ABCF->ab-angle b

Percentage Accurate: 18.3% → 57.4%

Time: 17.4s

Alternatives: 15

Speedup: 12.3×

• Could not converge on a ground truth

  1. A = -9.795469909002208e+88
  2. B = 1.7334934404164132e-293
  3. C = 2.2614478571167134e+238
  4. F = -3.2114752548772036e+94

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_2 := -\sqrt{2}\\ t_3 := C \cdot \left(A \cdot 4\right)\\ t_4 := t\_3 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{\left(\left(t\_4 \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} - \left(C + A\right)\right)}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - t\_0}{t\_1} \cdot F} \cdot t\_2\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(C, C, -{t\_0}^{2}\right), \frac{1}{t\_0 + C}, A\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_3\right)\right) \cdot 2\right)}}{t\_4}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_1\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{F}\right), \log \left(\mathsf{hypot}\left(C, B\_m\right) - C\right)\right)} \cdot \frac{t\_2}{B\_m}\\ \end{array} \end{array} \]

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 (* -4.0 A) C (* B_m B_m)))
        (t_2 (- (sqrt 2.0)))
        (t_3 (* C (* A 4.0)))
        (t_4 (- t_3 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt
           (*
            (* (* t_4 F) 2.0)
            (- (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))))
          t_4)))
   (if (<= t_5 (- INFINITY))
     (* (sqrt (* (/ (- (+ C A) t_0) t_1) F)) t_2)
     (if (<= t_5 -1e-214)
       (/
        (sqrt
         (*
          (fma (fma C C (- (pow t_0 2.0))) (/ 1.0 (+ t_0 C)) A)
          (* (* F (- (pow B_m 2.0) t_3)) 2.0)))
        t_4)
       (if (<= t_5 INFINITY)
         (/
          (sqrt (* (* (* F 2.0) t_1) (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))
          (- t_1))
         (*
          (exp (* 0.5 (fma -1.0 (log (/ -1.0 F)) (log (- (hypot C B_m) C)))))
          (/ t_2 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((-4.0 * A), C, (B_m * B_m));
	double t_2 = -sqrt(2.0);
	double t_3 = C * (A * 4.0);
	double t_4 = t_3 - pow(B_m, 2.0);
	double t_5 = sqrt((((t_4 * F) * 2.0) * (sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) - (C + A)))) / t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = sqrt(((((C + A) - t_0) / t_1) * F)) * t_2;
	} else if (t_5 <= -1e-214) {
		tmp = sqrt((fma(fma(C, C, -pow(t_0, 2.0)), (1.0 / (t_0 + C)), A) * ((F * (pow(B_m, 2.0) - t_3)) * 2.0))) / t_4;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((((F * 2.0) * t_1) * (fma(-0.5, ((B_m * B_m) / C), A) + A))) / -t_1;
	} else {
		tmp = exp((0.5 * fma(-1.0, log((-1.0 / F)), log((hypot(C, B_m) - C))))) * (t_2 / 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(-4.0 * A), C, Float64(B_m * B_m))
	t_2 = Float64(-sqrt(2.0))
	t_3 = Float64(C * Float64(A * 4.0))
	t_4 = Float64(t_3 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(Float64(Float64(t_4 * F) * 2.0) * Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) - Float64(C + A)))) / t_4)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - t_0) / t_1) * F)) * t_2);
	elseif (t_5 <= -1e-214)
		tmp = Float64(sqrt(Float64(fma(fma(C, C, Float64(-(t_0 ^ 2.0))), Float64(1.0 / Float64(t_0 + C)), A) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_3)) * 2.0))) / t_4);
	elseif (t_5 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_1) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A))) / Float64(-t_1));
	else
		tmp = Float64(exp(Float64(0.5 * fma(-1.0, log(Float64(-1.0 / F)), log(Float64(hypot(C, B_m) - C))))) * Float64(t_2 / 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[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$3 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(N[(t$95$4 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$5, -1e-214], N[(N[Sqrt[N[(N[(N[(C * C + (-N[Power[t$95$0, 2.0], $MachinePrecision])), $MachinePrecision] * N[(1.0 / N[(t$95$0 + C), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[Exp[N[(0.5 * N[(-1.0 * N[Log[N[(-1.0 / F), $MachinePrecision]], $MachinePrecision] + N[Log[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] - C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 / 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))) < -inf.0

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 56.3%

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

   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))) < -9.99999999999999913e-215

