ABCF->ab-angle b

Percentage Accurate: 20.0% → 53.7%

Time: 24.4s

Alternatives: 16

Speedup: 5.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 20.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\left(2 \cdot t\_3\right) \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)\right)\right)} \cdot \frac{-1}{t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= t_2 (- INFINITY))
     (-
      (sqrt
       (*
        2.0
        (*
         F
         (/
          (- (+ A C) (hypot B_m (- A C)))
          (fma -4.0 (* A C) (pow B_m 2.0)))))))
     (if (<= t_2 -2e-186)
       (*
        (sqrt (* (* 2.0 t_3) (* F (- (+ A C) (hypot (- A C) B_m)))))
        (/ -1.0 t_3))
       (if (<= t_2 INFINITY)
         (/
          (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
          (- t_0))
         (/ -1.0 (* B_m (pow (* (* 2.0 F) (- C (hypot C B_m))) -0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -sqrt((2.0 * (F * (((A + C) - hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))));
	} else if (t_2 <= -2e-186) {
		tmp = sqrt(((2.0 * t_3) * (F * ((A + C) - hypot((A - C), B_m))))) * (-1.0 / t_3);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) * (C - hypot(C, B_m))), -0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) - hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	elseif (t_2 <= -2e-186)
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_3) * Float64(F * Float64(Float64(A + C) - hypot(Float64(A - C), B_m))))) * Float64(-1.0 / t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) * Float64(C - hypot(C, B_m))) ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $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$2, -2e-186], N[(N[Sqrt[N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.0%

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

   3. Taylor expanded in F around 0 27.5%

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

   5. Simplified 45.0%

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

      1. sqrt-unprod 45.3%

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

      2. associate-/l* 54.3%

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

      3. associate-+r- 51.0%

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

      4. *-commutative 51.0%

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

   7. Applied egg-rr 51.0%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1.9999999999999998e-186

   1. Initial program 95.9%

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

   3. Simplified 87.4%

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

      1. div-inv 87.3%

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

   6. Applied egg-rr 96.0%

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

   if -1.9999999999999998e-186 < (/.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 24.7%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in C around inf 32.4%

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

      1. mul-1-neg 32.4%

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

   7. Simplified 32.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 3.4%

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

      1. mul-1-neg 3.4%

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

      2. +-commutative 3.4%

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

      3. unpow2 3.4%

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

      4. unpow2 3.4%

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

      5. hypot-define 18.7%

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

   5. Simplified 18.7%

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

      1. pow1/2 18.7%

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

      2. pow-to-exp 18.6%

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

   7. Applied egg-rr 18.6%

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

      1. exp-to-pow 18.7%

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

      2. pow1/2 18.7%

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

      3. associate-*l/ 18.7%

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

      4. sqrt-prod 18.7%

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

      5. distribute-frac-neg2 18.7%

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

      6. clear-num 18.7%

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

      7. associate-*r* 18.7%

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

   9. Applied egg-rr 18.7%

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

       1. div-inv 18.7%

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

       2. pow1/2 18.7%

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

       3. pow-flip 18.7%

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

       4. associate-*l* 18.7%

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

       5. metadata-eval 18.7%

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

   11. Applied egg-rr 18.7%

       \[\leadsto \frac{1}{\color{blue}{\left(-B\right) \cdot {\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{-0.5}}} \]
   12. Step-by-step derivation

       1. associate-*r* 18.7%

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

   13. Simplified 18.7%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\sqrt{\left(2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 51.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B\_m, A - C\right) - C\right)\right)} \cdot \sqrt{2 \cdot t\_3}}{-t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= t_2 -2e-186)
     (/
      (* (sqrt (* F (- A (- (hypot B_m (- A C)) C)))) (sqrt (* 2.0 t_3)))
      (- t_3))
     (if (<= t_2 INFINITY)
       (/
        (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
        (- t_0))
       (/ -1.0 (* B_m (pow (* (* 2.0 F) (- C (hypot C B_m))) -0.5)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -2e-186) {
		tmp = (sqrt((F * (A - (hypot(B_m, (A - C)) - C)))) * sqrt((2.0 * t_3))) / -t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) * (C - hypot(C, B_m))), -0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_2 <= -2e-186)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A - Float64(hypot(B_m, Float64(A - C)) - C)))) * sqrt(Float64(2.0 * t_3))) / Float64(-t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) * Float64(C - hypot(C, B_m))) ^ -0.5)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-186], N[(N[(N[Sqrt[N[(F * N[(A - N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 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))) < -1.9999999999999998e-186

   1. Initial program 44.0%

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

   3. Simplified 44.6%

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

      1. pow1/2 44.6%

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

      2. associate-*r* 49.5%

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

      3. unpow-prod-down 67.6%

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

      4. associate-+r- 65.7%

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

      5. hypot-undefine 57.7%

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

      6. unpow2 57.7%

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

      7. unpow2 57.7%

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

      8. +-commutative 57.7%

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

      9. unpow2 57.7%

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

      10. unpow2 57.7%

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

      11. hypot-define 65.7%

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

      12. pow1/2 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. unpow1/2 65.7%

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

      2. associate-+r- 67.6%

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

      3. hypot-undefine 57.7%

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

      4. unpow2 57.7%

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

      5. unpow2 57.7%

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

      6. +-commutative 57.7%

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

      7. unpow2 57.7%

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

      8. unpow2 57.7%

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

      9. hypot-undefine 67.6%

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

   8. Simplified 67.6%

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

   if -1.9999999999999998e-186 < (/.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 24.7%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in C around inf 32.4%

