ABCF->ab-angle a

Percentage Accurate: 19.0% → 60.4%

Time: 15.8s

Alternatives: 18

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := t\_0 \cdot 2\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_5 := \sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{t\_5}{-1} \cdot \frac{\sqrt{t\_1} \cdot \sqrt{F}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 10^{+228}:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\right)}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot t\_4}}{-1} \cdot \frac{t\_5}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\_m\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m)))
        (t_1 (* t_0 2.0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (fma -4.0 (* C A) (* B_m B_m)))
        (t_5 (sqrt (+ (+ (hypot B_m (- A C)) A) C))))
   (if (<= t_3 -5e-196)
     (* (/ t_5 -1.0) (/ (* (sqrt t_1) (sqrt F)) t_4))
     (if (<= t_3 1e+228)
       (/ (sqrt (* (* t_1 F) (+ (fma (/ (* B_m B_m) A) -0.5 C) C))) (- t_0))
       (if (<= t_3 INFINITY)
         (* (/ (sqrt (* (* 2.0 F) t_4)) -1.0) (/ t_5 t_4))
         (*
          (- (sqrt (+ (+ C A) (hypot (- A C) B_m))))
          (/ (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) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = t_0 * 2.0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = fma(-4.0, (C * A), (B_m * B_m));
	double t_5 = sqrt(((hypot(B_m, (A - C)) + A) + C));
	double tmp;
	if (t_3 <= -5e-196) {
		tmp = (t_5 / -1.0) * ((sqrt(t_1) * sqrt(F)) / t_4);
	} else if (t_3 <= 1e+228) {
		tmp = sqrt(((t_1 * F) * (fma(((B_m * B_m) / A), -0.5, C) + C))) / -t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(((2.0 * F) * t_4)) / -1.0) * (t_5 / t_4);
	} else {
		tmp = -sqrt(((C + A) + hypot((A - C), B_m))) * (sqrt((F * 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(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = Float64(t_0 * 2.0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_5 = sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C))
	tmp = 0.0
	if (t_3 <= -5e-196)
		tmp = Float64(Float64(t_5 / -1.0) * Float64(Float64(sqrt(t_1) * sqrt(F)) / t_4));
	elseif (t_3 <= 1e+228)
		tmp = Float64(sqrt(Float64(Float64(t_1 * F) * Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C))) / Float64(-t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_4)) / -1.0) * Float64(t_5 / t_4));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + A) + hypot(Float64(A - C), B_m)))) * Float64(sqrt(Float64(F * 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[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -5e-196], N[(N[(t$95$5 / -1.0), $MachinePrecision] * N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+228], N[(N[Sqrt[N[(N[(t$95$1 * F), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(t$95$5 / t$95$4), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $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))) < -5.0000000000000005e-196

   1. Initial program 34.8%

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

   4. Applied rewrites 60.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sqrt-prod N/A

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

      19. lift-*.f64 N/A

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

   6. Applied rewrites 77.1%

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

   if -5.0000000000000005e-196 < (/.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))) < 9.9999999999999992e227

   1. Initial program 25.4%

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.1

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

   7. Applied rewrites 26.1%

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

   9. Applied rewrites 33.5%

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

   if 9.9999999999999992e227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.7%

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

   4. Applied rewrites 68.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 20.0

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

   7. Applied rewrites 20.0%

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

   9. Applied rewrites 20.6%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 10^{+228}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_5 := -t\_4\\ t_6 := t\_1 \cdot 2\\ t_7 := \left(2 \cdot F\right) \cdot t\_4\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{t\_0} \cdot \frac{\sqrt{t\_6} \cdot \sqrt{F}}{t\_5}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot t\_7}}{t\_5}\\ \mathbf{elif}\;t\_3 \leq 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(t\_6 \cdot F\right) \cdot t\_0}}{-t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{t\_7}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\_m\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (fma (/ (* B_m B_m) A) -0.5 C) C))
        (t_1 (fma (* C A) -4.0 (* B_m B_m)))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (fma -4.0 (* C A) (* B_m B_m)))
        (t_5 (- t_4))
        (t_6 (* t_1 2.0))
        (t_7 (* (* 2.0 F) t_4)))
   (if (<= t_3 (- INFINITY))
     (* (sqrt t_0) (/ (* (sqrt t_6) (sqrt F)) t_5))
     (if (<= t_3 -5e-196)
       (/ (sqrt (* (+ (+ (hypot B_m (- A C)) A) C) t_7)) t_5)
       (if (<= t_3 1e-57)
         (/ (sqrt (* (* t_6 F) t_0)) (- t_1))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* 2.0 C)) -1.0) (/ (sqrt t_7) t_4))
           (*
            (- (sqrt (+ (+ C A) (hypot (- A C) B_m))))
            (/ (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) {
	double t_0 = fma(((B_m * B_m) / A), -0.5, C) + C;
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = fma(-4.0, (C * A), (B_m * B_m));
	double t_5 = -t_4;
	double t_6 = t_1 * 2.0;
	double t_7 = (2.0 * F) * t_4;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(t_0) * ((sqrt(t_6) * sqrt(F)) / t_5);
	} else if (t_3 <= -5e-196) {
		tmp = sqrt((((hypot(B_m, (A - C)) + A) + C) * t_7)) / t_5;
	} else if (t_3 <= 1e-57) {
		tmp = sqrt(((t_6 * F) * t_0)) / -t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * C)) / -1.0) * (sqrt(t_7) / t_4);
	} else {
		tmp = -sqrt(((C + A) + hypot((A - C), B_m))) * (sqrt((F * 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 = Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C)
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_5 = Float64(-t_4)
	t_6 = Float64(t_1 * 2.0)
	t_7 = Float64(Float64(2.0 * F) * t_4)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(t_0) * Float64(Float64(sqrt(t_6) * sqrt(F)) / t_5));
	elseif (t_3 <= -5e-196)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C) * t_7)) / t_5);
	elseif (t_3 <= 1e-57)
		tmp = Float64(sqrt(Float64(Float64(t_6 * F) * t_0)) / Float64(-t_1));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) / -1.0) * Float64(sqrt(t_7) / t_4));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + A) + hypot(Float64(A - C), B_m)))) * Float64(sqrt(Float64(F * 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[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-t$95$4)}, Block[{t$95$6 = N[(t$95$1 * 2.0), $MachinePrecision]}, Block[{t$95$7 = N[(N[(2.0 * F), $MachinePrecision] * t$95$4), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[(N[Sqrt[t$95$6], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-196], N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * t$95$7), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$3, 1e-57], N[(N[Sqrt[N[(N[(t$95$6 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[t$95$7], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.0%