   1. Initial program 99.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. flip-- N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 82.4%

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

   if -9.99999999999999913e-215 < (/.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 14.5%

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

   4. Applied rewrites 21.8%

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

   6. Applied rewrites 21.8%

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

   8. Applied rewrites 21.8%

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

   9. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 25.9

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

   11. Applied rewrites 25.9%

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

   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 0.0%

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

   5. Taylor expanded in A around 0

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

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 16.5

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

   7. Applied rewrites 16.5%

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

   9. Applied rewrites 15.6%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\log \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot F\right) \cdot 0.5} \]

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 22.1%

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\mathsf{fma}\left(-1, \log \left(\frac{-1}{F}\right), \log \left(-\left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(C, C, -{\left(\mathsf{hypot}\left(A - C, B\right)\right)}^{2}\right), \frac{1}{\mathsf{hypot}\left(A - C, B\right) + C}, A\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{F}\right), \log \left(\mathsf{hypot}\left(C, B\right) - C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := -\sqrt{2}\\ t_2 := C \cdot \left(A \cdot 4\right) - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(\left(t\_2 \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} - \left(C + A\right)\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_5 := \left(t\_4 \cdot 2\right) \cdot F\\ t_6 := -t\_4\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - t\_0}{t\_4} \cdot F} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(t\_5, C, \left(A - t\_0\right) \cdot t\_5\right)}}{t\_6}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_4\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{F}\right), \log \left(\mathsf{hypot}\left(C, B\_m\right) - C\right)\right)} \cdot \frac{t\_1}{B\_m}\\ \end{array} \end{array} \]

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 (hypot (- A C) B_m))
        (t_1 (- (sqrt 2.0)))
        (t_2 (- (* C (* A 4.0)) (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* (* t_2 F) 2.0)
            (- (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))))
          t_2))
        (t_4 (fma (* -4.0 A) C (* B_m B_m)))
        (t_5 (* (* t_4 2.0) F))
        (t_6 (- t_4)))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (* (/ (- (+ C A) t_0) t_4) F)) t_1)
     (if (<= t_3 -1e-214)
       (/ (sqrt (fma t_5 C (* (- A t_0) t_5))) t_6)
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (* (* F 2.0) t_4) (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))
          t_6)
         (*
          (exp (* 0.5 (fma -1.0 (log (/ -1.0 F)) (log (- (hypot C B_m) C)))))
          (/ 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 = hypot((A - C), B_m);
	double t_1 = -sqrt(2.0);
	double t_2 = (C * (A * 4.0)) - pow(B_m, 2.0);
	double t_3 = sqrt((((t_2 * F) * 2.0) * (sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) - (C + A)))) / t_2;
	double t_4 = fma((-4.0 * A), C, (B_m * B_m));
	double t_5 = (t_4 * 2.0) * F;
	double t_6 = -t_4;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(((((C + A) - t_0) / t_4) * F)) * t_1;
	} else if (t_3 <= -1e-214) {
		tmp = sqrt(fma(t_5, C, ((A - t_0) * t_5))) / t_6;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((F * 2.0) * t_4) * (fma(-0.5, ((B_m * B_m) / C), A) + A))) / t_6;
	} else {
		tmp = exp((0.5 * fma(-1.0, log((-1.0 / F)), log((hypot(C, B_m) - C))))) * (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 = hypot(Float64(A - C), B_m)
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(Float64(C * Float64(A * 4.0)) - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(Float64(t_2 * F) * 2.0) * Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) - Float64(C + A)))) / t_2)
	t_4 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_5 = Float64(Float64(t_4 * 2.0) * F)
	t_6 = Float64(-t_4)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - t_0) / t_4) * F)) * t_1);
	elseif (t_3 <= -1e-214)
		tmp = Float64(sqrt(fma(t_5, C, Float64(Float64(A - t_0) * t_5))) / t_6);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_4) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A))) / t_6);
	else
		tmp = Float64(exp(Float64(0.5 * fma(-1.0, log(Float64(-1.0 / F)), log(Float64(hypot(C, B_m) - C))))) * 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[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(t$95$2 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * 2.0), $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$6 = (-t$95$4)}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$4), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-214], N[(N[Sqrt[N[(t$95$5 * C + N[(N[(A - t$95$0), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], N[(N[Exp[N[(0.5 * N[(-1.0 * N[Log[N[(-1.0 / F), $MachinePrecision]], $MachinePrecision] + N[Log[N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] - C), $MachinePrecision]], $MachinePrecision]), $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))) < -inf.0