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

      1. mul-1-neg 32.4%

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

   7. Simplified 32.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 3.4%

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

      1. mul-1-neg 3.4%

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

      2. +-commutative 3.4%

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

      3. unpow2 3.4%

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

      4. unpow2 3.4%

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

      5. hypot-define 18.7%

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

   5. Simplified 18.7%

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

      1. pow1/2 18.7%

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

      2. pow-to-exp 18.6%

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

   7. Applied egg-rr 18.6%

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

      1. exp-to-pow 18.7%

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

      2. pow1/2 18.7%

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

      3. associate-*l/ 18.7%

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

      4. sqrt-prod 18.7%

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

      5. distribute-frac-neg2 18.7%

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

      6. clear-num 18.7%

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

      7. associate-*r* 18.7%

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

   9. Applied egg-rr 18.7%

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

       1. div-inv 18.7%

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

       2. pow1/2 18.7%

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

       3. pow-flip 18.7%

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

       4. associate-*l* 18.7%

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

       5. metadata-eval 18.7%

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

   11. Applied egg-rr 18.7%

       \[\leadsto \frac{1}{\color{blue}{\left(-B\right) \cdot {\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{-0.5}}} \]
   12. Step-by-step derivation

       1. associate-*r* 18.7%

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

   13. Simplified 18.7%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A - \left(\mathsf{hypot}\left(B, A - C\right) - C\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 10^{-245}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(A \cdot \left(--4\right) - \frac{{B\_m}^{2}}{C}\right)}\\ \mathbf{elif}\;B\_m \leq 2.05 \cdot 10^{+139}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

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 1e-245)
   (* (sqrt (* -0.5 (/ F C))) (- (sqrt 2.0)))
   (if (<= B_m 2.6e-87)
     (/
      (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
      (* C (- (* A (- -4.0)) (/ (pow B_m 2.0) C))))
     (if (<= B_m 2.05e+139)
       (-
        (sqrt
         (*
          2.0
          (*
           F
           (/
            (- (+ A C) (hypot B_m (- A C)))
            (fma -4.0 (* A C) (pow B_m 2.0)))))))
       (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e-245) {
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	} else if (B_m <= 2.6e-87) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * ((A * -(-4.0)) - (pow(B_m, 2.0) / C)));
	} else if (B_m <= 2.05e+139) {
		tmp = -sqrt((2.0 * (F * (((A + C) - hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1e-245)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 2.6e-87)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(C * Float64(Float64(A * Float64(-(-4.0))) - Float64((B_m ^ 2.0) / C))));
	elseif (B_m <= 2.05e+139)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) - hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	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[B$95$m, 1e-245], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.6e-87], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(N[(A * (--4.0)), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.05e+139], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $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 4 regimes

2. if B < 9.9999999999999993e-246

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 17.3%

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

   5. Simplified 27.5%

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

   6. Taylor expanded in A around -inf 19.1%

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

   if 9.9999999999999993e-246 < B < 2.60000000000000002e-87

   1. Initial program 28.3%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in C around inf 33.5%

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

      1. associate-*r* 33.5%

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

      2. mul-1-neg 33.5%

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

   7. Simplified 33.5%

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

   8. Taylor expanded in C around inf 33.5%

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

   if 2.60000000000000002e-87 < B < 2.0500000000000001e139

   1. Initial program 34.3%

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

   3. Taylor expanded in F around 0 33.5%

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

   5. Simplified 48.4%

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

      1. sqrt-unprod 48.6%

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

      2. associate-/l* 53.2%

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

      3. associate-+r- 55.1%

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

      4. *-commutative 55.1%

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

   7. Applied egg-rr 55.1%

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

   if 2.0500000000000001e139 < B

   1. Initial program 0.2%

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

   3. Taylor expanded in C around 0 9.2%

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

      1. mul-1-neg 9.2%

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

      2. +-commutative 9.2%

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

      3. unpow2 9.2%

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

      4. unpow2 9.2%

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

      5. hypot-define 49.7%

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

   5. Simplified 49.7%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10^{-245}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(A \cdot \left(--4\right) - \frac{{B}^{2}}{C}\right)}\\ \mathbf{elif}\;B \leq 2.05 \cdot 10^{+139}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.6 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(A \cdot \left(--4\right) - \frac{{B\_m}^{2}}{C}\right)}\\ \mathbf{elif}\;B\_m \leq 5.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{{B\_m}^{2} \cdot \frac{-F}{C}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

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 5.6e-247)
   (* (sqrt (* -0.5 (/ F C))) (- (sqrt 2.0)))
   (if (<= B_m 5e-119)
     (/
      (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
      (* C (- (* A (- -4.0)) (/ (pow B_m 2.0) C))))
     (if (<= B_m 5.7e+20)
       (/ (sqrt (* (pow B_m 2.0) (/ (- F) C))) (- B_m))
       (* (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 <= 5.6e-247) {
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	} else if (B_m <= 5e-119) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * ((A * -(-4.0)) - (pow(B_m, 2.0) / C)));
	} else if (B_m <= 5.7e+20) {
		tmp = sqrt((pow(B_m, 2.0) * (-F / C))) / -B_m;
	} 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 <= 5.6e-247) {
		tmp = Math.sqrt((-0.5 * (F / C))) * -Math.sqrt(2.0);
	} else if (B_m <= 5e-119) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * ((A * -(-4.0)) - (Math.pow(B_m, 2.0) / C)));
	} else if (B_m <= 5.7e+20) {
		tmp = Math.sqrt((Math.pow(B_m, 2.0) * (-F / C))) / -B_m;
	} 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 <= 5.6e-247:
		tmp = math.sqrt((-0.5 * (F / C))) * -math.sqrt(2.0)
	elif B_m <= 5e-119:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * ((A * -(-4.0)) - (math.pow(B_m, 2.0) / C)))
	elif B_m <= 5.7e+20:
		tmp = math.sqrt((math.pow(B_m, 2.0) * (-F / C))) / -B_m
	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 <= 5.6e-247)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 5e-119)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(C * Float64(Float64(A * Float64(-(-4.0))) - Float64((B_m ^ 2.0) / C))));
	elseif (B_m <= 5.7e+20)
		tmp = Float64(sqrt(Float64((B_m ^ 2.0) * Float64(Float64(-F) / C))) / Float64(-B_m));
	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 <= 5.6e-247)
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	elseif (B_m <= 5e-119)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * ((A * -(-4.0)) - ((B_m ^ 2.0) / C)));
	elseif (B_m <= 5.7e+20)
		tmp = sqrt(((B_m ^ 2.0) * (-F / C))) / -B_m;
	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, 5.6e-247], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 5e-119], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(N[(A * (--4.0)), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.7e+20], N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[((-F) / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $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 4 regimes

