ABCF->ab-angle b

Percentage Accurate: 18.7% → 57.5%

Time: 30.0s

Alternatives: 15

Speedup: 3.0×

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: 18.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := t\_2 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(F \cdot t\_3\right)\right) \cdot \left(\sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{C + \left(A - t\_1\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right)\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A - \left(t\_1 - C\right)\right)\right)}}{-t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\_m\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (hypot B_m (- A C)))
        (t_2 (* (* 4.0 A) C))
        (t_3 (- t_2 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* F t_3))
            (- (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))) (+ A C))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (*
      (sqrt 2.0)
      (- (sqrt (* F (/ (+ C (- A t_1)) (fma -4.0 (* A C) (pow B_m 2.0)))))))
     (if (<= t_4 -2e-213)
       (/ (sqrt (* (* F t_0) (* 2.0 (- A (- t_1 C))))) (- t_0))
       (if (<= t_4 INFINITY)
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C)))))
          t_3)
         (*
          (exp (* (+ (log (- F)) (log B_m)) 0.5))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = hypot(B_m, (A - C));
	double t_2 = (4.0 * A) * C;
	double t_3 = t_2 - pow(B_m, 2.0);
	double t_4 = sqrt(((2.0 * (F * t_3)) * (sqrt((pow(B_m, 2.0) + pow((A - C), 2.0))) - (A + C)))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = sqrt(2.0) * -sqrt((F * ((C + (A - t_1)) / fma(-4.0, (A * C), pow(B_m, 2.0)))));
	} else if (t_4 <= -2e-213) {
		tmp = sqrt(((F * t_0) * (2.0 * (A - (t_1 - C))))) / -t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * (A + (A + ((pow(B_m, 2.0) * -0.5) / C))))) / t_3;
	} else {
		tmp = exp(((log(-F) + log(B_m)) * 0.5)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = hypot(B_m, Float64(A - C))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(t_2 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(F * t_3)) * Float64(sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))) - Float64(A + C)))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(C + Float64(A - t_1)) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	elseif (t_4 <= -2e-213)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A - Float64(t_1 - C))))) / Float64(-t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C))))) / t_3);
	else
		tmp = Float64(exp(Float64(Float64(log(Float64(-F)) + log(B_m)) * 0.5)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(F * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(C + N[(A - t$95$1), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -2e-213], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A - N[(t$95$1 - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.6%

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

   3. Taylor expanded in F around 0 14.3%

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

   5. Simplified 68.6%

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

   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))) < -1.9999999999999999e-213

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   if -1.9999999999999999e-213 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 27.0%

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

   3. Taylor expanded in C around inf 32.8%

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

      1. mul-1-neg 32.8%

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

      2. associate--l+ 32.8%

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

      3. associate-*r/ 32.8%

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

   5. Simplified 32.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 2.0%

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

      1. mul-1-neg 2.0%

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

      2. unpow2 2.0%

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

      3. unpow2 2.0%

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

      4. hypot-define 20.1%

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

   5. Simplified 20.1%

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

      1. pow1/2 20.1%

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

      2. pow-to-exp 19.1%

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

   7. Applied egg-rr 19.1%

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

   8. Taylor expanded in B around inf 22.7%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-1 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. mul-1-neg 22.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(-F\right)} + -1 \cdot \log \left(\frac{1}{B}\right)\right) \cdot 0.5} \]

      2. mul-1-neg 22.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\left(-\log \left(\frac{1}{B}\right)\right)}\right) \cdot 0.5} \]

      3. log-rec 22.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \left(-\color{blue}{\left(-\log B\right)}\right)\right) \cdot 0.5} \]

      4. remove-double-neg 22.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\log B}\right) \cdot 0.5} \]

   10. Simplified 22.7%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{C + \left(A - \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(F \cdot \left(\left(4 \cdot A\right) \cdot C - {B}^{2}\right)\right)\right) \cdot \left(\sqrt{{B}^{2} + {\left(A - C\right)}^{2}} - \left(A + C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{2}}{B\_m} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\_m\right)}\right)\\ t_2 := {B\_m}^{2} + C \cdot \left(A \cdot -4\right)\\ t_3 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{-162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-106}:\\ \;\;\;\;\frac{-1}{\frac{t\_2}{\sqrt{2 \cdot \left(t\_2 \cdot \left(2 \cdot \left(A \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 20000000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_3\right) \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)\right)}}{-t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\_m\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (*
          (/ (sqrt 2.0) B_m)
          (- (exp (* 0.5 (+ (log (/ (* F -0.5) C)) (* 2.0 (log B_m))))))))
        (t_2 (+ (pow B_m 2.0) (* C (* A -4.0))))
        (t_3 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-184)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 5e-162)
       t_1
       (if (<= (pow B_m 2.0) 2e-106)
         (/ -1.0 (/ t_2 (sqrt (* 2.0 (* t_2 (* 2.0 (* A F)))))))
         (if (<= (pow B_m 2.0) 20000000.0)
           (/
            (sqrt (* (* F t_3) (* 2.0 (- A (- (hypot B_m (- A C)) C)))))
            (- t_3))
           (if (<= (pow B_m 2.0) 5e+90)
             t_1
             (*
              (exp (* (+ (log (- F)) (log B_m)) 0.5))
              (/ (sqrt 2.0) (- B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = (sqrt(2.0) / B_m) * -exp((0.5 * (log(((F * -0.5) / C)) + (2.0 * log(B_m)))));
	double t_2 = pow(B_m, 2.0) + (C * (A * -4.0));
	double t_3 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-184) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e-162) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 2e-106) {
		tmp = -1.0 / (t_2 / sqrt((2.0 * (t_2 * (2.0 * (A * F))))));
	} else if (pow(B_m, 2.0) <= 20000000.0) {
		tmp = sqrt(((F * t_3) * (2.0 * (A - (hypot(B_m, (A - C)) - C))))) / -t_3;
	} else if (pow(B_m, 2.0) <= 5e+90) {
		tmp = t_1;
	} else {
		tmp = exp(((log(-F) + log(B_m)) * 0.5)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(sqrt(2.0) / B_m) * Float64(-exp(Float64(0.5 * Float64(log(Float64(Float64(F * -0.5) / C)) + Float64(2.0 * log(B_m)))))))
	t_2 = Float64((B_m ^ 2.0) + Float64(C * Float64(A * -4.0)))
	t_3 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-184)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e-162)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 2e-106)
		tmp = Float64(-1.0 / Float64(t_2 / sqrt(Float64(2.0 * Float64(t_2 * Float64(2.0 * Float64(A * F)))))));
	elseif ((B_m ^ 2.0) <= 20000000.0)
		tmp = Float64(sqrt(Float64(Float64(F * t_3) * Float64(2.0 * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C))))) / Float64(-t_3));
	elseif ((B_m ^ 2.0) <= 5e+90)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(Float64(log(Float64(-F)) + log(B_m)) * 0.5)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(0.5 * N[(N[Log[N[(N[(F * -0.5), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-184], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-162], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-106], N[(-1.0 / N[(t$95$2 / N[Sqrt[N[(2.0 * N[(t$95$2 * N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 20000000.0], N[(N[Sqrt[N[(N[(F * t$95$3), $MachinePrecision] * N[(2.0 * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+90], t$95$1, N[(N[Exp[N[(N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e-184