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 56.3%

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

   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))) < -9.99999999999999913e-215

   1. Initial program 99.2%

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 99.3%

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

   if -9.99999999999999913e-215 < (/.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 14.5%

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

   4. Applied rewrites 21.8%

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

   6. Applied rewrites 21.8%

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

   8. Applied rewrites 21.8%

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

   9. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 25.9

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

   11. Applied rewrites 25.9%

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

   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 0.0%

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

   5. Taylor expanded in A around 0

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

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 16.5

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

   7. Applied rewrites 16.5%

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

   9. Applied rewrites 15.6%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\log \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot F\right) \cdot 0.5} \]

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 22.1%

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\mathsf{fma}\left(-1, \log \left(\frac{-1}{F}\right), \log \left(-\left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2\right) \cdot F, C, \left(A - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \mathsf{fma}\left(-1, \log \left(\frac{-1}{F}\right), \log \left(\mathsf{hypot}\left(C, B\right) - C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := -\sqrt{2}\\ t_2 := C \cdot \left(A \cdot 4\right) - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(\left(t\_2 \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} - \left(C + A\right)\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_5 := \left(t\_4 \cdot 2\right) \cdot F\\ t_6 := -t\_4\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - t\_0}{t\_4} \cdot F} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(t\_5, C, \left(A - t\_0\right) \cdot t\_5\right)}}{t\_6}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_4\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-1, \frac{C}{B\_m} - \log B\_m, \log \left(-F\right)\right) \cdot 0.5} \cdot \frac{t\_1}{B\_m}\\ \end{array} \end{array} \]

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 (hypot (- A C) B_m))
        (t_1 (- (sqrt 2.0)))
        (t_2 (- (* C (* A 4.0)) (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* (* t_2 F) 2.0)
            (- (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))))
          t_2))
        (t_4 (fma (* -4.0 A) C (* B_m B_m)))
        (t_5 (* (* t_4 2.0) F))
        (t_6 (- t_4)))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (* (/ (- (+ C A) t_0) t_4) F)) t_1)
     (if (<= t_3 -1e-214)
       (/ (sqrt (fma t_5 C (* (- A t_0) t_5))) t_6)
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (* (* F 2.0) t_4) (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))
          t_6)
         (*
          (exp (* (fma -1.0 (- (/ C B_m) (log B_m)) (log (- F))) 0.5))
          (/ 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 = hypot((A - C), B_m);
	double t_1 = -sqrt(2.0);
	double t_2 = (C * (A * 4.0)) - pow(B_m, 2.0);
	double t_3 = sqrt((((t_2 * F) * 2.0) * (sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) - (C + A)))) / t_2;
	double t_4 = fma((-4.0 * A), C, (B_m * B_m));
	double t_5 = (t_4 * 2.0) * F;
	double t_6 = -t_4;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(((((C + A) - t_0) / t_4) * F)) * t_1;
	} else if (t_3 <= -1e-214) {
		tmp = sqrt(fma(t_5, C, ((A - t_0) * t_5))) / t_6;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((F * 2.0) * t_4) * (fma(-0.5, ((B_m * B_m) / C), A) + A))) / t_6;
	} else {
		tmp = exp((fma(-1.0, ((C / B_m) - log(B_m)), log(-F)) * 0.5)) * (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 = hypot(Float64(A - C), B_m)
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(Float64(C * Float64(A * 4.0)) - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(Float64(t_2 * F) * 2.0) * Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) - Float64(C + A)))) / t_2)
	t_4 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_5 = Float64(Float64(t_4 * 2.0) * F)
	t_6 = Float64(-t_4)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - t_0) / t_4) * F)) * t_1);
	elseif (t_3 <= -1e-214)
		tmp = Float64(sqrt(fma(t_5, C, Float64(Float64(A - t_0) * t_5))) / t_6);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_4) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A))) / t_6);
	else
		tmp = Float64(exp(Float64(fma(-1.0, Float64(Float64(C / B_m) - log(B_m)), log(Float64(-F))) * 0.5)) * 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[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(t$95$2 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * 2.0), $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$6 = (-t$95$4)}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$4), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-214], N[(N[Sqrt[N[(t$95$5 * C + N[(N[(A - t$95$0), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], N[(N[Exp[N[(N[(-1.0 * N[(N[(C / B$95$m), $MachinePrecision] - N[Log[B$95$m], $MachinePrecision]), $MachinePrecision] + N[Log[(-F)], $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -inf.0