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

   4. Applied rewrites 40.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 23.7

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

   7. Applied rewrites 23.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. sqrt-prod N/A

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

   9. Applied rewrites 35.0%

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

       1. lift-/.f64 N/A

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

       2. frac-2neg N/A

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

       3. metadata-eval N/A

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

       4. /-rgt-identity N/A

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

       5. lower-neg.f64 35.0

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

   11. Applied rewrites 35.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000005e-196

   1. Initial program 99.5%

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

   4. Applied rewrites 99.5%

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

   if -5.0000000000000005e-196 < (/.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))) < 9.99999999999999955e-58

   1. Initial program 15.5%

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

   4. Applied rewrites 17.1%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   9. Applied rewrites 32.9%

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

   if 9.99999999999999955e-58 < (/.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 27.7%

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 13.2

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

   7. Applied rewrites 13.2%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 20.0

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

   7. Applied rewrites 20.0%

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

   9. Applied rewrites 20.6%

      \[\leadsto \color{blue}{\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -\infty:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C} \cdot \frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := t\_0 \cdot 2\\ t_2 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_2}\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_5 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{t\_5} \cdot \frac{\sqrt{t\_1} \cdot \sqrt{F}}{-t\_4}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(B\_m, C\right) + C\right) \cdot F}\\ \mathbf{elif}\;t\_3 \leq 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 \cdot F\right) \cdot t\_5}}{-t\_0}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot t\_4}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* C A) -4.0 (* B_m B_m)))
        (t_1 (* t_0 2.0))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* t_2 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_2)))
        (t_4 (fma -4.0 (* C A) (* B_m B_m)))
        (t_5 (+ (fma (/ (* B_m B_m) A) -0.5 C) C)))
   (if (<= t_3 -2e+216)
     (* (sqrt t_5) (/ (* (sqrt t_1) (sqrt F)) (- t_4)))
     (if (<= t_3 -5e-196)
       (* (/ (sqrt 2.0) (- B_m)) (sqrt (* (+ (hypot B_m C) C) F)))
       (if (<= t_3 1e-57)
         (/ (sqrt (* (* t_1 F) t_5)) (- t_0))
         (if (<= t_3 INFINITY)
           (* (/ (sqrt (* 2.0 C)) -1.0) (/ (sqrt (* (* 2.0 F) t_4)) t_4))
           (/ (sqrt (* F 2.0)) (- (sqrt 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((C * A), -4.0, (B_m * B_m));
	double t_1 = t_0 * 2.0;
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = sqrt(((2.0 * (t_2 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_2;
	double t_4 = fma(-4.0, (C * A), (B_m * B_m));
	double t_5 = fma(((B_m * B_m) / A), -0.5, C) + C;
	double tmp;
	if (t_3 <= -2e+216) {
		tmp = sqrt(t_5) * ((sqrt(t_1) * sqrt(F)) / -t_4);
	} else if (t_3 <= -5e-196) {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((hypot(B_m, C) + C) * F));
	} else if (t_3 <= 1e-57) {
		tmp = sqrt(((t_1 * F) * t_5)) / -t_0;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * C)) / -1.0) * (sqrt(((2.0 * F) * t_4)) / t_4);
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(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(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = Float64(t_0 * 2.0)
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_2))
	t_4 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_5 = Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C)
	tmp = 0.0
	if (t_3 <= -2e+216)
		tmp = Float64(sqrt(t_5) * Float64(Float64(sqrt(t_1) * sqrt(F)) / Float64(-t_4)));
	elseif (t_3 <= -5e-196)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(Float64(hypot(B_m, C) + C) * F)));
	elseif (t_3 <= 1e-57)
		tmp = Float64(sqrt(Float64(Float64(t_1 * F) * t_5)) / Float64(-t_0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) / -1.0) * Float64(sqrt(Float64(Float64(2.0 * F) * t_4)) / t_4));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(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[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+216], N[(N[Sqrt[t$95$5], $MachinePrecision] * N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / (-t$95$4)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-196], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e-57], N[(N[Sqrt[N[(N[(t$95$1 * F), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 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))) < -2e216