   1. Initial program 29.0%

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

   3. Taylor expanded in A around -inf 26.6%

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

   if 2.0000000000000001e-184 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000014e-162 or 2e7 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e90

   1. Initial program 20.4%

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

   3. Taylor expanded in A around 0 12.6%

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

      1. mul-1-neg 12.6%

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

      2. unpow2 12.6%

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

      3. unpow2 12.6%

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

      4. hypot-define 12.7%

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

   5. Simplified 12.7%

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

      1. pow1/2 12.8%

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

      2. pow-to-exp 12.1%

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

   7. Applied egg-rr 12.1%

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

   8. Taylor expanded in B around 0 10.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-0.5 \cdot \frac{F}{C}\right) + 2 \cdot \log B\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. associate-*r/ 10.5%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(\frac{-0.5 \cdot F}{C}\right)} + 2 \cdot \log B\right) \cdot 0.5} \]

   10. Simplified 10.5%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(\frac{-0.5 \cdot F}{C}\right) + 2 \cdot \log B\right)} \cdot 0.5} \]

   if 5.00000000000000014e-162 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999988e-106

   1. Initial program 14.7%

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

   3. Taylor expanded in C around inf 2.5%

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

      1. mul-1-neg 2.5%

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

      2. associate--l+ 2.5%

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

      3. associate-*r/ 2.5%

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

   5. Simplified 2.5%

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

      1. clear-num 2.5%

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

      2. inv-pow 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. unpow-1 2.5%

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

      2. cancel-sign-sub-inv 2.5%

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

      3. metadata-eval 2.5%

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

      4. associate-*r* 2.5%

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

      5. associate-*l* 14.6%

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

      6. cancel-sign-sub-inv 14.6%

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

      7. metadata-eval 14.6%

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

      8. associate-*r* 14.6%

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

   9. Simplified 14.6%

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

   10. Taylor expanded in A around inf 14.6%

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

       1. *-commutative 14.6%

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

   12. Simplified 14.6%

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

   if 1.99999999999999988e-106 < (pow.f64 B #s(literal 2 binary64)) < 2e7

   1. Initial program 47.1%

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

   3. Simplified 56.8%

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

   if 5.0000000000000004e90 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.7%

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

   3. Taylor expanded in A around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. unpow2 10.7%

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

      3. unpow2 10.7%

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

      4. hypot-define 29.5%

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

   5. Simplified 29.5%

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

      1. pow1/2 29.5%

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

      2. pow-to-exp 28.0%

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

   7. Applied egg-rr 28.0%

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

   8. Taylor expanded in B around inf 30.0%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-1 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. mul-1-neg 30.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(-F\right)} + -1 \cdot \log \left(\frac{1}{B}\right)\right) \cdot 0.5} \]

      2. mul-1-neg 30.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\left(-\log \left(\frac{1}{B}\right)\right)}\right) \cdot 0.5} \]

      3. log-rec 30.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \left(-\color{blue}{\left(-\log B\right)}\right)\right) \cdot 0.5} \]

      4. remove-double-neg 30.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\log B}\right) \cdot 0.5} \]

   10. Simplified 30.0%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-F\right) + \log B\right)} \cdot 0.5} \]
3. Recombined 5 regimes into one program.

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\right)}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-106}:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} + C \cdot \left(A \cdot -4\right)}{\sqrt{2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(A \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 20000000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ t_2 := F \cdot t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-179}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 20000000:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\_m\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (- t_0)) (t_2 (* F t_0)))
   (if (<= (pow B_m 2.0) 1e-179)
     (/ (sqrt (* t_2 (* 2.0 (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C)))))) t_1)
     (if (<= (pow B_m 2.0) 20000000.0)
       (/ (sqrt (* t_2 (* 2.0 (- A (- (hypot B_m (- A C)) C))))) t_1)
       (if (<= (pow B_m 2.0) 5e+90)
         (*
          (/ (sqrt 2.0) B_m)
          (- (exp (* 0.5 (+ (log (/ (* F -0.5) C)) (* 2.0 (log B_m)))))))
         (*
          (exp (* (+ (log (- F)) (log B_m)) 0.5))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -t_0;
	double t_2 = F * t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-179) {
		tmp = sqrt((t_2 * (2.0 * (A + (A + ((pow(B_m, 2.0) * -0.5) / C)))))) / t_1;
	} else if (pow(B_m, 2.0) <= 20000000.0) {
		tmp = sqrt((t_2 * (2.0 * (A - (hypot(B_m, (A - C)) - C))))) / t_1;
	} else if (pow(B_m, 2.0) <= 5e+90) {
		tmp = (sqrt(2.0) / B_m) * -exp((0.5 * (log(((F * -0.5) / C)) + (2.0 * log(B_m)))));
	} else {
		tmp = exp(((log(-F) + log(B_m)) * 0.5)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-t_0)
	t_2 = Float64(F * t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-179)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C)))))) / t_1);
	elseif ((B_m ^ 2.0) <= 20000000.0)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C))))) / t_1);
	elseif ((B_m ^ 2.0) <= 5e+90)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-exp(Float64(0.5 * Float64(log(Float64(Float64(F * -0.5) / C)) + Float64(2.0 * log(B_m)))))));
	else
		tmp = Float64(exp(Float64(Float64(log(Float64(-F)) + log(B_m)) * 0.5)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-179], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 20000000.0], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+90], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(0.5 * N[(N[Log[N[(N[(F * -0.5), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-179