2. if B < 5.59999999999999973e-247

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 17.3%

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

   5. Simplified 27.5%

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

   6. Taylor expanded in A around -inf 19.1%

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

   if 5.59999999999999973e-247 < B < 4.99999999999999993e-119

   1. Initial program 24.8%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in C around inf 34.4%

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

      1. associate-*r* 34.4%

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

      2. mul-1-neg 34.4%

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

   7. Simplified 34.4%

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

   8. Taylor expanded in C around inf 34.4%

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

   if 4.99999999999999993e-119 < B < 5.7e20

   1. Initial program 31.7%

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

   3. Taylor expanded in A around 0 25.4%

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

      1. mul-1-neg 25.4%

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

      2. +-commutative 25.4%

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

      3. unpow2 25.4%

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

      4. unpow2 25.4%

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

      5. hypot-define 29.3%

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

   5. Simplified 29.3%

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

      1. neg-sub0 29.3%

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

      2. associate-*l/ 29.3%

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

      3. pow1/2 29.3%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 29.3%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 29.4%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 29.4%

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

      2. distribute-neg-frac2 29.4%

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

      3. unpow1/2 29.3%

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

   9. Simplified 29.3%

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

   10. Taylor expanded in C around inf 24.7%

       \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \frac{{B}^{2} \cdot F}{C}}}}{-B} \]
   11. Step-by-step derivation

       1. mul-1-neg 24.7%

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

       2. associate-/l* 28.3%

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

   12. Simplified 28.3%

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

   if 5.7e20 < B

   1. Initial program 17.8%

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

   3. Taylor expanded in C around 0 27.0%

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

      1. mul-1-neg 27.0%

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

      2. +-commutative 27.0%

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

      3. unpow2 27.0%

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

      4. unpow2 27.0%

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

      5. hypot-define 51.5%

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

   5. Simplified 51.5%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.6 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 5 \cdot 10^{-119}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(A \cdot \left(--4\right) - \frac{{B}^{2}}{C}\right)}\\ \mathbf{elif}\;B \leq 5.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{{B}^{2} \cdot \frac{-F}{C}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.25 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(\frac{{B\_m}^{2}}{-C} - A \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.25e-246)
   (* (sqrt (* -0.5 (/ F C))) (- (sqrt 2.0)))
   (if (<= B_m 5.5e-86)
     (/
      (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
      (* C (- (/ (pow B_m 2.0) (- C)) (* A -4.0))))
     (/ -1.0 (* B_m (pow (* (* 2.0 F) (- C (hypot C B_m))) -0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.25e-246) {
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	} else if (B_m <= 5.5e-86) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * ((pow(B_m, 2.0) / -C) - (A * -4.0)));
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) * (C - hypot(C, B_m))), -0.5));
	}
	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 <= 2.25e-246) {
		tmp = Math.sqrt((-0.5 * (F / C))) * -Math.sqrt(2.0);
	} else if (B_m <= 5.5e-86) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * ((Math.pow(B_m, 2.0) / -C) - (A * -4.0)));
	} else {
		tmp = -1.0 / (B_m * Math.pow(((2.0 * F) * (C - Math.hypot(C, B_m))), -0.5));
	}
	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 <= 2.25e-246:
		tmp = math.sqrt((-0.5 * (F / C))) * -math.sqrt(2.0)
	elif B_m <= 5.5e-86:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * ((math.pow(B_m, 2.0) / -C) - (A * -4.0)))
	else:
		tmp = -1.0 / (B_m * math.pow(((2.0 * F) * (C - math.hypot(C, B_m))), -0.5))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.25e-246)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 5.5e-86)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(C * Float64(Float64((B_m ^ 2.0) / Float64(-C)) - Float64(A * -4.0))));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) * Float64(C - hypot(C, B_m))) ^ -0.5)));
	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 <= 2.25e-246)
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	elseif (B_m <= 5.5e-86)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / (C * (((B_m ^ 2.0) / -C) - (A * -4.0)));
	else
		tmp = -1.0 / (B_m * (((2.0 * F) * (C - hypot(C, B_m))) ^ -0.5));
	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, 2.25e-246], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 5.5e-86], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] / (-C)), $MachinePrecision] - N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.25e-246