   1. Initial program 6.1%

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

   4. Applied rewrites 42.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.1

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

   7. Applied rewrites 26.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. sqrt-prod N/A

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

   9. Applied rewrites 37.1%

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

       1. lift-/.f64 N/A

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

       2. frac-2neg N/A

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

       3. metadata-eval N/A

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

       4. /-rgt-identity N/A

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

       5. lower-neg.f64 37.1

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

   11. Applied rewrites 37.1%

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

   if -2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000005e-196

   1. Initial program 99.5%

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

   3. Taylor expanded in A around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 35.2

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

   5. Applied rewrites 35.2%

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

   if -5.0000000000000005e-196 < (/.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))) < 9.99999999999999955e-58

   1. Initial program 15.5%

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

   4. Applied rewrites 17.1%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   9. Applied rewrites 32.9%

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

   if 9.99999999999999955e-58 < (/.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 27.7%

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 13.2

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

   7. Applied rewrites 13.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 12.8

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

   5. Applied rewrites 12.8%

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

   7. Applied rewrites 14.9%

      \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C} \cdot \frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := t\_0 \cdot 2\\ t_2 := \sqrt{F \cdot 2}\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ t_5 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_6 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{t\_6} \cdot \frac{\sqrt{t\_1} \cdot \sqrt{F}}{-t\_5}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{B\_m \cdot \left(\left(\frac{A}{B\_m} + 1\right) + \frac{C}{B\_m}\right)} \cdot \left(-\frac{t\_2}{B\_m}\right)\\ \mathbf{elif}\;t\_4 \leq 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(t\_1 \cdot F\right) \cdot t\_6}}{-t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot t\_5}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 (* C A) -4.0 (* B_m B_m)))
        (t_1 (* t_0 2.0))
        (t_2 (sqrt (* F 2.0)))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3)))
        (t_5 (fma -4.0 (* C A) (* B_m B_m)))
        (t_6 (+ (fma (/ (* B_m B_m) A) -0.5 C) C)))
   (if (<= t_4 -2e+216)
     (* (sqrt t_6) (/ (* (sqrt t_1) (sqrt F)) (- t_5)))
     (if (<= t_4 -5e-196)
       (* (sqrt (* B_m (+ (+ (/ A B_m) 1.0) (/ C B_m)))) (- (/ t_2 B_m)))
       (if (<= t_4 1e-57)
         (/ (sqrt (* (* t_1 F) t_6)) (- t_0))
         (if (<= t_4 INFINITY)
           (* (/ (sqrt (* 2.0 C)) -1.0) (/ (sqrt (* (* 2.0 F) t_5)) t_5))
           (/ t_2 (- (sqrt 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((C * A), -4.0, (B_m * B_m));
	double t_1 = t_0 * 2.0;
	double t_2 = sqrt((F * 2.0));
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_3;
	double t_5 = fma(-4.0, (C * A), (B_m * B_m));
	double t_6 = fma(((B_m * B_m) / A), -0.5, C) + C;
	double tmp;
	if (t_4 <= -2e+216) {
		tmp = sqrt(t_6) * ((sqrt(t_1) * sqrt(F)) / -t_5);
	} else if (t_4 <= -5e-196) {
		tmp = sqrt((B_m * (((A / B_m) + 1.0) + (C / B_m)))) * -(t_2 / B_m);
	} else if (t_4 <= 1e-57) {
		tmp = sqrt(((t_1 * F) * t_6)) / -t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * C)) / -1.0) * (sqrt(((2.0 * F) * t_5)) / t_5);
	} else {
		tmp = t_2 / -sqrt(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(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = Float64(t_0 * 2.0)
	t_2 = sqrt(Float64(F * 2.0))
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_3))
	t_5 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_6 = Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C)
	tmp = 0.0
	if (t_4 <= -2e+216)
		tmp = Float64(sqrt(t_6) * Float64(Float64(sqrt(t_1) * sqrt(F)) / Float64(-t_5)));
	elseif (t_4 <= -5e-196)
		tmp = Float64(sqrt(Float64(B_m * Float64(Float64(Float64(A / B_m) + 1.0) + Float64(C / B_m)))) * Float64(-Float64(t_2 / B_m)));
	elseif (t_4 <= 1e-57)
		tmp = Float64(sqrt(Float64(Float64(t_1 * F) * t_6)) / Float64(-t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) / -1.0) * Float64(sqrt(Float64(Float64(2.0 * F) * t_5)) / t_5));
	else
		tmp = Float64(t_2 / Float64(-sqrt(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[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision]}, Block[{t$95$5 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]}, If[LessEqual[t$95$4, -2e+216], N[(N[Sqrt[t$95$6], $MachinePrecision] * N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / (-t$95$5)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-196], N[(N[Sqrt[N[(B$95$m * N[(N[(N[(A / B$95$m), $MachinePrecision] + 1.0), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(t$95$2 / B$95$m), $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, 1e-57], N[(N[Sqrt[N[(N[(t$95$1 * F), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], N[(t$95$2 / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 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))) < -2e216