   1. Initial program 27.8%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in C around inf 26.0%

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

      1. associate-*r/ 26.0%

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

      2. mul-1-neg 26.0%

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

   7. Simplified 26.0%

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

   if 1e-179 < (pow.f64 B #s(literal 2 binary64)) < 2e7

   1. Initial program 38.0%

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

   3. Simplified 44.1%

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

   if 2e7 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e90

   1. Initial program 19.3%

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

   3. Taylor expanded in A around 0 13.6%

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

      1. mul-1-neg 13.6%

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

      2. unpow2 13.6%

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

      3. unpow2 13.6%

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

      4. hypot-define 13.7%

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

   5. Simplified 13.7%

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

      1. pow1/2 13.7%

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

      2. pow-to-exp 12.8%

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

   7. Applied egg-rr 12.8%

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

   8. Taylor expanded in B around 0 15.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-0.5 \cdot \frac{F}{C}\right) + 2 \cdot \log B\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. associate-*r/ 15.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(\frac{-0.5 \cdot F}{C}\right)} + 2 \cdot \log B\right) \cdot 0.5} \]

   10. Simplified 15.9%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(\frac{-0.5 \cdot F}{C}\right) + 2 \cdot \log B\right)} \cdot 0.5} \]

   if 5.0000000000000004e90 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.7%

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

   3. Taylor expanded in A around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. unpow2 10.7%

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

      3. unpow2 10.7%

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

      4. hypot-define 29.5%

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

   5. Simplified 29.5%

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

      1. pow1/2 29.5%

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

      2. pow-to-exp 28.0%

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

   7. Applied egg-rr 28.0%

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

   8. Taylor expanded in B around inf 30.0%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-1 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. mul-1-neg 30.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(-F\right)} + -1 \cdot \log \left(\frac{1}{B}\right)\right) \cdot 0.5} \]

      2. mul-1-neg 30.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\left(-\log \left(\frac{1}{B}\right)\right)}\right) \cdot 0.5} \]

      3. log-rec 30.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \left(-\color{blue}{\left(-\log B\right)}\right)\right) \cdot 0.5} \]

      4. remove-double-neg 30.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\log B}\right) \cdot 0.5} \]

   10. Simplified 30.0%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-179}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 20000000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 20000000:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B\_m}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 20000000.0)
   (/
    (sqrt (* -8.0 (* (* F (* A C)) (+ A A))))
    (- (fma -4.0 (* A C) (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 5e+90)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) C)))))
     (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 20000000.0) {
		tmp = sqrt((-8.0 * ((F * (A * C)) * (A + A)))) / -fma(-4.0, (A * C), pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+90) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * ((pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 20000000.0)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(F * Float64(A * C)) * Float64(A + A)))) / Float64(-fma(-4.0, Float64(A * C), (B_m ^ 2.0))));
	elseif ((B_m ^ 2.0) <= 5e+90)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 20000000.0], N[(N[Sqrt[N[(-8.0 * N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+90], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e7

   1. Initial program 30.7%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in C around inf 18.9%

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

      1. associate-*r* 19.6%

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

      2. *-commutative 19.6%

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

      3. mul-1-neg 19.6%

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

   7. Simplified 19.6%

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

      1. *-un-lft-identity 19.6%

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

      2. associate-*r* 19.5%

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

      3. *-commutative 19.5%

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

      4. fma-undefine 19.5%

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

      5. *-commutative 19.5%

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

      6. *-commutative 19.5%

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

      7. associate-*l* 19.5%

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

      8. *-commutative 19.5%

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

      9. fma-define 19.5%

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

      10. *-commutative 19.5%

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

   9. Applied egg-rr 19.5%

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

       1. *-lft-identity 19.5%

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

       2. sub-neg 19.5%

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

       3. remove-double-neg 19.5%

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

       4. *-commutative 19.5%

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

       5. *-commutative 19.5%

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

       6. *-commutative 19.5%

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

   11. Simplified 19.5%

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

   if 2e7 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000004e90

   1. Initial program 19.3%

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

   3. Taylor expanded in A around 0 13.6%

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

      1. mul-1-neg 13.6%

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

      2. unpow2 13.6%

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

      3. unpow2 13.6%

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

      4. hypot-define 13.7%

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

   5. Simplified 13.7%

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

   6. Taylor expanded in C around inf 23.8%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}} \]

   if 5.0000000000000004e90 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 11.7%