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 17.3%

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

   5. Simplified 27.5%

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

   6. Taylor expanded in A around -inf 19.1%

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

   if 2.25e-246 < B < 5.5e-86

   1. Initial program 28.3%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in C around inf 33.5%

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

      1. associate-*r* 33.5%

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

      2. mul-1-neg 33.5%

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

   7. Simplified 33.5%

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

   8. Taylor expanded in C around inf 33.5%

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

   if 5.5e-86 < B

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 26.5%

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

      1. mul-1-neg 26.5%

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

      2. +-commutative 26.5%

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

      3. unpow2 26.5%

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

      4. unpow2 26.5%

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

      5. hypot-define 46.6%

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

   5. Simplified 46.6%

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

      1. pow1/2 46.6%

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

      2. pow-to-exp 46.6%

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

   7. Applied egg-rr 46.6%

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

      1. exp-to-pow 46.6%

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

      2. pow1/2 46.6%

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

      3. associate-*l/ 46.6%

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

      4. sqrt-prod 46.7%

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

      5. distribute-frac-neg2 46.7%

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

      6. clear-num 46.7%

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

      7. associate-*r* 46.7%

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

   9. Applied egg-rr 46.7%

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

       1. div-inv 46.7%

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

       2. pow1/2 46.7%

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

       3. pow-flip 46.7%

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

       4. associate-*l* 46.7%

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

       5. metadata-eval 46.7%

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

   11. Applied egg-rr 46.7%

       \[\leadsto \frac{1}{\color{blue}{\left(-B\right) \cdot {\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{-0.5}}} \]
   12. Step-by-step derivation

       1. associate-*r* 46.7%

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

   13. Simplified 46.7%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.25 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{C \cdot \left(\frac{{B}^{2}}{-C} - A \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot t\_0\\ \mathbf{elif}\;B\_m \leq 2.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\sqrt{{B\_m}^{2} \cdot \frac{-F}{C}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 1.75e-246)
     (* (sqrt (* -0.5 (/ F C))) t_0)
     (if (<= B_m 2.4e-118)
       (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
       (if (<= B_m 5.5e+25)
         (/ (sqrt (* (pow B_m 2.0) (/ (- F) C))) (- B_m))
         (* (sqrt (/ F (- B_m))) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 1.75e-246) {
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	} else if (B_m <= 2.4e-118) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else if (B_m <= 5.5e+25) {
		tmp = sqrt((pow(B_m, 2.0) * (-F / C))) / -B_m;
	} else {
		tmp = sqrt((F / -B_m)) * t_0;
	}
	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 = -sqrt(2.0d0)
    if (b_m <= 1.75d-246) then
        tmp = sqrt(((-0.5d0) * (f / c))) * t_0
    else if (b_m <= 2.4d-118) then
        tmp = sqrt(((a * (-8.0d0)) * (c * (f * (a + a))))) / ((4.0d0 * a) * c)
    else if (b_m <= 5.5d+25) then
        tmp = sqrt(((b_m ** 2.0d0) * (-f / c))) / -b_m
    else
        tmp = sqrt((f / -b_m)) * t_0
    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 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 1.75e-246) {
		tmp = Math.sqrt((-0.5 * (F / C))) * t_0;
	} else if (B_m <= 2.4e-118) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else if (B_m <= 5.5e+25) {
		tmp = Math.sqrt((Math.pow(B_m, 2.0) * (-F / C))) / -B_m;
	} else {
		tmp = Math.sqrt((F / -B_m)) * t_0;
	}
	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 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 1.75e-246:
		tmp = math.sqrt((-0.5 * (F / C))) * t_0
	elif B_m <= 2.4e-118:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	elif B_m <= 5.5e+25:
		tmp = math.sqrt((math.pow(B_m, 2.0) * (-F / C))) / -B_m
	else:
		tmp = math.sqrt((F / -B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.75e-246)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_0);
	elseif (B_m <= 2.4e-118)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 5.5e+25)
		tmp = Float64(sqrt(Float64((B_m ^ 2.0) * Float64(Float64(-F) / C))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(F / Float64(-B_m))) * t_0);
	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 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 1.75e-246)
		tmp = sqrt((-0.5 * (F / C))) * t_0;
	elseif (B_m <= 2.4e-118)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	elseif (B_m <= 5.5e+25)
		tmp = sqrt(((B_m ^ 2.0) * (-F / C))) / -B_m;
	else
		tmp = sqrt((F / -B_m)) * t_0;
	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[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 1.75e-246], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 2.4e-118], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.5e+25], N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[((-F) / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(F / (-B$95$m)), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.7500000000000001e-246

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 17.3%

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

   5. Simplified 27.5%

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

   6. Taylor expanded in A around -inf 19.1%

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

   if 1.7500000000000001e-246 < B < 2.4000000000000001e-118

   1. Initial program 24.8%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in C around inf 26.9%

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

      1. associate-*r* 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in C around inf 34.1%

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

      1. associate-*r* 34.4%

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

      2. mul-1-neg 34.4%

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

   10. Simplified 34.1%

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

   if 2.4000000000000001e-118 < B < 5.50000000000000018e25

   1. Initial program 30.6%

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

   3. Taylor expanded in A around 0 24.5%

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

      1. mul-1-neg 24.5%

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

      2. +-commutative 24.5%

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

      3. unpow2 24.5%

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

      4. unpow2 24.5%

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

      5. hypot-define 28.4%

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

   5. Simplified 28.4%

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

      1. neg-sub0 28.4%

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

      2. associate-*l/ 28.4%

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

      3. pow1/2 28.4%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 28.4%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 28.5%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 28.5%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 28.5%

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

      2. distribute-neg-frac2 28.5%

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

      3. unpow1/2 28.4%

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

   9. Simplified 28.4%

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

   10. Taylor expanded in C around inf 23.9%

       \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \frac{{B}^{2} \cdot F}{C}}}}{-B} \]
   11. Step-by-step derivation