   1. Initial program 6.1%

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

   4. Applied rewrites 42.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.1

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

   7. Applied rewrites 26.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. sqrt-prod N/A

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

   9. Applied rewrites 37.1%

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

       1. lift-/.f64 N/A

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

       2. frac-2neg N/A

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

       3. metadata-eval N/A

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

       4. /-rgt-identity N/A

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

       5. lower-neg.f64 37.1

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

   11. Applied rewrites 37.1%

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

   if -2e216 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000005e-196

   1. Initial program 99.5%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 39.2

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

   7. Applied rewrites 39.2%

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

   9. Applied rewrites 39.3%

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

   10. Taylor expanded in B around inf

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

       1. lower-*.f64 N/A

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

       2. associate-+r+ N/A

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

       3. lower-+.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 34.6

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

   12. Applied rewrites 34.6%

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

   if -5.0000000000000005e-196 < (/.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))) < 9.99999999999999955e-58

   1. Initial program 15.5%

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

   4. Applied rewrites 17.1%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 24.6

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

   7. Applied rewrites 24.6%

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

   9. Applied rewrites 32.9%

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

   if 9.99999999999999955e-58 < (/.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 27.7%

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 13.2

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

   7. Applied rewrites 13.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 12.8

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

   5. Applied rewrites 12.8%

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

   7. Applied rewrites 14.9%

      \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -2 \cdot 10^{+216}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C} \cdot \frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2} \cdot \sqrt{F}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{B \cdot \left(\left(\frac{A}{B} + 1\right) + \frac{C}{B}\right)} \cdot \left(-\frac{\sqrt{F \cdot 2}}{B}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \sqrt{F \cdot 2}\\ t_2 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_3 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(t\_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{-t\_3}\\ t_5 := \sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{t\_5}{-1} \cdot \left(t\_1 \cdot \frac{\sqrt{t\_2}}{t\_2}\right)\\ \mathbf{elif}\;t\_4 \leq 10^{+228}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_2 \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\right)}}{-t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot t\_0}}{-1} \cdot \frac{t\_5}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\_m\right)}\right) \cdot \frac{t\_1}{B\_m}\\ \end{array} \end{array} \]

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 -4.0 (* C A) (* B_m B_m)))
        (t_1 (sqrt (* F 2.0)))
        (t_2 (fma (* C A) -4.0 (* B_m B_m)))
        (t_3 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* t_3 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          (- t_3)))
        (t_5 (sqrt (+ (+ (hypot B_m (- A C)) A) C))))
   (if (<= t_4 -5e-196)
     (* (/ t_5 -1.0) (* t_1 (/ (sqrt t_2) t_2)))
     (if (<= t_4 1e+228)
       (/
        (sqrt (* (* (* t_2 2.0) F) (+ (fma (/ (* B_m B_m) A) -0.5 C) C)))
        (- t_2))
       (if (<= t_4 INFINITY)
         (* (/ (sqrt (* (* 2.0 F) t_0)) -1.0) (/ t_5 t_0))
         (* (- (sqrt (+ (+ C A) (hypot (- A C) B_m)))) (/ t_1 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(-4.0, (C * A), (B_m * B_m));
	double t_1 = sqrt((F * 2.0));
	double t_2 = fma((C * A), -4.0, (B_m * B_m));
	double t_3 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_4 = sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / -t_3;
	double t_5 = sqrt(((hypot(B_m, (A - C)) + A) + C));
	double tmp;
	if (t_4 <= -5e-196) {
		tmp = (t_5 / -1.0) * (t_1 * (sqrt(t_2) / t_2));
	} else if (t_4 <= 1e+228) {
		tmp = sqrt((((t_2 * 2.0) * F) * (fma(((B_m * B_m) / A), -0.5, C) + C))) / -t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(((2.0 * F) * t_0)) / -1.0) * (t_5 / t_0);
	} else {
		tmp = -sqrt(((C + A) + hypot((A - C), B_m))) * (t_1 / 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(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = sqrt(Float64(F * 2.0))
	t_2 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_3 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / Float64(-t_3))
	t_5 = sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C))
	tmp = 0.0
	if (t_4 <= -5e-196)
		tmp = Float64(Float64(t_5 / -1.0) * Float64(t_1 * Float64(sqrt(t_2) / t_2)));
	elseif (t_4 <= 1e+228)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_2 * 2.0) * F) * Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C))) / Float64(-t_2));
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) / -1.0) * Float64(t_5 / t_0));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + A) + hypot(Float64(A - C), B_m)))) * Float64(t_1 / 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -5e-196], N[(N[(t$95$5 / -1.0), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[t$95$2], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e+228], N[(N[Sqrt[N[(N[(N[(t$95$2 * 2.0), $MachinePrecision] * F), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$2)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(t$95$5 / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[(t$95$1 / B$95$m), $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))) < -5.0000000000000005e-196

   1. Initial program 34.8%

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

   4. Applied rewrites 60.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift-sqrt.f64 N/A

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

      16. pow1/2 N/A

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

      17. lift-*.f64 N/A

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

      18. unpow-prod-down N/A

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

   6. Applied rewrites 77.0%

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

   if -5.0000000000000005e-196 < (/.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))) < 9.9999999999999992e227