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

   3. Taylor expanded in C around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. +-commutative 11.8%

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

      3. unpow2 11.8%

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

      4. unpow2 11.8%

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

      5. hypot-define 29.7%

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

   5. Simplified 29.7%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 20000000:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 4.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(A + \left(A + \frac{{B\_m}^{2} \cdot -0.5}{C}\right)\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 55000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_1\right) \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)\right)}}{-t\_1}\\ \mathbf{elif}\;B\_m \leq 1.05 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\_m\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 4.7e-88)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (pow B_m 2.0) t_0) F))
        (+ A (+ A (/ (* (pow B_m 2.0) -0.5) C)))))
      (- t_0 (pow B_m 2.0)))
     (if (<= B_m 55000.0)
       (/ (sqrt (* (* F t_1) (* 2.0 (- A (- (hypot B_m (- A C)) C))))) (- t_1))
       (if (<= B_m 1.05e+50)
         (*
          (/ (sqrt 2.0) B_m)
          (- (exp (* 0.5 (+ (log (/ (* F -0.5) C)) (* 2.0 (log B_m)))))))
         (*
          (exp (* (+ (log (- F)) (log B_m)) 0.5))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 4.7e-88) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (A + (A + ((pow(B_m, 2.0) * -0.5) / C))))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 55000.0) {
		tmp = sqrt(((F * t_1) * (2.0 * (A - (hypot(B_m, (A - C)) - C))))) / -t_1;
	} else if (B_m <= 1.05e+50) {
		tmp = (sqrt(2.0) / B_m) * -exp((0.5 * (log(((F * -0.5) / C)) + (2.0 * log(B_m)))));
	} else {
		tmp = exp(((log(-F) + log(B_m)) * 0.5)) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 4.7e-88)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(A + Float64(A + Float64(Float64((B_m ^ 2.0) * -0.5) / C))))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 55000.0)
		tmp = Float64(sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C))))) / Float64(-t_1));
	elseif (B_m <= 1.05e+50)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-exp(Float64(0.5 * Float64(log(Float64(Float64(F * -0.5) / C)) + Float64(2.0 * log(B_m)))))));
	else
		tmp = Float64(exp(Float64(Float64(log(Float64(-F)) + log(B_m)) * 0.5)) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.7e-88], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 55000.0], N[(N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$1)), $MachinePrecision], If[LessEqual[B$95$m, 1.05e+50], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(0.5 * N[(N[Log[N[(N[(F * -0.5), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.7e-88

   1. Initial program 22.3%

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

   3. Taylor expanded in C around inf 14.9%

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

      1. mul-1-neg 14.9%

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

      2. associate--l+ 14.9%

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

      3. associate-*r/ 14.9%

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

   5. Simplified 14.9%

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

   if 4.7e-88 < B < 55000

   1. Initial program 40.1%

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

   3. Simplified 45.7%

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

   if 55000 < B < 1.05e50

   1. Initial program 27.9%

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

   3. Taylor expanded in A around 0 28.7%

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

      1. mul-1-neg 28.7%

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

      2. unpow2 28.7%

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

      3. unpow2 28.7%

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

      4. hypot-define 28.7%

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

   5. Simplified 28.7%

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

      1. pow1/2 28.7%

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

      2. pow-to-exp 26.5%

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

   7. Applied egg-rr 26.5%

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

   8. Taylor expanded in B around 0 35.8%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-0.5 \cdot \frac{F}{C}\right) + 2 \cdot \log B\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. associate-*r/ 35.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(\frac{-0.5 \cdot F}{C}\right)} + 2 \cdot \log B\right) \cdot 0.5} \]

   10. Simplified 35.8%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(\frac{-0.5 \cdot F}{C}\right) + 2 \cdot \log B\right)} \cdot 0.5} \]

   if 1.05e50 < B

   1. Initial program 12.6%

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

   3. Taylor expanded in A around 0 18.7%

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

      1. mul-1-neg 18.7%

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

      2. unpow2 18.7%

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

      3. unpow2 18.7%

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

      4. hypot-define 52.7%

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

   5. Simplified 52.7%

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

      1. pow1/2 52.7%

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

      2. pow-to-exp 49.8%

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

   7. Applied egg-rr 49.8%

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

   8. Taylor expanded in B around inf 56.0%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-1 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. mul-1-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(-F\right)} + -1 \cdot \log \left(\frac{1}{B}\right)\right) \cdot 0.5} \]

      2. mul-1-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\left(-\log \left(\frac{1}{B}\right)\right)}\right) \cdot 0.5} \]

      3. log-rec 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \left(-\color{blue}{\left(-\log B\right)}\right)\right) \cdot 0.5} \]

      4. remove-double-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\log B}\right) \cdot 0.5} \]