       1. mul-1-neg 23.9%

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

       2. associate-/l* 27.4%

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

   12. Simplified 27.4%

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

   if 5.50000000000000018e25 < B

   1. Initial program 18.2%

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

   3. Taylor expanded in F around 0 21.9%

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

   5. Simplified 31.3%

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

   6. Taylor expanded in B around inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. mul-1-neg 56.4%

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

   8. Simplified 56.4%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.75 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 2.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{\sqrt{{B}^{2} \cdot \frac{-F}{C}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \sqrt{-0.5 \cdot \frac{F}{C}} \cdot t\_0\\ \mathbf{if}\;B\_m \leq 1.85 \cdot 10^{-247}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 4.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 7.4 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))) (t_1 (* (sqrt (* -0.5 (/ F C))) t_0)))
   (if (<= B_m 1.85e-247)
     t_1
     (if (<= B_m 4.5e-120)
       (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
       (if (<= B_m 7.4e+25) t_1 (* (sqrt (/ F (- B_m))) t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = sqrt((-0.5 * (F / C))) * t_0;
	double tmp;
	if (B_m <= 1.85e-247) {
		tmp = t_1;
	} else if (B_m <= 4.5e-120) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else if (B_m <= 7.4e+25) {
		tmp = t_1;
	} else {
		tmp = sqrt((F / -B_m)) * t_0;
	}
	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) :: tmp
    t_0 = -sqrt(2.0d0)
    t_1 = sqrt(((-0.5d0) * (f / c))) * t_0
    if (b_m <= 1.85d-247) then
        tmp = t_1
    else if (b_m <= 4.5d-120) then
        tmp = sqrt(((a * (-8.0d0)) * (c * (f * (a + a))))) / ((4.0d0 * a) * c)
    else if (b_m <= 7.4d+25) then
        tmp = t_1
    else
        tmp = sqrt((f / -b_m)) * t_0
    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 = -Math.sqrt(2.0);
	double t_1 = Math.sqrt((-0.5 * (F / C))) * t_0;
	double tmp;
	if (B_m <= 1.85e-247) {
		tmp = t_1;
	} else if (B_m <= 4.5e-120) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else if (B_m <= 7.4e+25) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((F / -B_m)) * t_0;
	}
	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 = -math.sqrt(2.0)
	t_1 = math.sqrt((-0.5 * (F / C))) * t_0
	tmp = 0
	if B_m <= 1.85e-247:
		tmp = t_1
	elif B_m <= 4.5e-120:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	elif B_m <= 7.4e+25:
		tmp = t_1
	else:
		tmp = math.sqrt((F / -B_m)) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * t_0)
	tmp = 0.0
	if (B_m <= 1.85e-247)
		tmp = t_1;
	elseif (B_m <= 4.5e-120)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	elseif (B_m <= 7.4e+25)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(F / Float64(-B_m))) * t_0);
	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 = -sqrt(2.0);
	t_1 = sqrt((-0.5 * (F / C))) * t_0;
	tmp = 0.0;
	if (B_m <= 1.85e-247)
		tmp = t_1;
	elseif (B_m <= 4.5e-120)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	elseif (B_m <= 7.4e+25)
		tmp = t_1;
	else
		tmp = sqrt((F / -B_m)) * t_0;
	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[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[B$95$m, 1.85e-247], t$95$1, If[LessEqual[B$95$m, 4.5e-120], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.4e+25], t$95$1, N[(N[Sqrt[N[(F / (-B$95$m)), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.85000000000000005e-247 or 4.5e-120 < B < 7.3999999999999998e25

   1. Initial program 19.0%

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

   3. Taylor expanded in F around 0 19.2%

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

   5. Simplified 29.2%

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

   6. Taylor expanded in A around -inf 19.6%

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

   if 1.85000000000000005e-247 < B < 4.5e-120

   1. Initial program 24.8%

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

   3. Simplified 27.2%

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

   5. Taylor expanded in C around inf 26.9%

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

      1. associate-*r* 26.9%

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

   7. Simplified 26.9%

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

   8. Taylor expanded in C around inf 34.1%

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

      1. associate-*r* 34.4%

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

      2. mul-1-neg 34.4%

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

   10. Simplified 34.1%

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

   if 7.3999999999999998e25 < B

   1. Initial program 18.2%

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

   3. Taylor expanded in F around 0 21.9%

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

   5. Simplified 31.3%

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

   6. Taylor expanded in B around inf 56.4%

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

      1. associate-*r/ 56.4%

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

      2. mul-1-neg 56.4%

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

   8. Simplified 56.4%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.85 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 4.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 7.4 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7.6 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 3 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B\_m \cdot {\left(\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.6e-246)
   (* (sqrt (* -0.5 (/ F C))) (- (sqrt 2.0)))
   (if (<= B_m 3e-86)
     (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
     (/ -1.0 (* B_m (pow (* (* 2.0 F) (- C (hypot C B_m))) -0.5))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.6e-246) {
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	} else if (B_m <= 3e-86) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / (B_m * pow(((2.0 * F) * (C - hypot(C, B_m))), -0.5));
	}
	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 <= 7.6e-246) {
		tmp = Math.sqrt((-0.5 * (F / C))) * -Math.sqrt(2.0);
	} else if (B_m <= 3e-86) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / (B_m * Math.pow(((2.0 * F) * (C - Math.hypot(C, B_m))), -0.5));
	}
	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 <= 7.6e-246:
		tmp = math.sqrt((-0.5 * (F / C))) * -math.sqrt(2.0)
	elif B_m <= 3e-86:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	else:
		tmp = -1.0 / (B_m * math.pow(((2.0 * F) * (C - math.hypot(C, B_m))), -0.5))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.6e-246)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 3e-86)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) * Float64(C - hypot(C, B_m))) ^ -0.5)));
	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 <= 7.6e-246)
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	elseif (B_m <= 3e-86)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	else
		tmp = -1.0 / (B_m * (((2.0 * F) * (C - hypot(C, B_m))) ^ -0.5));
	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, 7.6e-246], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3e-86], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.59999999999999951e-246