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 56.3%

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

   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))) < -9.99999999999999913e-215

   1. Initial program 99.2%

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 99.3%

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

   if -9.99999999999999913e-215 < (/.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 14.5%

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

   4. Applied rewrites 21.8%

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

   6. Applied rewrites 21.8%

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

   8. Applied rewrites 21.8%

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

   9. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 25.9

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

   11. Applied rewrites 25.9%

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

   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 0.0%

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

   5. Taylor expanded in A around 0

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

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 16.5

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

   7. Applied rewrites 16.5%

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

   9. Applied rewrites 15.6%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\log \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot F\right) \cdot 0.5} \]

   10. Taylor expanded in B around inf

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

   12. Applied rewrites 19.4%

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\mathsf{fma}\left(-1, \left(-\log B\right) + \frac{C}{B}, \log \left(-F\right)\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2\right) \cdot F, C, \left(A - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-1, \frac{C}{B} - \log B, \log \left(-F\right)\right) \cdot 0.5} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := -\sqrt{2}\\ t_2 := C \cdot \left(A \cdot 4\right) - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(\left(t\_2 \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}} - \left(C + A\right)\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ t_5 := \left(t\_4 \cdot 2\right) \cdot F\\ t_6 := -t\_4\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - t\_0}{t\_4} \cdot F} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(t\_5, C, \left(A - t\_0\right) \cdot t\_5\right)}}{t\_6}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_4\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right)}}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-1, -\log B\_m, \log \left(-F\right)\right) \cdot 0.5} \cdot \frac{t\_1}{B\_m}\\ \end{array} \end{array} \]

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 (hypot (- A C) B_m))
        (t_1 (- (sqrt 2.0)))
        (t_2 (- (* C (* A 4.0)) (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (* (* t_2 F) 2.0)
            (- (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))) (+ C A))))
          t_2))
        (t_4 (fma (* -4.0 A) C (* B_m B_m)))
        (t_5 (* (* t_4 2.0) F))
        (t_6 (- t_4)))
   (if (<= t_3 (- INFINITY))
     (* (sqrt (* (/ (- (+ C A) t_0) t_4) F)) t_1)
     (if (<= t_3 -1e-214)
       (/ (sqrt (fma t_5 C (* (- A t_0) t_5))) t_6)
       (if (<= t_3 INFINITY)
         (/
          (sqrt (* (* (* F 2.0) t_4) (+ (fma -0.5 (/ (* B_m B_m) C) A) A)))
          t_6)
         (*
          (exp (* (fma -1.0 (- (log B_m)) (log (- F))) 0.5))
          (/ 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 = hypot((A - C), B_m);
	double t_1 = -sqrt(2.0);
	double t_2 = (C * (A * 4.0)) - pow(B_m, 2.0);
	double t_3 = sqrt((((t_2 * F) * 2.0) * (sqrt((pow((A - C), 2.0) + pow(B_m, 2.0))) - (C + A)))) / t_2;
	double t_4 = fma((-4.0 * A), C, (B_m * B_m));
	double t_5 = (t_4 * 2.0) * F;
	double t_6 = -t_4;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(((((C + A) - t_0) / t_4) * F)) * t_1;
	} else if (t_3 <= -1e-214) {
		tmp = sqrt(fma(t_5, C, ((A - t_0) * t_5))) / t_6;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((F * 2.0) * t_4) * (fma(-0.5, ((B_m * B_m) / C), A) + A))) / t_6;
	} else {
		tmp = exp((fma(-1.0, -log(B_m), log(-F)) * 0.5)) * (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 = hypot(Float64(A - C), B_m)
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(Float64(C * Float64(A * 4.0)) - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(Float64(t_2 * F) * 2.0) * Float64(sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))) - Float64(C + A)))) / t_2)
	t_4 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	t_5 = Float64(Float64(t_4 * 2.0) * F)
	t_6 = Float64(-t_4)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - t_0) / t_4) * F)) * t_1);
	elseif (t_3 <= -1e-214)
		tmp = Float64(sqrt(fma(t_5, C, Float64(Float64(A - t_0) * t_5))) / t_6);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(F * 2.0) * t_4) * Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A))) / t_6);
	else
		tmp = Float64(exp(Float64(fma(-1.0, Float64(-log(B_m)), log(Float64(-F))) * 0.5)) * 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[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(t$95$2 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(C + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * 2.0), $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$6 = (-t$95$4)}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - t$95$0), $MachinePrecision] / t$95$4), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -1e-214], N[(N[Sqrt[N[(t$95$5 * C + N[(N[(A - t$95$0), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision] * N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], N[(N[Exp[N[(N[(-1.0 * (-N[Log[B$95$m], $MachinePrecision]) + N[Log[(-F)], $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -inf.0

