A * A))))) / Float64(-t_0))
	tmp = 0.0
	if (A <= -5.8e+142)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / t_0)));
	elseif (A <= -4.6e-126)
		tmp = t_1;
	elseif (A <= 1.1e-150)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - sqrt(fma(B_m, B_m, Float64(C * C))))))) / B_m));
	else
		tmp = t_1;
	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[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(-16.0 * N[(F * N[(C * N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]}, If[LessEqual[A, -5.8e+142], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, -4.6e-126], t$95$1, If[LessEqual[A, 1.1e-150], (-N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if A < -5.80000000000000027e142

   1. Initial program 7.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 24.5

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

   5. Simplified 24.5%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 22.5

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

   8. Simplified 22.5%

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

   if -5.80000000000000027e142 < A < -4.60000000000000021e-126 or 1.1e-150 < A

   1. Initial program 23.0%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.2

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

   5. Simplified 21.2%

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

   7. Applied egg-rr 21.2%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 17.7

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

   10. Simplified 17.7%

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

   if -4.60000000000000021e-126 < A < 1.1e-150

   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

   4. Applied egg-rr 26.7%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 25.1

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

   7. Simplified 25.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. neg-mul-1 N/A

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

      11. lower-/.f64 N/A

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

   9. Applied egg-rr 25.2%

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

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5.8 \cdot 10^{+142}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\\ \mathbf{elif}\;A \leq -4.6 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{elif}\;A \leq 1.1 \cdot 10^{-150}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(F \cdot \left(C \cdot \left(A \cdot A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\\ \mathbf{if}\;B\_m \leq 6.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{2 \cdot \sqrt{t\_0 \cdot \left(A \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \end{array} \end{array} \]

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 (* (* A C) -4.0))))
   (if (<= B_m 6.8e+37)
     (/ (* 2.0 (sqrt (* t_0 (* A F)))) (- t_0))
     (*
      (sqrt (* F (- A (sqrt (fma B_m B_m (* A 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, ((A * C) * -4.0));
	double tmp;
	if (B_m <= 6.8e+37) {
		tmp = (2.0 * sqrt((t_0 * (A * F)))) / -t_0;
	} else {
		tmp = sqrt((F * (A - sqrt(fma(B_m, B_m, (A * 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(Float64(A * C) * -4.0))
	tmp = 0.0
	if (B_m <= 6.8e+37)
		tmp = Float64(Float64(2.0 * sqrt(Float64(t_0 * Float64(A * F)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A)))))) * Float64(-Float64(sqrt(2.0) / 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[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6.8e+37], N[(N[(2.0 * N[Sqrt[N[(t$95$0 * N[(A * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 6.80000000000000011e37

   1. Initial program 23.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.3

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 20.4

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

   10. Simplified 20.4%

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

   if 6.80000000000000011e37 < B

   1. Initial program 18.2%

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

   3. Taylor expanded in C around 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. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Simplified 24.2%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{2 \cdot \sqrt{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right) \cdot \left(A \cdot F\right)}}{-\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B\_m, B\_m, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \end{array} \end{array} \]

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 6.8e+37)
   (/
    (sqrt (* (+ A A) (* 2.0 (* F (fma B_m B_m (* (* A C) -4.0))))))
    (* 4.0 (* A C)))
   (* (sqrt (* F (- A (sqrt (fma B_m B_m (* A 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 tmp;
	if (B_m <= 6.8e+37) {
		tmp = sqrt(((A + A) * (2.0 * (F * fma(B_m, B_m, ((A * C) * -4.0)))))) / (4.0 * (A * C));
	} else {
		tmp = sqrt((F * (A - sqrt(fma(B_m, B_m, (A * 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)
	tmp = 0.0
	if (B_m <= 6.8e+37)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(Float64(A * C) * -4.0)))))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B_m, B_m, Float64(A * A)))))) * Float64(-Float64(sqrt(2.0) / 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, 6.8e+37], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B$95$m * B$95$m + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.80000000000000011e37

   1. Initial program 23.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.3

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 18.4

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

   10. Simplified 18.4%

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

   if 6.80000000000000011e37 < B

   1. Initial program 18.2%

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

   3. Taylor expanded in C around 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. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Simplified 24.2%

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

4. Final simplification 19.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.3% accurate, 6.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)}}{B\_m}\\ \end{array} \end{array} \]

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
 (if (<= B_m 1.8e+38)
   (/
    (sqrt (* (+ A A) (* 2.0 (* F (fma B_m B_m (* (* A C) -4.0))))))
    (* 4.0 (* A C)))
   (- (/ (sqrt (* 2.0 (* F (- C (sqrt (fma B_m B_m (* C C))))))) 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 <= 1.8e+38) {
		tmp = sqrt(((A + A) * (2.0 * (F * fma(B_m, B_m, ((A * C) * -4.0)))))) / (4.0 * (A * C));
	} else {
		tmp = -(sqrt((2.0 * (F * (C - sqrt(fma(B_m, B_m, (C * C))))))) / 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 <= 1.8e+38)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(Float64(A * C) * -4.0)))))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - sqrt(fma(B_m, B_m, Float64(C * C))))))) / 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, 1.8e+38], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.79999999999999985e38