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 17.3%

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

   5. Simplified 27.5%

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

   6. Taylor expanded in A around -inf 19.1%

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

   if 7.59999999999999951e-246 < B < 3.0000000000000001e-86

   1. Initial program 28.3%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in C around inf 27.4%

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

      1. associate-*r* 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in C around inf 33.3%

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

      1. associate-*r* 33.5%

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

      2. mul-1-neg 33.5%

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

   10. Simplified 33.3%

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

   if 3.0000000000000001e-86 < B

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 26.5%

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

      1. mul-1-neg 26.5%

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

      2. +-commutative 26.5%

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

      3. unpow2 26.5%

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

      4. unpow2 26.5%

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

      5. hypot-define 46.6%

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

   5. Simplified 46.6%

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

      1. pow1/2 46.6%

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

      2. pow-to-exp 46.6%

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

   7. Applied egg-rr 46.6%

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

      1. exp-to-pow 46.6%

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

      2. pow1/2 46.6%

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

      3. associate-*l/ 46.6%

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

      4. sqrt-prod 46.7%

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

      5. distribute-frac-neg2 46.7%

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

      6. clear-num 46.7%

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

      7. associate-*r* 46.7%

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

   9. Applied egg-rr 46.7%

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

       1. div-inv 46.7%

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

       2. pow1/2 46.7%

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

       3. pow-flip 46.7%

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

       4. associate-*l* 46.7%

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

       5. metadata-eval 46.7%

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

   11. Applied egg-rr 46.7%

       \[\leadsto \frac{1}{\color{blue}{\left(-B\right) \cdot {\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{-0.5}}} \]
   12. Step-by-step derivation

       1. associate-*r* 46.7%

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

   13. Simplified 46.7%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.6 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 3 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B \cdot {\left(\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 1.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 6.2e-246)
   (* (sqrt (* -0.5 (/ F C))) (- (sqrt 2.0)))
   (if (<= B_m 1.5e-86)
     (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
     (/ -1.0 (/ B_m (sqrt (* (* 2.0 F) (- C (hypot C B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 6.2e-246) {
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	} else if (B_m <= 1.5e-86) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / (B_m / sqrt(((2.0 * F) * (C - hypot(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 (B_m <= 6.2e-246) {
		tmp = Math.sqrt((-0.5 * (F / C))) * -Math.sqrt(2.0);
	} else if (B_m <= 1.5e-86) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / (B_m / Math.sqrt(((2.0 * F) * (C - Math.hypot(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 B_m <= 6.2e-246:
		tmp = math.sqrt((-0.5 * (F / C))) * -math.sqrt(2.0)
	elif B_m <= 1.5e-86:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	else:
		tmp = -1.0 / (B_m / math.sqrt(((2.0 * F) * (C - math.hypot(C, B_m)))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 6.2e-246)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 1.5e-86)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(2.0 * F) * Float64(C - hypot(C, 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 <= 6.2e-246)
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	elseif (B_m <= 1.5e-86)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	else
		tmp = -1.0 / (B_m / sqrt(((2.0 * F) * (C - hypot(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[B$95$m, 6.2e-246], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.5e-86], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.2000000000000001e-246

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 17.3%

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

   5. Simplified 27.5%

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

   6. Taylor expanded in A around -inf 19.1%

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

   if 6.2000000000000001e-246 < B < 1.5e-86

   1. Initial program 28.3%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in C around inf 27.4%

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

      1. associate-*r* 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in C around inf 33.3%

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

      1. associate-*r* 33.5%

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

      2. mul-1-neg 33.5%

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

   10. Simplified 33.3%

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

   if 1.5e-86 < B

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 26.5%

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

      1. mul-1-neg 26.5%

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

      2. +-commutative 26.5%

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

      3. unpow2 26.5%

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

      4. unpow2 26.5%

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

      5. hypot-define 46.6%

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

   5. Simplified 46.6%

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

      1. pow1/2 46.6%

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

      2. pow-to-exp 46.6%

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

   7. Applied egg-rr 46.6%

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

      1. exp-to-pow 46.6%

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

      2. pow1/2 46.6%

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

      3. associate-*l/ 46.6%

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

      4. sqrt-prod 46.7%

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

      5. distribute-frac-neg2 46.7%

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

      6. clear-num 46.7%

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

      7. associate-*r* 46.7%

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

   9. Applied egg-rr 46.7%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.7 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B\_m \leq 1.35 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.7e-246)
   (* (sqrt (* -0.5 (/ F C))) (- (sqrt 2.0)))
   (if (<= B_m 1.35e-86)
     (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
     (/ (sqrt (* 2.0 (* F (- C (hypot C B_m))))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.7e-246) {
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	} else if (B_m <= 1.35e-86) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -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 <= 3.7e-246) {
		tmp = Math.sqrt((-0.5 * (F / C))) * -Math.sqrt(2.0);
	} else if (B_m <= 1.35e-86) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = Math.sqrt((2.0 * (F * (C - Math.hypot(C, B_m))))) / -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 <= 3.7e-246:
		tmp = math.sqrt((-0.5 * (F / C))) * -math.sqrt(2.0)
	elif B_m <= 1.35e-86:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	else:
		tmp = math.sqrt((2.0 * (F * (C - math.hypot(C, B_m))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.7e-246)
		tmp = Float64(sqrt(Float64(-0.5 * Float64(F / C))) * Float64(-sqrt(2.0)));
	elseif (B_m <= 1.35e-86)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C - hypot(C, B_m))))) / 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 <= 3.7e-246)
		tmp = sqrt((-0.5 * (F / C))) * -sqrt(2.0);
	elseif (B_m <= 1.35e-86)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	else
		tmp = sqrt((2.0 * (F * (C - hypot(C, B_m))))) / -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, 3.7e-246], N[(N[Sqrt[N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.35e-86], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.7e-246