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 56.3%

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

   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))) < -9.99999999999999913e-215

   1. Initial program 99.2%

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

   4. Applied rewrites 98.9%

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 99.3%

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

   if -9.99999999999999913e-215 < (/.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 14.5%

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

   4. Applied rewrites 21.8%

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

   6. Applied rewrites 21.8%

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

   8. Applied rewrites 21.8%

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

   9. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lower-neg.f64 25.9

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

   11. Applied rewrites 25.9%

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

   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 0.0%

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

   5. Taylor expanded in A around 0

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

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 16.5

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

   7. Applied rewrites 16.5%

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

   9. Applied rewrites 15.6%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\log \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot F\right) \cdot 0.5} \]

   10. Taylor expanded in B around inf

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\left(\log \left(-1 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right) \cdot \frac{1}{2}} \]
   11. Step-by-step derivation

   12. Applied rewrites 18.8%

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot e^{\mathsf{fma}\left(-1, -\log B, \log \left(-F\right)\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2\right) \cdot F, C, \left(A - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot 2\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(\left(C \cdot \left(A \cdot 4\right) - {B}^{2}\right) \cdot F\right) \cdot 2\right) \cdot \left(\sqrt{{\left(A - C\right)}^{2} + {B}^{2}} - \left(C + A\right)\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right) \cdot \left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(-1, -\log B, \log \left(-F\right)\right) \cdot 0.5} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.2% accurate, 2.0× speedup

Math

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

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 A) C (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-58)
     (/ (sqrt (* (+ A A) (* (* F 2.0) t_0))) (- t_0))
     (/ (sqrt (* (* (- A (hypot A B_m)) 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 * A), C, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-58) {
		tmp = sqrt(((A + A) * ((F * 2.0) * t_0))) / -t_0;
	} else {
		tmp = sqrt((((A - hypot(A, B_m)) * 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(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-58)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(Float64(F * 2.0) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(A, B_m)) * F) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-58], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-58

   1. Initial program 20.8%

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 27.8%

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

   8. Applied rewrites 27.9%

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

   9. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 23.9

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

   11. Applied rewrites 23.9%

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

   if 2.0000000000000001e-58 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 23.2%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 24.2

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 24.3%

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

4. Final simplification 24.1%

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

Alternative 6: 49.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(F \cdot 2\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 5.8 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\_m\right)}{t\_0} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\_m\right)\right) \cdot F\right) \cdot 2}}{-B\_m}\\ \end{array} \end{array} \]

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 A) C (* B_m B_m))))
   (if (<= B_m 1.25e-29)
     (/ (sqrt (* (+ A A) (* (* F 2.0) t_0))) (- t_0))
     (if (<= B_m 5.8e+118)
       (* (sqrt (* (/ (- (+ C A) (hypot (- A C) B_m)) t_0) F)) (- (sqrt 2.0)))
       (/ (sqrt (* (* (- A (hypot A B_m)) 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 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 1.25e-29) {
		tmp = sqrt(((A + A) * ((F * 2.0) * t_0))) / -t_0;
	} else if (B_m <= 5.8e+118) {
		tmp = sqrt(((((C + A) - hypot((A - C), B_m)) / t_0) * F)) * -sqrt(2.0);
	} else {
		tmp = sqrt((((A - hypot(A, B_m)) * 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(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 1.25e-29)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(Float64(F * 2.0) * t_0))) / Float64(-t_0));
	elseif (B_m <= 5.8e+118)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B_m)) / t_0) * F)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(A - hypot(A, B_m)) * F) * 2.0)) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.25e-29], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 5.8e+118], N[(N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.24999999999999996e-29