   10. Simplified 56.0%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 55000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.05 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{2}}{B\_m} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\_m\right)}\right)\\ t_2 := {B\_m}^{2} + C \cdot \left(A \cdot -4\right)\\ \mathbf{if}\;B\_m \leq 2.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 1.1 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 680000:\\ \;\;\;\;\frac{-1}{\frac{t\_2}{\sqrt{2 \cdot \left(t\_2 \cdot \left(2 \cdot \left(A \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 4.1 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\_m\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (*
          (/ (sqrt 2.0) B_m)
          (- (exp (* 0.5 (+ (log (/ (* F -0.5) C)) (* 2.0 (log B_m))))))))
        (t_2 (+ (pow B_m 2.0) (* C (* A -4.0)))))
   (if (<= B_m 2.7e-92)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= B_m 1.1e-77)
       t_1
       (if (<= B_m 680000.0)
         (/ -1.0 (/ t_2 (sqrt (* 2.0 (* t_2 (* 2.0 (* A F)))))))
         (if (<= B_m 4.1e+46)
           t_1
           (*
            (exp (* (+ (log (- F)) (log B_m)) 0.5))
            (/ (sqrt 2.0) (- B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = (sqrt(2.0) / B_m) * -exp((0.5 * (log(((F * -0.5) / C)) + (2.0 * log(B_m)))));
	double t_2 = pow(B_m, 2.0) + (C * (A * -4.0));
	double tmp;
	if (B_m <= 2.7e-92) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 1.1e-77) {
		tmp = t_1;
	} else if (B_m <= 680000.0) {
		tmp = -1.0 / (t_2 / sqrt((2.0 * (t_2 * (2.0 * (A * F))))));
	} else if (B_m <= 4.1e+46) {
		tmp = t_1;
	} else {
		tmp = exp(((log(-F) + log(B_m)) * 0.5)) * (sqrt(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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    t_1 = (sqrt(2.0d0) / b_m) * -exp((0.5d0 * (log(((f * (-0.5d0)) / c)) + (2.0d0 * log(b_m)))))
    t_2 = (b_m ** 2.0d0) + (c * (a * (-4.0d0)))
    if (b_m <= 2.7d-92) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * a))) / (t_0 - (b_m ** 2.0d0))
    else if (b_m <= 1.1d-77) then
        tmp = t_1
    else if (b_m <= 680000.0d0) then
        tmp = (-1.0d0) / (t_2 / sqrt((2.0d0 * (t_2 * (2.0d0 * (a * f))))))
    else if (b_m <= 4.1d+46) then
        tmp = t_1
    else
        tmp = exp(((log(-f) + log(b_m)) * 0.5d0)) * (sqrt(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 t_0 = (4.0 * A) * C;
	double t_1 = (Math.sqrt(2.0) / B_m) * -Math.exp((0.5 * (Math.log(((F * -0.5) / C)) + (2.0 * Math.log(B_m)))));
	double t_2 = Math.pow(B_m, 2.0) + (C * (A * -4.0));
	double tmp;
	if (B_m <= 2.7e-92) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 1.1e-77) {
		tmp = t_1;
	} else if (B_m <= 680000.0) {
		tmp = -1.0 / (t_2 / Math.sqrt((2.0 * (t_2 * (2.0 * (A * F))))));
	} else if (B_m <= 4.1e+46) {
		tmp = t_1;
	} else {
		tmp = Math.exp(((Math.log(-F) + Math.log(B_m)) * 0.5)) * (Math.sqrt(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):
	t_0 = (4.0 * A) * C
	t_1 = (math.sqrt(2.0) / B_m) * -math.exp((0.5 * (math.log(((F * -0.5) / C)) + (2.0 * math.log(B_m)))))
	t_2 = math.pow(B_m, 2.0) + (C * (A * -4.0))
	tmp = 0
	if B_m <= 2.7e-92:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 1.1e-77:
		tmp = t_1
	elif B_m <= 680000.0:
		tmp = -1.0 / (t_2 / math.sqrt((2.0 * (t_2 * (2.0 * (A * F))))))
	elif B_m <= 4.1e+46:
		tmp = t_1
	else:
		tmp = math.exp(((math.log(-F) + math.log(B_m)) * 0.5)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(sqrt(2.0) / B_m) * Float64(-exp(Float64(0.5 * Float64(log(Float64(Float64(F * -0.5) / C)) + Float64(2.0 * log(B_m)))))))
	t_2 = Float64((B_m ^ 2.0) + Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (B_m <= 2.7e-92)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 1.1e-77)
		tmp = t_1;
	elseif (B_m <= 680000.0)
		tmp = Float64(-1.0 / Float64(t_2 / sqrt(Float64(2.0 * Float64(t_2 * Float64(2.0 * Float64(A * F)))))));
	elseif (B_m <= 4.1e+46)
		tmp = t_1;
	else
		tmp = Float64(exp(Float64(Float64(log(Float64(-F)) + log(B_m)) * 0.5)) * Float64(sqrt(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)
	t_0 = (4.0 * A) * C;
	t_1 = (sqrt(2.0) / B_m) * -exp((0.5 * (log(((F * -0.5) / C)) + (2.0 * log(B_m)))));
	t_2 = (B_m ^ 2.0) + (C * (A * -4.0));
	tmp = 0.0;
	if (B_m <= 2.7e-92)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 1.1e-77)
		tmp = t_1;
	elseif (B_m <= 680000.0)
		tmp = -1.0 / (t_2 / sqrt((2.0 * (t_2 * (2.0 * (A * F))))));
	elseif (B_m <= 4.1e+46)
		tmp = t_1;
	else
		tmp = exp(((log(-F) + log(B_m)) * 0.5)) * (sqrt(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(0.5 * N[(N[Log[N[(N[(F * -0.5), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision] + N[(2.0 * N[Log[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.7e-92], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.1e-77], t$95$1, If[LessEqual[B$95$m, 680000.0], N[(-1.0 / N[(t$95$2 / N[Sqrt[N[(2.0 * N[(t$95$2 * N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 4.1e+46], t$95$1, N[(N[Exp[N[(N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 2.69999999999999995e-92

   1. Initial program 22.3%

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

   3. Taylor expanded in A around -inf 15.8%

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

   if 2.69999999999999995e-92 < B < 1.10000000000000003e-77 or 6.8e5 < B < 4.1e46

   1. Initial program 30.0%

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

   3. Taylor expanded in A around 0 29.0%

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

      1. mul-1-neg 29.0%

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

      2. unpow2 29.0%

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

      3. unpow2 29.0%

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

      4. hypot-define 29.0%

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

   5. Simplified 29.0%

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

      1. pow1/2 29.0%

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

      2. pow-to-exp 27.4%

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

   7. Applied egg-rr 27.4%

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

   8. Taylor expanded in B around 0 26.7%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-0.5 \cdot \frac{F}{C}\right) + 2 \cdot \log B\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. associate-*r/ 26.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(\frac{-0.5 \cdot F}{C}\right)} + 2 \cdot \log B\right) \cdot 0.5} \]