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 17.3%

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

   5. Simplified 27.5%

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

   6. Taylor expanded in A around -inf 19.1%

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

   if 3.7e-246 < B < 1.34999999999999996e-86

   1. Initial program 28.3%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in C around inf 27.4%

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

      1. associate-*r* 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in C around inf 33.3%

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

      1. associate-*r* 33.5%

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

      2. mul-1-neg 33.5%

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

   10. Simplified 33.3%

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

   if 1.34999999999999996e-86 < B

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 26.5%

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

      1. mul-1-neg 26.5%

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

      2. +-commutative 26.5%

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

      3. unpow2 26.5%

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

      4. unpow2 26.5%

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

      5. hypot-define 46.6%

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

   5. Simplified 46.6%

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

      1. neg-sub0 46.6%

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

      2. associate-*l/ 46.6%

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

      3. pow1/2 46.6%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

      4. pow1/2 46.6%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 46.7%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 46.7%

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

      2. distribute-neg-frac2 46.7%

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

      3. unpow1/2 46.7%

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

   9. Simplified 46.7%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.7 \cdot 10^{-246}:\\ \;\;\;\;\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;B \leq 1.35 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 0.00016:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 0.00016)
   (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
   (* (sqrt (/ F (- B_m))) (- (sqrt 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.00016) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = sqrt((F / -B_m)) * -sqrt(2.0);
	}
	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 (b_m <= 0.00016d0) then
        tmp = sqrt(((a * (-8.0d0)) * (c * (f * (a + a))))) / ((4.0d0 * a) * c)
    else
        tmp = sqrt((f / -b_m)) * -sqrt(2.0d0)
    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 (B_m <= 0.00016) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = Math.sqrt((F / -B_m)) * -Math.sqrt(2.0);
	}
	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 <= 0.00016:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	else:
		tmp = math.sqrt((F / -B_m)) * -math.sqrt(2.0)
	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 <= 0.00016)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(sqrt(Float64(F / Float64(-B_m))) * Float64(-sqrt(2.0)));
	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 <= 0.00016)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	else
		tmp = sqrt((F / -B_m)) * -sqrt(2.0);
	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, 0.00016], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / (-B$95$m)), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.60000000000000013e-4

   1. Initial program 20.2%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in C around inf 18.4%

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

      1. associate-*r* 18.4%

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

   7. Simplified 18.4%

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

   8. Taylor expanded in C around inf 19.1%

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

      1. associate-*r* 18.8%

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

      2. mul-1-neg 18.8%

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

   10. Simplified 19.1%

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

   if 1.60000000000000013e-4 < B

   1. Initial program 17.0%

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

   3. Taylor expanded in F around 0 20.5%

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

   5. Simplified 29.7%

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

   6. Taylor expanded in B around inf 52.7%

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

      1. associate-*r/ 52.7%

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

      2. mul-1-neg 52.7%

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

   8. Simplified 52.7%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.00016:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{-B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.8e-86)
   (/ (sqrt (* (* A -8.0) (* C (* F (+ A A))))) (* (* 4.0 A) C))
   (/ -1.0 (/ B_m (sqrt (* -2.0 (* B_m F)))))))

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 <= 4.8e-86) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / (B_m / sqrt((-2.0 * (B_m * F))));
	}
	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 (b_m <= 4.8d-86) then
        tmp = sqrt(((a * (-8.0d0)) * (c * (f * (a + a))))) / ((4.0d0 * a) * c)
    else
        tmp = (-1.0d0) / (b_m / sqrt(((-2.0d0) * (b_m * f))))
    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 (B_m <= 4.8e-86) {
		tmp = Math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / (B_m / Math.sqrt((-2.0 * (B_m * F))));
	}
	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 <= 4.8e-86:
		tmp = math.sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C)
	else:
		tmp = -1.0 / (B_m / math.sqrt((-2.0 * (B_m * F))))
	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 <= 4.8e-86)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(-2.0 * Float64(B_m * F)))));
	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 <= 4.8e-86)
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / ((4.0 * A) * C);
	else
		tmp = -1.0 / (B_m / sqrt((-2.0 * (B_m * F))));
	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, 4.8e-86], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.80000000000000026e-86

   1. Initial program 19.1%

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

   3. Simplified 24.3%

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

   5. Taylor expanded in C around inf 17.6%

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

      1. associate-*r* 17.6%

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

   7. Simplified 17.6%

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

   8. Taylor expanded in C around inf 19.1%

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

      1. associate-*r* 18.7%

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

      2. mul-1-neg 18.7%

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

   10. Simplified 19.1%

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

   if 4.80000000000000026e-86 < B

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 26.5%

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

      1. mul-1-neg 26.5%

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

      2. +-commutative 26.5%

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

      3. unpow2 26.5%

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

      4. unpow2 26.5%

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

      5. hypot-define 46.6%

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

   5. Simplified 46.6%

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

      1. pow1/2 46.6%

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

      2. pow-to-exp 46.6%

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

   7. Applied egg-rr 46.6%

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

      1. exp-to-pow 46.6%

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

      2. pow1/2 46.6%

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

      3. associate-*l/ 46.6%

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

      4. sqrt-prod 46.7%

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

      5. distribute-frac-neg2 46.7%

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

      6. clear-num 46.7%

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

      7. associate-*r* 46.7%

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

   9. Applied egg-rr 46.7%

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

   10. Taylor expanded in C around 0 40.1%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{-2 \cdot \left(B \cdot F\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.95 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.95e-54)
   (/ (sqrt (* F (* -8.0 (* A (* C (+ A A)))))) (* (* 4.0 A) C))
   (/ -1.0 (/ B_m (sqrt (* -2.0 (* B_m F)))))))