   1. Initial program 25.4%

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

   4. Applied rewrites 26.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 26.1

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

   7. Applied rewrites 26.1%

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

   9. Applied rewrites 33.5%

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

   if 9.9999999999999992e227 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 3.7%

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

   4. Applied rewrites 68.1%

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

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

   1. Initial program 0.0%

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 20.0

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

   7. Applied rewrites 20.0%

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

   9. Applied rewrites 20.6%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -5 \cdot 10^{-196}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{-1} \cdot \left(\sqrt{F \cdot 2} \cdot \frac{\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 10^{+228}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_1 \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\right)}}{-t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+119}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\_m\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-171)
     (/
      (sqrt (* (* (* t_1 2.0) F) (+ (fma (/ (* B_m B_m) A) -0.5 C) C)))
      (- t_1))
     (if (<= (pow B_m 2.0) 4e+119)
       (*
        (/ (sqrt (+ (+ (hypot B_m (- A C)) A) C)) -1.0)
        (/ (sqrt (* (* 2.0 F) t_0)) t_0))
       (*
        (- (sqrt (+ (+ C A) (hypot (- A C) B_m))))
        (/ (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) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-171) {
		tmp = sqrt((((t_1 * 2.0) * F) * (fma(((B_m * B_m) / A), -0.5, C) + C))) / -t_1;
	} else if (pow(B_m, 2.0) <= 4e+119) {
		tmp = (sqrt(((hypot(B_m, (A - C)) + A) + C)) / -1.0) * (sqrt(((2.0 * F) * t_0)) / t_0);
	} else {
		tmp = -sqrt(((C + A) + hypot((A - C), B_m))) * (sqrt((F * 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(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-171)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_1 * 2.0) * F) * Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C))) / Float64(-t_1));
	elseif ((B_m ^ 2.0) <= 4e+119)
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / -1.0) * Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) / t_0));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + A) + hypot(Float64(A - C), B_m)))) * Float64(sqrt(Float64(F * 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-171], N[(N[Sqrt[N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] * F), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+119], N[(N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-172

   1. Initial program 16.6%

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   9. Applied rewrites 22.8%

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

   if 9.9999999999999998e-172 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999978e119

   1. Initial program 36.5%

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

   4. Applied rewrites 61.9%

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

   if 3.99999999999999978e119 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.6%

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

   4. Applied rewrites 21.6%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 28.6

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

   7. Applied rewrites 28.6%

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

   9. Applied rewrites 28.9%

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

Alternative 7: 54.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_1 \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\right)}}{-t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+119}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot t\_0}}{-1} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\_m\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-171)
     (/
      (sqrt (* (* (* t_1 2.0) F) (+ (fma (/ (* B_m B_m) A) -0.5 C) C)))
      (- t_1))
     (if (<= (pow B_m 2.0) 4e+119)
       (*
        (/ (sqrt (* (* 2.0 F) t_0)) -1.0)
        (/ (sqrt (+ (+ (hypot B_m (- A C)) A) C)) t_0))
       (*
        (- (sqrt (+ (+ C A) (hypot (- A C) B_m))))
        (/ (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) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-171) {
		tmp = sqrt((((t_1 * 2.0) * F) * (fma(((B_m * B_m) / A), -0.5, C) + C))) / -t_1;
	} else if (pow(B_m, 2.0) <= 4e+119) {
		tmp = (sqrt(((2.0 * F) * t_0)) / -1.0) * (sqrt(((hypot(B_m, (A - C)) + A) + C)) / t_0);
	} else {
		tmp = -sqrt(((C + A) + hypot((A - C), B_m))) * (sqrt((F * 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(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-171)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_1 * 2.0) * F) * Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C))) / Float64(-t_1));
	elseif ((B_m ^ 2.0) <= 4e+119)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) / -1.0) * Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) / t_0));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + A) + hypot(Float64(A - C), B_m)))) * Float64(sqrt(Float64(F * 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-171], N[(N[Sqrt[N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] * F), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+119], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-172

   1. Initial program 16.6%

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   9. Applied rewrites 22.8%

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

   if 9.9999999999999998e-172 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999978e119

   1. Initial program 36.5%

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

   4. Applied rewrites 61.9%

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

   if 3.99999999999999978e119 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.6%