   1. Initial program 22.9%

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

   4. Applied rewrites 27.5%

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

   6. Applied rewrites 27.5%

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

   8. Applied rewrites 27.6%

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

   9. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 16.6

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

   11. Applied rewrites 16.6%

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

   if 1.24999999999999996e-29 < B < 5.80000000000000032e118

   1. Initial program 38.5%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 52.7%

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

   if 5.80000000000000032e118 < B

   1. Initial program 5.5%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 52.0

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

   5. Applied rewrites 52.0%

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

   7. Applied rewrites 52.3%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot \left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 5.8 \cdot 10^{+118}:\\ \;\;\;\;\sqrt{\frac{\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot F\right) \cdot 2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.5% accurate, 2.7× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-58)
     (/ (sqrt (* (+ A A) (* (* F 2.0) t_0))) (- t_0))
     (/ (sqrt (fma -2.0 (* F B_m) (* (* F A) 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 * A), C, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-58) {
		tmp = sqrt(((A + A) * ((F * 2.0) * t_0))) / -t_0;
	} else {
		tmp = sqrt(fma(-2.0, (F * B_m), ((F * A) * 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(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-58)
		tmp = Float64(sqrt(Float64(Float64(A + A) * Float64(Float64(F * 2.0) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(F * B_m), Float64(Float64(F * A) * 2.0))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-58], N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(F * B$95$m), $MachinePrecision] + N[(N[(F * A), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-58

   1. Initial program 20.8%

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 27.8%

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

   8. Applied rewrites 27.9%

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

   9. Taylor expanded in C around inf

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

       1. lower--.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 23.9

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

   11. Applied rewrites 23.9%

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

   if 2.0000000000000001e-58 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 23.2%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 24.2

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 20.5%

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

4. Final simplification 21.8%

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

Alternative 8: 42.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}{\sqrt{\left(\left(\left(\left(A + A\right) \cdot C\right) \cdot F\right) \cdot A\right) \cdot -8}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B\_m, \left(F \cdot A\right) \cdot 2\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-58)
   (/
    -1.0
    (/
     (fma (* C A) -4.0 (* B_m B_m))
     (sqrt (* (* (* (* (+ A A) C) F) A) -8.0))))
   (/ (sqrt (fma -2.0 (* F B_m) (* (* F A) 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 (pow(B_m, 2.0) <= 2e-58) {
		tmp = -1.0 / (fma((C * A), -4.0, (B_m * B_m)) / sqrt((((((A + A) * C) * F) * A) * -8.0)));
	} else {
		tmp = sqrt(fma(-2.0, (F * B_m), ((F * A) * 2.0))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-58)
		tmp = Float64(-1.0 / Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) / sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * C) * F) * A) * -8.0))));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(F * B_m), Float64(Float64(F * A) * 2.0))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-58], N[(-1.0 / N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(F * B$95$m), $MachinePrecision] + N[(N[(F * A), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-58

   1. Initial program 20.8%

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 27.8%

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

   7. Taylor expanded in C around inf

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

      1. rem-square-sqrt N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}{\sqrt{\color{blue}{{\left(\sqrt{-8}\right)}^{2}} \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}{\sqrt{\color{blue}{{\left(\sqrt{-8}\right)}^{2} \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}} \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 19.0

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

   9. Applied rewrites 19.0%

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

   11. Applied rewrites 22.2%

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

   if 2.0000000000000001e-58 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 23.2%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 24.2

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 20.5%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}{\sqrt{\left(\left(\left(\left(A + A\right) \cdot C\right) \cdot F\right) \cdot A\right) \cdot -8}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B, \left(F \cdot A\right) \cdot 2\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\frac{\left(C \cdot A\right) \cdot -4}{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B\_m, \left(F \cdot A\right) \cdot 2\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-58)
   (/ -1.0 (/ (* (* C A) -4.0) (sqrt (* (* (* (* (+ A A) F) C) A) -8.0))))
   (/ (sqrt (fma -2.0 (* F B_m) (* (* F A) 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 (pow(B_m, 2.0) <= 2e-58) {
		tmp = -1.0 / (((C * A) * -4.0) / sqrt((((((A + A) * F) * C) * A) * -8.0)));
	} else {
		tmp = sqrt(fma(-2.0, (F * B_m), ((F * A) * 2.0))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-58)
		tmp = Float64(-1.0 / Float64(Float64(Float64(C * A) * -4.0) / sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0))));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(F * B_m), Float64(Float64(F * A) * 2.0))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-58], N[(-1.0 / N[(N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(F * B$95$m), $MachinePrecision] + N[(N[(F * A), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-58