   1. Initial program 23.5%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 21.3

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

   5. Simplified 21.3%

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

   7. Applied egg-rr 21.3%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 18.4

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

   10. Simplified 18.4%

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

   if 1.79999999999999985e38 < B

   1. Initial program 18.2%

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

   4. Applied egg-rr 18.1%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 23.3

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

   7. Simplified 23.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. neg-mul-1 N/A

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

      11. lower-/.f64 N/A

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

   9. Applied egg-rr 23.4%

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

4. Final simplification 19.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1.2 \cdot 10^{-99}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)\right)}}{B\_m}\\ \end{array} \end{array} \]

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
 (if (<= A -1.2e-99)
   (* -2.0 (sqrt (/ (* A F) (fma B_m B_m (* (* A C) -4.0)))))
   (- (/ (sqrt (* 2.0 (* F (- C (sqrt (fma B_m B_m (* C C))))))) 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 <= -1.2e-99) {
		tmp = -2.0 * sqrt(((A * F) / fma(B_m, B_m, ((A * C) * -4.0))));
	} else {
		tmp = -(sqrt((2.0 * (F * (C - sqrt(fma(B_m, B_m, (C * C))))))) / 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 <= -1.2e-99)
		tmp = Float64(-2.0 * sqrt(Float64(Float64(A * F) / fma(B_m, B_m, Float64(Float64(A * C) * -4.0)))));
	else
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - sqrt(fma(B_m, B_m, Float64(C * C))))))) / 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[A, -1.2e-99], N[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if A < -1.2e-99

   1. Initial program 29.1%

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

   3. Taylor expanded in C around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower-+.f64 33.7

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

   5. Simplified 33.7%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. unpow2 N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 24.3

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

   8. Simplified 24.3%

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

   if -1.2e-99 < A

   1. Initial program 18.4%

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

   4. Applied egg-rr 17.8%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 14.3

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

   7. Simplified 14.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. associate-*l/ N/A

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

      9. frac-times N/A

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

      10. neg-mul-1 N/A

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

      11. lower-/.f64 N/A

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

   9. Applied egg-rr 14.3%

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

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.2 \cdot 10^{-99}:\\ \;\;\;\;-2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \sqrt{\mathsf{fma}\left(B, B, C \cdot C\right)}\right)\right)}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 19.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -2 \cdot \sqrt{\frac{A \cdot F}{\mathsf{fma}\left(B\_m, B\_m, \left(A \cdot C\right) \cdot -4\right)}} \end{array} \]

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
 (* -2.0 (sqrt (/ (* A F) (fma B_m B_m (* (* A C) -4.0))))))

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 -2.0 * sqrt(((A * F) / fma(B_m, B_m, ((A * C) * -4.0))));
}

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(-2.0 * sqrt(Float64(Float64(A * F) / fma(B_m, B_m, Float64(Float64(A * C) * -4.0)))))
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[(-2.0 * N[Sqrt[N[(N[(A * F), $MachinePrecision] / N[(B$95$m * B$95$m + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.3%

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

3. Taylor expanded in C around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-+.f64 18.2

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

5. Simplified 18.2%

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

6. Taylor expanded in F around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

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

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

   6. unpow2 N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 10.8

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

8. Simplified 10.8%

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

9. Final simplification 10.8%

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

Alternative 13: 8.8% accurate, 15.3× speedup

Math

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

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 (* -2.0 (/ (sqrt (* 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) {
	return -2.0 * (sqrt((A * F)) / 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 = (-2.0d0) * (sqrt((a * f)) / 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 -2.0 * (Math.sqrt((A * F)) / 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 -2.0 * (math.sqrt((A * F)) / 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(-2.0 * Float64(sqrt(Float64(A * F)) / 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 = -2.0 * (sqrt((A * F)) / 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[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.3%

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

3. Taylor expanded in C around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower-+.f64 18.2

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

5. Simplified 18.2%

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

7. Applied egg-rr 18.2%

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

8. Taylor expanded in B around inf

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 2.8

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

10. Simplified 2.8%

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

11. Final simplification 2.8%

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

Alternative 14: 1.6% accurate, 18.2× speedup

Math

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

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 (* 2.0 (/ 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) {
	return sqrt((2.0 * (F / 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((2.0d0 * (f / 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((2.0 * (F / 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((2.0 * (F / 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(2.0 * Float64(F / 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((2.0 * (F / 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[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 22.3%

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

3. Taylor expanded in 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. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 2.1

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

5. Simplified 2.1%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. lift-sqrt.f64 N/A

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

   11. sqrt-unprod N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-*.f64 2.1

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

7. Applied egg-rr 2.1%

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

8. Final simplification 2.1%

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

Alternative 15: 1.6% 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 22.3%

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

3. Taylor expanded in 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. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 2.1

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

5. Simplified 2.1%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. remove-double-neg N/A

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

   10. lift-sqrt.f64 N/A

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

   11. sqrt-unprod N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower-*.f64 2.1

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

7. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
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 egg-rr 2.0%

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

Reproduce

herbie shell --seed 2024207 
(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))))