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

   4. Applied rewrites 21.6%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 28.6

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

   7. Applied rewrites 28.6%

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

   9. Applied rewrites 28.9%

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

Alternative 8: 54.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_2 := -t\_1\\ t_3 := \left(t\_1 \cdot 2\right) \cdot F\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-171}:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+64}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 + C\right) + A} \cdot \sqrt{t\_3}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + t\_0}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot (- A C) B_m))
        (t_1 (fma (* C A) -4.0 (* B_m B_m)))
        (t_2 (- t_1))
        (t_3 (* (* t_1 2.0) F)))
   (if (<= (pow B_m 2.0) 1e-171)
     (/ (sqrt (* t_3 (+ (fma (/ (* B_m B_m) A) -0.5 C) C))) t_2)
     (if (<= (pow B_m 2.0) 1e+64)
       (/ (* (sqrt (+ (+ t_0 C) A)) (sqrt t_3)) t_2)
       (* (- (sqrt (+ (+ C A) t_0))) (/ (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) {
	double t_0 = hypot((A - C), B_m);
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double t_2 = -t_1;
	double t_3 = (t_1 * 2.0) * F;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-171) {
		tmp = sqrt((t_3 * (fma(((B_m * B_m) / A), -0.5, C) + C))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e+64) {
		tmp = (sqrt(((t_0 + C) + A)) * sqrt(t_3)) / t_2;
	} else {
		tmp = -sqrt(((C + A) + t_0)) * (sqrt((F * 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 = hypot(Float64(A - C), B_m)
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_2 = Float64(-t_1)
	t_3 = Float64(Float64(t_1 * 2.0) * F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-171)
		tmp = Float64(sqrt(Float64(t_3 * Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e+64)
		tmp = Float64(Float64(sqrt(Float64(Float64(t_0 + C) + A)) * sqrt(t_3)) / t_2);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + A) + t_0))) * Float64(sqrt(Float64(F * 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[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[(N[(t$95$1 * 2.0), $MachinePrecision] * F), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-171], N[(N[Sqrt[N[(t$95$3 * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+64], N[(N[(N[Sqrt[N[(N[(t$95$0 + C), $MachinePrecision] + A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-172

   1. Initial program 16.6%

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   9. Applied rewrites 22.8%

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

   if 9.9999999999999998e-172 < (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e64

   1. Initial program 36.9%

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

   4. Applied rewrites 67.9%

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

   6. Applied rewrites 67.5%

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

   if 1.00000000000000002e64 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.6%

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

   4. Applied rewrites 22.6%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 27.4

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

   7. Applied rewrites 27.4%

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

   9. Applied rewrites 27.7%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+64}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + C\right) + A} \cdot \sqrt{\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(C + A\right) + \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_1 \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\right)}}{-t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(-\sqrt{t\_1 \cdot F}\right) \cdot \sqrt{2 \cdot t\_0}}{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{t\_0}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (+ C A) (hypot (- A C) B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-171)
     (/
      (sqrt (* (* (* t_1 2.0) F) (+ (fma (/ (* B_m B_m) A) -0.5 C) C)))
      (- t_1))
     (if (<= (pow B_m 2.0) 2e+78)
       (/
        (* (- (sqrt (* t_1 F))) (sqrt (* 2.0 t_0)))
        (fma -4.0 (* C A) (* B_m B_m)))
       (* (- (sqrt t_0)) (/ (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) {
	double t_0 = (C + A) + hypot((A - C), B_m);
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-171) {
		tmp = sqrt((((t_1 * 2.0) * F) * (fma(((B_m * B_m) / A), -0.5, C) + C))) / -t_1;
	} else if (pow(B_m, 2.0) <= 2e+78) {
		tmp = (-sqrt((t_1 * F)) * sqrt((2.0 * t_0))) / fma(-4.0, (C * A), (B_m * B_m));
	} else {
		tmp = -sqrt(t_0) * (sqrt((F * 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 = Float64(Float64(C + A) + hypot(Float64(A - C), B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-171)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_1 * 2.0) * F) * Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C))) / Float64(-t_1));
	elseif ((B_m ^ 2.0) <= 2e+78)
		tmp = Float64(Float64(Float64(-sqrt(Float64(t_1 * F))) * sqrt(Float64(2.0 * t_0))) / fma(-4.0, Float64(C * A), Float64(B_m * B_m)));
	else
		tmp = Float64(Float64(-sqrt(t_0)) * Float64(sqrt(Float64(F * 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[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-171], N[(N[Sqrt[N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] * F), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+78], N[(N[((-N[Sqrt[N[(t$95$1 * F), $MachinePrecision]], $MachinePrecision]) * N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[t$95$0], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-172

   1. Initial program 16.6%

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

   4. Applied rewrites 24.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 19.9

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

   7. Applied rewrites 19.9%

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

   9. Applied rewrites 22.8%

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

   if 9.9999999999999998e-172 < (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000002e78

   1. Initial program 38.9%

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 67.0%

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

   if 2.00000000000000002e78 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.1%

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

   4. Applied rewrites 21.4%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 27.3

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

   7. Applied rewrites 27.3%

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

   9. Applied rewrites 27.5%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(-\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2 \cdot \left(\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-169}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_1 \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\right)}}{-t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\_m\right)}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-169)
     (/
      (sqrt (* (* (* t_1 2.0) F) (+ (fma (/ (* B_m B_m) A) -0.5 C) C)))
      (- t_1))
     (if (<= (pow B_m 2.0) 2e-22)
       (* (/ (sqrt (* 2.0 C)) -1.0) (/ (sqrt (* (* 2.0 F) t_0)) t_0))
       (*
        (- (sqrt (+ (+ C A) (hypot (- A C) B_m))))
        (/ (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) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-169) {
		tmp = sqrt((((t_1 * 2.0) * F) * (fma(((B_m * B_m) / A), -0.5, C) + C))) / -t_1;
	} else if (pow(B_m, 2.0) <= 2e-22) {
		tmp = (sqrt((2.0 * C)) / -1.0) * (sqrt(((2.0 * F) * t_0)) / t_0);
	} else {
		tmp = -sqrt(((C + A) + hypot((A - C), B_m))) * (sqrt((F * 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(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-169)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_1 * 2.0) * F) * Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C))) / Float64(-t_1));
	elseif ((B_m ^ 2.0) <= 2e-22)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) / -1.0) * Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) / t_0));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(C + A) + hypot(Float64(A - C), B_m)))) * Float64(sqrt(Float64(F * 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-169], N[(N[Sqrt[N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] * F), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-22], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e-169