   10. Simplified 26.7%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(\frac{-0.5 \cdot F}{C}\right) + 2 \cdot \log B\right)} \cdot 0.5} \]

   if 1.10000000000000003e-77 < B < 6.8e5

   1. Initial program 40.8%

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

   3. Taylor expanded in C around inf 7.7%

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

      1. mul-1-neg 7.7%

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

      2. associate--l+ 7.7%

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

      3. associate-*r/ 7.7%

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

   5. Simplified 7.7%

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

      1. clear-num 7.7%

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

      2. inv-pow 7.7%

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

   7. Applied egg-rr 7.7%

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

      1. unpow-1 7.7%

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

      2. cancel-sign-sub-inv 7.7%

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

      3. metadata-eval 7.7%

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

      4. associate-*r* 7.7%

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

      5. associate-*l* 13.2%

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

      6. cancel-sign-sub-inv 13.2%

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

      7. metadata-eval 13.2%

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

      8. associate-*r* 13.2%

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

   9. Simplified 13.2%

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

   10. Taylor expanded in A around inf 12.6%

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

       1. *-commutative 12.6%

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

   12. Simplified 12.6%

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

   if 4.1e46 < B

   1. Initial program 12.6%

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

   3. Taylor expanded in A around 0 18.7%

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

      1. mul-1-neg 18.7%

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

      2. unpow2 18.7%

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

      3. unpow2 18.7%

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

      4. hypot-define 52.7%

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

   5. Simplified 52.7%

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

      1. pow1/2 52.7%

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

      2. pow-to-exp 49.8%

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

   7. Applied egg-rr 49.8%

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

   8. Taylor expanded in B around inf 56.0%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-1 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. mul-1-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(-F\right)} + -1 \cdot \log \left(\frac{1}{B}\right)\right) \cdot 0.5} \]

      2. mul-1-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\left(-\log \left(\frac{1}{B}\right)\right)}\right) \cdot 0.5} \]

      3. log-rec 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \left(-\color{blue}{\left(-\log B\right)}\right)\right) \cdot 0.5} \]

      4. remove-double-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\log B}\right) \cdot 0.5} \]

   10. Simplified 56.0%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.1 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\right)}\right)\\ \mathbf{elif}\;B \leq 680000:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} + C \cdot \left(A \cdot -4\right)}{\sqrt{2 \cdot \left(\left({B}^{2} + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(A \cdot F\right)\right)\right)}}}\\ \mathbf{elif}\;B \leq 4.1 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(\frac{F \cdot -0.5}{C}\right) + 2 \cdot \log B\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 28500:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 1.9 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B\_m}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\_m\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 28500.0)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= B_m 1.9e+48)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) C)))))
       (* (exp (* (+ (log (- F)) (log B_m)) 0.5)) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 28500.0) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 1.9e+48) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * ((pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = exp(((log(-F) + log(B_m)) * 0.5)) * (sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if (b_m <= 28500.0d0) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * a))) / (t_0 - (b_m ** 2.0d0))
    else if (b_m <= 1.9d+48) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt(((-0.5d0) * (((b_m ** 2.0d0) * f) / c)))
    else
        tmp = exp(((log(-f) + log(b_m)) * 0.5d0)) * (sqrt(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 t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 28500.0) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 1.9e+48) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((-0.5 * ((Math.pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = Math.exp(((Math.log(-F) + Math.log(B_m)) * 0.5)) * (Math.sqrt(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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 28500.0:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 1.9e+48:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((-0.5 * ((math.pow(B_m, 2.0) * F) / C)))
	else:
		tmp = math.exp(((math.log(-F) + math.log(B_m)) * 0.5)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 28500.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 1.9e+48)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / C)))));
	else
		tmp = Float64(exp(Float64(Float64(log(Float64(-F)) + log(B_m)) * 0.5)) * Float64(sqrt(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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 28500.0)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 1.9e+48)
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * (((B_m ^ 2.0) * F) / C)));
	else
		tmp = exp(((log(-F) + log(B_m)) * 0.5)) * (sqrt(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 28500.0], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.9e+48], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 28500

   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 A around -inf 14.7%

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

   if 28500 < B < 1.9e48

   1. Initial program 27.9%

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

   3. Taylor expanded in A around 0 28.7%

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

      1. mul-1-neg 28.7%

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

      2. unpow2 28.7%

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

      3. unpow2 28.7%

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

      4. hypot-define 28.7%

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

   5. Simplified 28.7%

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

   6. Taylor expanded in C around inf 39.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}} \]

   if 1.9e48 < B

   1. Initial program 12.6%

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

   3. Taylor expanded in A around 0 18.7%

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

      1. mul-1-neg 18.7%

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

      2. unpow2 18.7%

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

      3. unpow2 18.7%

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

      4. hypot-define 52.7%

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

   5. Simplified 52.7%

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

      1. pow1/2 52.7%

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

      2. pow-to-exp 49.8%

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

   7. Applied egg-rr 49.8%

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

   8. Taylor expanded in B around inf 56.0%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-1 \cdot F\right) + -1 \cdot \log \left(\frac{1}{B}\right)\right)} \cdot 0.5} \]
   9. Step-by-step derivation

      1. mul-1-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \color{blue}{\left(-F\right)} + -1 \cdot \log \left(\frac{1}{B}\right)\right) \cdot 0.5} \]

      2. mul-1-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\left(-\log \left(\frac{1}{B}\right)\right)}\right) \cdot 0.5} \]

      3. log-rec 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \left(-\color{blue}{\left(-\log B\right)}\right)\right) \cdot 0.5} \]

      4. remove-double-neg 56.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-F\right) + \color{blue}{\log B}\right) \cdot 0.5} \]

   10. Simplified 56.0%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{\left(\log \left(-F\right) + \log B\right)} \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 28500:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(-F\right) + \log B\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 39000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 1.85 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B\_m}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 39000.0)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= B_m 1.85e+47)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) C)))))
       (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 39000.0) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 1.85e+47) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * ((pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

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 t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 39000.0) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 1.85e+47) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((-0.5 * ((Math.pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = Math.sqrt((F * (A - Math.hypot(B_m, A)))) * (Math.sqrt(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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 39000.0:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 1.85e+47:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((-0.5 * ((math.pow(B_m, 2.0) * F) / C)))
	else:
		tmp = math.sqrt((F * (A - math.hypot(B_m, A)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 39000.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 1.85e+47)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 39000.0)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 1.85e+47)
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * (((B_m ^ 2.0) * F) / C)));
	else
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 39000.0], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.85e+47], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 39000