   1. Initial program 20.8%

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 27.8%

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

   7. Taylor expanded in C around inf

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

      1. rem-square-sqrt N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}{\sqrt{\color{blue}{{\left(\sqrt{-8}\right)}^{2}} \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}{\sqrt{\color{blue}{{\left(\sqrt{-8}\right)}^{2} \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}} \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 19.0

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

   9. Applied rewrites 19.0%

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

   10. Taylor expanded in C around inf

       \[\leadsto \frac{-1}{\frac{\color{blue}{-4 \cdot \left(A \cdot C\right)}}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

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

       2. lower-*.f64 18.7

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

   12. Applied rewrites 18.7%

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

   if 2.0000000000000001e-58 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 23.2%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 24.2

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 20.5%

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

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{\frac{\left(C \cdot A\right) \cdot -4}{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B, \left(F \cdot A\right) \cdot 2\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B\_m, \left(F \cdot A\right) \cdot 2\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-58)
   (/
    (sqrt (* (* (* (* (+ A A) F) C) A) -8.0))
    (- (fma (* -4.0 A) C (* B_m B_m))))
   (/ (sqrt (fma -2.0 (* F B_m) (* (* F A) 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 (pow(B_m, 2.0) <= 2e-58) {
		tmp = sqrt((((((A + A) * F) * C) * A) * -8.0)) / -fma((-4.0 * A), C, (B_m * B_m));
	} else {
		tmp = sqrt(fma(-2.0, (F * B_m), ((F * A) * 2.0))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-58)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(Float64(A + A) * F) * C) * A) * -8.0)) / Float64(-fma(Float64(-4.0 * A), C, Float64(B_m * B_m))));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(F * B_m), Float64(Float64(F * A) * 2.0))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-58], N[(N[Sqrt[N[(N[(N[(N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision] * C), $MachinePrecision] * A), $MachinePrecision] * -8.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(F * B$95$m), $MachinePrecision] + N[(N[(F * A), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-58

   1. Initial program 20.8%

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 27.8%

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

   7. Taylor expanded in C around inf

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

      1. rem-square-sqrt N/A

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

      2. unpow2 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}{\sqrt{\color{blue}{{\left(\sqrt{-8}\right)}^{2}} \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}{\sqrt{\color{blue}{{\left(\sqrt{-8}\right)}^{2} \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}} \]

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 19.0

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

   9. Applied rewrites 19.0%

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

   11. Applied rewrites 19.0%

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

   if 2.0000000000000001e-58 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 23.2%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 24.2

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 20.5%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(\left(A + A\right) \cdot F\right) \cdot C\right) \cdot A\right) \cdot -8}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B, \left(F \cdot A\right) \cdot 2\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot C\right) \cdot \left(\left(A \cdot A\right) \cdot -16\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B\_m, \left(F \cdot A\right) \cdot 2\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-58)
   (/ (sqrt (* (* F C) (* (* A A) -16.0))) (- (fma (* -4.0 A) C (* B_m B_m))))
   (/ (sqrt (fma -2.0 (* F B_m) (* (* F A) 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 (pow(B_m, 2.0) <= 2e-58) {
		tmp = sqrt(((F * C) * ((A * A) * -16.0))) / -fma((-4.0 * A), C, (B_m * B_m));
	} else {
		tmp = sqrt(fma(-2.0, (F * B_m), ((F * A) * 2.0))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-58)
		tmp = Float64(sqrt(Float64(Float64(F * C) * Float64(Float64(A * A) * -16.0))) / Float64(-fma(Float64(-4.0 * A), C, Float64(B_m * B_m))));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(F * B_m), Float64(Float64(F * A) * 2.0))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-58], N[(N[Sqrt[N[(N[(F * C), $MachinePrecision] * N[(N[(A * A), $MachinePrecision] * -16.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(F * B$95$m), $MachinePrecision] + N[(N[(F * A), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-58

   1. Initial program 20.8%

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

   4. Applied rewrites 27.7%

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

   6. Applied rewrites 27.8%

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

   8. Applied rewrites 27.9%

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

   9. Taylor expanded in A around -inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 15.0

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

   11. Applied rewrites 15.0%

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

   if 2.0000000000000001e-58 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 23.2%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 24.2