   1. Initial program 17.4%

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 19.7

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

   7. Applied rewrites 19.7%

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

   9. Applied rewrites 22.5%

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

   if 1.00000000000000002e-169 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-22

   1. Initial program 25.7%

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

   4. Applied rewrites 61.4%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

   if 2.0000000000000001e-22 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.5%

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

   4. Applied rewrites 29.1%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 28.0

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

   7. Applied rewrites 28.0%

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

   9. Applied rewrites 28.3%

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

Alternative 11: 51.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-169}:\\ \;\;\;\;\frac{\sqrt{\left(\left(t\_1 \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C\right) + C\right)}}{-t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 -4.0 (* C A) (* B_m B_m)))
        (t_1 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e-169)
     (/
      (sqrt (* (* (* t_1 2.0) F) (+ (fma (/ (* B_m B_m) A) -0.5 C) C)))
      (- t_1))
     (if (<= (pow B_m 2.0) 2e-7)
       (* (/ (sqrt (* 2.0 C)) -1.0) (/ (sqrt (* (* 2.0 F) t_0)) t_0))
       (/ (- (sqrt F)) (sqrt (* B_m 0.5)))))))

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(-4.0, (C * A), (B_m * B_m));
	double t_1 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-169) {
		tmp = sqrt((((t_1 * 2.0) * F) * (fma(((B_m * B_m) / A), -0.5, C) + C))) / -t_1;
	} else if (pow(B_m, 2.0) <= 2e-7) {
		tmp = (sqrt((2.0 * C)) / -1.0) * (sqrt(((2.0 * F) * t_0)) / t_0);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-169)
		tmp = Float64(sqrt(Float64(Float64(Float64(t_1 * 2.0) * F) * Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, C) + C))) / Float64(-t_1));
	elseif ((B_m ^ 2.0) <= 2e-7)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) / -1.0) * Float64(sqrt(Float64(Float64(2.0 * F) * t_0)) / t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-169], N[(N[Sqrt[N[(N[(N[(t$95$1 * 2.0), $MachinePrecision] * F), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-7], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000002e-169

   1. Initial program 17.4%

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

   4. Applied rewrites 25.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 19.7

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

   7. Applied rewrites 19.7%

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

   9. Applied rewrites 22.5%

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

   if 1.00000000000000002e-169 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-7

   1. Initial program 27.2%

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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 21.5

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

   7. Applied rewrites 21.5%

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

   if 1.9999999999999999e-7 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 16.7%

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

   9. Applied rewrites 16.7%

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

   11. Applied rewrites 18.7%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-169}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right) \cdot F\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C\right) + C\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := \left(2 \cdot F\right) \cdot t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot t\_1}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{t\_1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 -4.0 (* C A) (* B_m B_m))) (t_1 (* (* 2.0 F) t_0)))
   (if (<= (pow B_m 2.0) 1e-171)
     (/ (sqrt (* (* 2.0 C) t_1)) (- t_0))
     (if (<= (pow B_m 2.0) 2e-7)
       (* (/ (sqrt (* 2.0 C)) -1.0) (/ (sqrt t_1) t_0))
       (/ (- (sqrt F)) (sqrt (* B_m 0.5)))))))

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(-4.0, (C * A), (B_m * B_m));
	double t_1 = (2.0 * F) * t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-171) {
		tmp = sqrt(((2.0 * C) * t_1)) / -t_0;
	} else if (pow(B_m, 2.0) <= 2e-7) {
		tmp = (sqrt((2.0 * C)) / -1.0) * (sqrt(t_1) / t_0);
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-171)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * t_1)) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 2e-7)
		tmp = Float64(Float64(sqrt(Float64(2.0 * C)) / -1.0) * Float64(sqrt(t_1) / t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-171], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-7], N[(N[(N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision] * N[(N[Sqrt[t$95$1], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e-172

   1. Initial program 16.6%

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

   4. Applied rewrites 22.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 22.8

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

   7. Applied rewrites 22.8%

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

   if 9.9999999999999998e-172 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-7

   1. Initial program 29.5%

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

   4. Applied rewrites 61.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 20.9

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

   7. Applied rewrites 20.9%

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

   if 1.9999999999999999e-7 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 16.7%

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

   9. Applied rewrites 16.7%

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

   11. Applied rewrites 18.7%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-171}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{2 \cdot C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \end{array} \end{array} \]

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 -4.0 (* C A) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-7)
     (/ (sqrt (* (* 2.0 C) (* (* 2.0 F) t_0))) (- t_0))
     (/ (- (sqrt F)) (sqrt (* B_m 0.5))))))

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(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-7) {
		tmp = sqrt(((2.0 * C) * ((2.0 * F) * t_0))) / -t_0;
	} else {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	}
	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(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-7)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-7], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-7