   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 A around -inf 14.7%

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

   if 39000 < B < 1.8500000000000002e47

   1. Initial program 27.9%

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

   3. Taylor expanded in A around 0 28.7%

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

      1. mul-1-neg 28.7%

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

      2. unpow2 28.7%

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

      3. unpow2 28.7%

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

      4. hypot-define 28.7%

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

   5. Simplified 28.7%

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

   6. Taylor expanded in C around inf 39.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}} \]

   if 1.8500000000000002e47 < B

   1. Initial program 12.6%

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

   3. Taylor expanded in C around 0 20.8%

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

      1. mul-1-neg 20.8%

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

      2. +-commutative 20.8%

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

      3. unpow2 20.8%

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

      4. unpow2 20.8%

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

      5. hypot-define 52.9%

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

   5. Simplified 52.9%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 39000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.85 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B\_m}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.4e+47)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) C)))))
   (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.4e+47) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * ((pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

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 (B_m <= 1.4e+47) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((-0.5 * ((Math.pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = Math.sqrt((F * (A - Math.hypot(B_m, A)))) * (Math.sqrt(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 B_m <= 1.4e+47:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((-0.5 * ((math.pow(B_m, 2.0) * F) / C)))
	else:
		tmp = math.sqrt((F * (A - math.hypot(B_m, A)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.4e+47)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(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 (B_m <= 1.4e+47)
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * (((B_m ^ 2.0) * F) / C)));
	else
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(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[B$95$m, 1.4e+47], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.39999999999999994e47

   1. Initial program 24.4%

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

   3. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. unpow2 7.0%

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

      3. unpow2 7.0%

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

      4. hypot-define 7.7%

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

   5. Simplified 7.7%

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

   6. Taylor expanded in C around inf 7.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \sqrt{\color{blue}{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}} \]

   if 1.39999999999999994e47 < B

   1. Initial program 12.6%

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

   3. Taylor expanded in C around 0 20.8%

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

      1. mul-1-neg 20.8%

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

      2. +-commutative 20.8%

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

      3. unpow2 20.8%

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

      4. unpow2 20.8%

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

      5. hypot-define 52.9%

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

   5. Simplified 52.9%

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

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4.7 \cdot 10^{+197}:\\ \;\;\;\;-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{A \cdot F}}{B\_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -4.7e+197)
   (* -2.0 (expm1 (log1p (/ (sqrt (* A F)) B_m))))
   (/ (sqrt (* 2.0 (* F (- C (hypot B_m C))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -4.7e+197) {
		tmp = -2.0 * expm1(log1p((sqrt((A * F)) / B_m)));
	} else {
		tmp = sqrt((2.0 * (F * (C - hypot(B_m, C))))) / -B_m;
	}
	return tmp;
}

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 <= -4.7e+197) {
		tmp = -2.0 * Math.expm1(Math.log1p((Math.sqrt((A * F)) / B_m)));
	} else {
		tmp = Math.sqrt((2.0 * (F * (C - Math.hypot(B_m, C))))) / -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 <= -4.7e+197:
		tmp = -2.0 * math.expm1(math.log1p((math.sqrt((A * F)) / B_m)))
	else:
		tmp = math.sqrt((2.0 * (F * (C - math.hypot(B_m, C))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -4.7e+197)
		tmp = Float64(-2.0 * expm1(log1p(Float64(sqrt(Float64(A * F)) / B_m))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C))))) / 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, -4.7e+197], N[(-2.0 * N[(Exp[N[Log[1 + N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.6999999999999999e197

   1. Initial program 2.1%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in A around -inf 1.9%

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

   6. Taylor expanded in A around 0 8.8%

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

      1. expm1-log1p-u 8.5%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)\right)} \]

      2. expm1-undefine 2.7%

         \[\leadsto -2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{A \cdot F} \cdot \frac{1}{B}\right)} - 1\right)} \]

      3. un-div-inv 2.7%

         \[\leadsto -2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{A \cdot F}}{B}}\right)} - 1\right) \]

   8. Applied egg-rr 2.7%

      \[\leadsto -2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{A \cdot F}}{B}\right)} - 1\right)} \]
   9. Step-by-step derivation

      1. expm1-define 8.6%

         \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{A \cdot F}}{B}\right)\right)} \]

      2. *-commutative 8.6%

         \[\leadsto -2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{F \cdot A}}}{B}\right)\right) \]

   10. Simplified 8.6%

       \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{F \cdot A}}{B}\right)\right)} \]

   if -4.6999999999999999e197 < A

   1. Initial program 23.1%

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

   3. Taylor expanded in A around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. unpow2 10.2%

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

      3. unpow2 10.2%

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

      4. hypot-define 18.8%

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

   5. Simplified 18.8%

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

      1. associate-*l/ 18.8%

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

      2. pow1/2 18.8%

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

      3. pow1/2 18.8%

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

      4. pow-prod-down 18.9%

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

   7. Applied egg-rr 18.9%

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

      1. unpow1/2 18.9%

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

   9. Simplified 18.9%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.7 \cdot 10^{+197}:\\ \;\;\;\;-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{A \cdot F}}{B}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 * (A - Math.hypot(B_m, A)))) * (Math.sqrt(2.0) / -B_m);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.7%

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

3. Taylor expanded in C around 0 9.8%

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

   1. mul-1-neg 9.8%

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

   2. +-commutative 9.8%

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

   3. unpow2 9.8%

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

   4. unpow2 9.8%

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

   5. hypot-define 17.7%

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

5. Simplified 17.7%

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

6. Final simplification 17.7%

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

Alternative 12: 27.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{+198}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -1e+198)
   (* -2.0 (/ (sqrt (* A F)) B_m))
   (/ (sqrt (* 2.0 (* F (- C (hypot B_m C))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -1e+198) {
		tmp = -2.0 * (sqrt((A * F)) / B_m);
	} else {
		tmp = sqrt((2.0 * (F * (C - hypot(B_m, C))))) / -B_m;
	}
	return tmp;
}