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 24.3%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 20.5%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot C\right) \cdot \left(\left(A \cdot A\right) \cdot -16\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B, \left(F \cdot A\right) \cdot 2\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -3.4 \cdot 10^{+147}:\\ \;\;\;\;\frac{-2}{B\_m} \cdot \sqrt{F \cdot A}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B\_m, \left(F \cdot A\right) \cdot 2\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -3.4e+147)
   (* (/ -2.0 B_m) (sqrt (* F A)))
   (/ (sqrt (fma -2.0 (* F B_m) (* (* F A) 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 (A <= -3.4e+147) {
		tmp = (-2.0 / B_m) * sqrt((F * A));
	} else {
		tmp = sqrt(fma(-2.0, (F * B_m), ((F * A) * 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 (A <= -3.4e+147)
		tmp = Float64(Float64(-2.0 / B_m) * sqrt(Float64(F * A)));
	else
		tmp = Float64(sqrt(fma(-2.0, Float64(F * B_m), Float64(Float64(F * A) * 2.0))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -3.4e+147], N[(N[(-2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(F * B$95$m), $MachinePrecision] + N[(N[(F * A), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -3.4e147

   1. Initial program 4.9%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 7.9

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

   5. Applied rewrites 7.9%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 8.0%

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

   if -3.4e147 < A

   1. Initial program 24.6%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 16.7

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

   5. Applied rewrites 16.7%

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

   7. Applied rewrites 16.7%

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

   8. Taylor expanded in A around 0

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

   10. Applied rewrites 14.1%

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

4. Final simplification 13.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -3.4 \cdot 10^{+147}:\\ \;\;\;\;\frac{-2}{B} \cdot \sqrt{F \cdot A}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-2, F \cdot B, \left(F \cdot A\right) \cdot 2\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{-2}{B\_m} \cdot \sqrt{F \cdot A}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot B\_m\right) \cdot -2}}{-B\_m}\\ \end{array} \end{array} \]

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 (<= A -4.8e+131)
   (* (/ -2.0 B_m) (sqrt (* F A)))
   (/ (sqrt (* (* F B_m) -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 (A <= -4.8e+131) {
		tmp = (-2.0 / B_m) * sqrt((F * A));
	} else {
		tmp = sqrt(((F * B_m) * -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 (a <= (-4.8d+131)) then
        tmp = ((-2.0d0) / b_m) * sqrt((f * a))
    else
        tmp = sqrt(((f * b_m) * (-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 (A <= -4.8e+131) {
		tmp = (-2.0 / B_m) * Math.sqrt((F * A));
	} else {
		tmp = Math.sqrt(((F * B_m) * -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 A <= -4.8e+131:
		tmp = (-2.0 / B_m) * math.sqrt((F * A))
	else:
		tmp = math.sqrt(((F * B_m) * -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 (A <= -4.8e+131)
		tmp = Float64(Float64(-2.0 / B_m) * sqrt(Float64(F * A)));
	else
		tmp = Float64(sqrt(Float64(Float64(F * B_m) * -2.0)) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -4.8e+131)
		tmp = (-2.0 / B_m) * sqrt((F * A));
	else
		tmp = sqrt(((F * B_m) * -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[A, -4.8e+131], N[(N[(-2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(F * B$95$m), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.7999999999999999e131

   1. Initial program 7.8%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 7.7

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

   5. Applied rewrites 7.7%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 7.9%

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

   if -4.7999999999999999e131 < A

   1. Initial program 24.3%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 16.8

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

   5. Applied rewrites 16.8%

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

   7. Applied rewrites 16.8%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 15.1%

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

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.8 \cdot 10^{+131}:\\ \;\;\;\;\frac{-2}{B} \cdot \sqrt{F \cdot A}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot B\right) \cdot -2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 9.2% accurate, 15.3× speedup

Math

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

FPCore

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

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = ((-2.0d0) / b_m) * sqrt((f * a))
end function

Java

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

Python

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

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(-2.0 / B_m) * sqrt(Float64(F * A)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (-2.0 / B_m) * sqrt((F * A));
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[(-2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in C around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. +-commutative N/A

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

   12. unpow2 N/A

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

   13. unpow2 N/A

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

   14. lower-hypot.f64 15.6

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

5. Applied rewrites 15.6%

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

6. Taylor expanded in A around -inf

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

8. Applied rewrites 2.4%

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

9. Final simplification 2.4%

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

Alternative 15: 1.6% accurate, 18.2× speedup

Math

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

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 B_m) 2.0)))

C

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

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 / b_m) * 2.0d0))
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 / B_m) * 2.0));
}

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 / B_m) * 2.0))

Julia

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 1.9

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

5. Applied rewrites 1.9%

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

7. Applied rewrites 1.9%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))