   1. Initial program 19.7%

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

   4. Applied rewrites 26.6%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 19.7

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

   7. Applied rewrites 19.7%

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

   if 1.9999999999999999e-7 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 16.7%

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

   9. Applied rewrites 16.7%

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

   11. Applied rewrites 18.7%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 1.1e-11)
   (/ (sqrt (* -16.0 (* A (* (* C C) F)))) (- (fma -4.0 (* C A) (* B_m B_m))))
   (/ (sqrt (* F 2.0)) (- (sqrt 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 <= 1.1e-11) {
		tmp = sqrt((-16.0 * (A * ((C * C) * F)))) / -fma(-4.0, (C * A), (B_m * B_m));
	} else {
		tmp = sqrt((F * 2.0)) / -sqrt(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 <= 1.1e-11)
		tmp = Float64(sqrt(Float64(-16.0 * Float64(A * Float64(Float64(C * C) * F)))) / Float64(-fma(-4.0, Float64(C * A), Float64(B_m * B_m))));
	else
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(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, 1.1e-11], N[(N[Sqrt[N[(-16.0 * N[(A * N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.1000000000000001e-11

   1. Initial program 17.9%

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

   4. Applied rewrites 23.7%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 9.7

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

   7. Applied rewrites 9.7%

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

   if 1.1000000000000001e-11 < B

   1. Initial program 14.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 41.8

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

   5. Applied rewrites 41.8%

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

   7. Applied rewrites 48.5%

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

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.5% accurate, 8.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 3.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2 \cdot C}\right) \cdot \frac{\sqrt{F \cdot 2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 3.4e+135)
   (/ (- (sqrt F)) (sqrt (* B_m 0.5)))
   (* (- (sqrt (* 2.0 C))) (/ (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) {
	double tmp;
	if (C <= 3.4e+135) {
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	} else {
		tmp = -sqrt((2.0 * C)) * (sqrt((F * 2.0)) / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 3.4d+135) then
        tmp = -sqrt(f) / sqrt((b_m * 0.5d0))
    else
        tmp = -sqrt((2.0d0 * c)) * (sqrt((f * 2.0d0)) / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 3.4e+135) {
		tmp = -Math.sqrt(F) / Math.sqrt((B_m * 0.5));
	} else {
		tmp = -Math.sqrt((2.0 * C)) * (Math.sqrt((F * 2.0)) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 3.4e+135:
		tmp = -math.sqrt(F) / math.sqrt((B_m * 0.5))
	else:
		tmp = -math.sqrt((2.0 * C)) * (math.sqrt((F * 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 (C <= 3.4e+135)
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)));
	else
		tmp = Float64(Float64(-sqrt(Float64(2.0 * C))) * Float64(sqrt(Float64(F * 2.0)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 3.4e+135)
		tmp = -sqrt(F) / sqrt((B_m * 0.5));
	else
		tmp = -sqrt((2.0 * C)) * (sqrt((F * 2.0)) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 3.4e+135], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]) * N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 3.4000000000000001e135

   1. Initial program 19.0%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 12.1

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

   5. Applied rewrites 12.1%

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

   7. Applied rewrites 12.2%

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

   9. Applied rewrites 12.2%

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

   11. Applied rewrites 13.3%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]

   if 3.4000000000000001e135 < C

   1. Initial program 7.2%

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

   4. Applied rewrites 35.6%

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

   5. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 22.8

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

   7. Applied rewrites 22.8%

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

   9. Applied rewrites 22.8%

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

   10. Taylor expanded in A around -inf

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

       1. lower-*.f64 20.5

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

   12. Applied rewrites 20.5%

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

4. Final simplification 14.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3.4 \cdot 10^{+135}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2 \cdot C}\right) \cdot \frac{\sqrt{F \cdot 2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-\sqrt{F}}{\sqrt{B\_m \cdot 0.5}} \end{array} \]

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)) (sqrt (* B_m 0.5))))

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) / sqrt((B_m * 0.5));
}

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) / sqrt((b_m * 0.5d0))
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) / Math.sqrt((B_m * 0.5));
}

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) / math.sqrt((B_m * 0.5))

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(Float64(-sqrt(F)) / sqrt(Float64(B_m * 0.5)))
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) / sqrt((B_m * 0.5));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(B$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 11.1

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

5. Applied rewrites 11.1%

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

7. Applied rewrites 11.1%

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

9. Applied rewrites 11.1%

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

11. Applied rewrites 12.1%

    \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]

12. Final simplification 12.1%

    \[\leadsto \frac{-\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
13. Add Preprocessing

Alternative 17: 35.2% accurate, 12.6× speedup

Math

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

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)) (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) {
	return -sqrt(F) * sqrt((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) * sqrt((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) * Math.sqrt((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) * math.sqrt((2.0 / B_m))

Julia

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

Wolfram

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

Derivation

1. Initial program 17.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 11.1

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

5. Applied rewrites 11.1%

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

7. Applied rewrites 11.1%

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

9. Applied rewrites 12.1%

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

10. Final simplification 12.1%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 18: 27.2% accurate, 16.9× 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 Float64(-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 17.3%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 11.1

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

5. Applied rewrites 11.1%

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

7. Applied rewrites 11.1%

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

9. Applied rewrites 11.1%

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

Reproduce

herbie shell --seed 2024322 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))