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 <= -1e+198) {
		tmp = -2.0 * (Math.sqrt((A * F)) / B_m);
	} else {
		tmp = Math.sqrt((2.0 * (F * (C - Math.hypot(B_m, C))))) / -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 <= -1e+198:
		tmp = -2.0 * (math.sqrt((A * F)) / B_m)
	else:
		tmp = math.sqrt((2.0 * (F * (C - math.hypot(B_m, C))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -1e+198)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B_m));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C))))) / 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 <= -1e+198)
		tmp = -2.0 * (sqrt((A * F)) / B_m);
	else
		tmp = sqrt((2.0 * (F * (C - hypot(B_m, C))))) / -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, -1e+198], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.00000000000000002e198

   1. Initial program 2.1%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in A around -inf 1.9%

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

   6. Taylor expanded in A around 0 8.8%

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

   7. Taylor expanded in A around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. distribute-neg-frac 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 8.9%

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

      7. distribute-lft-neg-in 8.9%

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

      8. metadata-eval 8.9%

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

      9. *-lft-identity 8.9%

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

      10. *-commutative 8.9%

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

   9. Simplified 8.9%

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

   if -1.00000000000000002e198 < A

   1. Initial program 23.1%

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

   3. Taylor expanded in A around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. unpow2 10.2%

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

      3. unpow2 10.2%

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

      4. hypot-define 18.8%

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

   5. Simplified 18.8%

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

      1. associate-*l/ 18.8%

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

      2. pow1/2 18.8%

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

      3. pow1/2 18.8%

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

      4. pow-prod-down 18.9%

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

   7. Applied egg-rr 18.9%

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

      1. unpow1/2 18.9%

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

   9. Simplified 18.9%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1 \cdot 10^{+198}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2.7 \cdot 10^{+197}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{B\_m \cdot \left(-F\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -2.7e+197)
   (* -2.0 (/ (sqrt (* A F)) B_m))
   (* (sqrt (* B_m (- F))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -2.7e+197) {
		tmp = -2.0 * (sqrt((A * F)) / B_m);
	} else {
		tmp = sqrt((B_m * -F)) * (sqrt(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 <= (-2.7d+197)) then
        tmp = (-2.0d0) * (sqrt((a * f)) / b_m)
    else
        tmp = sqrt((b_m * -f)) * (sqrt(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 <= -2.7e+197) {
		tmp = -2.0 * (Math.sqrt((A * F)) / B_m);
	} else {
		tmp = Math.sqrt((B_m * -F)) * (Math.sqrt(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 <= -2.7e+197:
		tmp = -2.0 * (math.sqrt((A * F)) / B_m)
	else:
		tmp = math.sqrt((B_m * -F)) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -2.7e+197)
		tmp = Float64(-2.0 * Float64(sqrt(Float64(A * F)) / B_m));
	else
		tmp = Float64(sqrt(Float64(B_m * Float64(-F))) * Float64(sqrt(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 <= -2.7e+197)
		tmp = -2.0 * (sqrt((A * F)) / B_m);
	else
		tmp = sqrt((B_m * -F)) * (sqrt(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, -2.7e+197], N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -2.7e197

   1. Initial program 2.1%

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

   3. Simplified 13.4%

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

   5. Taylor expanded in A around -inf 1.9%

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

   6. Taylor expanded in A around 0 8.8%

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

   7. Taylor expanded in A around -inf 0.0%

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

      1. mul-1-neg 0.0%

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

      2. associate-*r/ 0.0%

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

      3. distribute-neg-frac 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 8.9%

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

      7. distribute-lft-neg-in 8.9%

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

      8. metadata-eval 8.9%

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

      9. *-lft-identity 8.9%

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

      10. *-commutative 8.9%

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

   9. Simplified 8.9%

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

   if -2.7e197 < A

   1. Initial program 23.1%

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

   3. Taylor expanded in A around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. unpow2 10.2%

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

      3. unpow2 10.2%

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

      4. hypot-define 18.8%

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

   5. Simplified 18.8%

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

   6. Taylor expanded in C around 0 16.6%

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

      1. associate-*r* 16.6%

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

      2. mul-1-neg 16.6%

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

   8. Simplified 16.6%

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

4. Final simplification 16.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.7 \cdot 10^{+197}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{A \cdot F}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 9.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -2 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{1}{B\_m}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* -2.0 (* (pow (* A F) 0.5) (/ 1.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -2.0 * (pow((A * F), 0.5) * (1.0 / B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-2.0d0) * (((a * f) ** 0.5d0) * (1.0d0 / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -2.0 * (Math.pow((A * F), 0.5) * (1.0 / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -2.0 * (math.pow((A * F), 0.5) * (1.0 / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) * Float64(1.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -2.0 * (((A * F) ^ 0.5) * (1.0 / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] * N[(1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.7%

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

3. Simplified 22.9%

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

5. Taylor expanded in A around -inf 7.3%

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

6. Taylor expanded in A around 0 2.4%

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

   1. pow1/2 2.6%

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

8. Applied egg-rr 2.6%

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

9. Final simplification 2.6%

   \[\leadsto -2 \cdot \left({\left(A \cdot F\right)}^{0.5} \cdot \frac{1}{B}\right) \]
10. Add Preprocessing

Alternative 15: 9.2% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.7%

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

3. Simplified 22.9%

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

5. Taylor expanded in A around -inf 7.3%

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

6. Taylor expanded in A around 0 2.4%

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

7. Taylor expanded in A around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. associate-*r/ 0.0%

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

   3. distribute-neg-frac 0.0%

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

   4. *-commutative 0.0%

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

   5. unpow2 0.0%

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

   6. rem-square-sqrt 2.4%

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

   7. distribute-lft-neg-in 2.4%

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

   8. metadata-eval 2.4%

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

   9. *-lft-identity 2.4%

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

   10. *-commutative 2.4%

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

9. Simplified 2.4%

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

10. Final simplification 2.4%

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

Reproduce

herbie shell --seed 2024077 
(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))))