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 <= 1.95e-54) {
		tmp = sqrt((F * (-8.0 * (A * (C * (A + A)))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / (B_m / sqrt((-2.0 * (B_m * F))));
	}
	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 (b_m <= 1.95d-54) then
        tmp = sqrt((f * ((-8.0d0) * (a * (c * (a + a)))))) / ((4.0d0 * a) * c)
    else
        tmp = (-1.0d0) / (b_m / sqrt(((-2.0d0) * (b_m * f))))
    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 (B_m <= 1.95e-54) {
		tmp = Math.sqrt((F * (-8.0 * (A * (C * (A + A)))))) / ((4.0 * A) * C);
	} else {
		tmp = -1.0 / (B_m / Math.sqrt((-2.0 * (B_m * F))));
	}
	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 <= 1.95e-54:
		tmp = math.sqrt((F * (-8.0 * (A * (C * (A + A)))))) / ((4.0 * A) * C)
	else:
		tmp = -1.0 / (B_m / math.sqrt((-2.0 * (B_m * F))))
	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 <= 1.95e-54)
		tmp = Float64(sqrt(Float64(F * Float64(-8.0 * Float64(A * Float64(C * Float64(A + A)))))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(-2.0 * Float64(B_m * F)))));
	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 <= 1.95e-54)
		tmp = sqrt((F * (-8.0 * (A * (C * (A + A)))))) / ((4.0 * A) * C);
	else
		tmp = -1.0 / (B_m / sqrt((-2.0 * (B_m * F))));
	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, 1.95e-54], N[(N[Sqrt[N[(F * N[(-8.0 * N[(A * N[(C * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.95e-54

   1. Initial program 19.1%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in C around inf 17.7%

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

      1. associate-*r* 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in C around inf 17.4%

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

      1. mul-1-neg 17.4%

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

   10. Simplified 17.4%

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

   if 1.95e-54 < B

   1. Initial program 20.8%

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

   3. Taylor expanded in A around 0 25.7%

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

      1. mul-1-neg 25.7%

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

      2. +-commutative 25.7%

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

      3. unpow2 25.7%

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

      4. unpow2 25.7%

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

      5. hypot-define 47.6%

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

   5. Simplified 47.6%

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

      1. pow1/2 47.6%

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

      2. pow-to-exp 47.6%

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

   7. Applied egg-rr 47.6%

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

      1. exp-to-pow 47.6%

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

      2. pow1/2 47.6%

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

      3. associate-*l/ 47.6%

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

      4. sqrt-prod 47.8%

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

      5. distribute-frac-neg2 47.8%

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

      6. clear-num 47.8%

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

      7. associate-*r* 47.8%

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

   9. Applied egg-rr 47.8%

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

   10. Taylor expanded in C around 0 40.5%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.95 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot \left(A + A\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{-2 \cdot \left(B \cdot F\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.9% accurate, 5.8× speedup

Math

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

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
 (/ -1.0 (/ B_m (sqrt (* -2.0 (* B_m F))))))

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 -1.0 / (B_m / sqrt((-2.0 * (B_m * F))));
}

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 = (-1.0d0) / (b_m / sqrt(((-2.0d0) * (b_m * f))))
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 -1.0 / (B_m / Math.sqrt((-2.0 * (B_m * F))));
}

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 -1.0 / (B_m / math.sqrt((-2.0 * (B_m * F))))

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(-1.0 / Float64(B_m / sqrt(Float64(-2.0 * Float64(B_m * F)))))
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 = -1.0 / (B_m / sqrt((-2.0 * (B_m * F))));
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[(-1.0 / N[(B$95$m / N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in A around 0 11.2%

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

   1. mul-1-neg 11.2%

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

   2. +-commutative 11.2%

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

   3. unpow2 11.2%

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

   4. unpow2 11.2%

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

   5. hypot-define 18.6%

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

5. Simplified 18.6%

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

   1. pow1/2 18.6%

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

   2. pow-to-exp 18.5%

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

7. Applied egg-rr 18.5%

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

   1. exp-to-pow 18.6%

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

   2. pow1/2 18.6%

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

   3. associate-*l/ 18.5%

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

   4. sqrt-prod 18.6%

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

   5. distribute-frac-neg2 18.6%

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

   6. clear-num 18.6%

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

   7. associate-*r* 18.6%

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

9. Applied egg-rr 18.6%

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

10. Taylor expanded in C around 0 15.8%

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

11. Final simplification 15.8%

    \[\leadsto \frac{-1}{\frac{B}{\sqrt{-2 \cdot \left(B \cdot F\right)}}} \]
12. Add Preprocessing

Alternative 15: 25.9% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((-2.0 * (B_m * 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(sqrt(Float64(-2.0 * Float64(B_m * F))) / 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((-2.0 * (B_m * 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[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in A around 0 11.2%

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

   1. mul-1-neg 11.2%

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

   2. +-commutative 11.2%

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

   3. unpow2 11.2%

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

   4. unpow2 11.2%

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

   5. hypot-define 18.6%

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

5. Simplified 18.6%

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

   1. neg-sub0 18.6%

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

   2. associate-*l/ 18.5%

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

   3. pow1/2 18.5%

      \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{B} \]

   4. pow1/2 18.6%

      \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}^{0.5}}}{B} \]

   5. pow-prod-down 18.6%

      \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}}{B} \]

7. Applied egg-rr 18.6%

   \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}} \]
8. Step-by-step derivation

   1. neg-sub0 18.6%

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

   2. distribute-neg-frac2 18.6%

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

   3. unpow1/2 18.6%

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

9. Simplified 18.6%

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

10. Taylor expanded in C around 0 15.8%

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

Alternative 16: 1.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.5%

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

3. Taylor expanded in A around 0 11.2%

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

   1. mul-1-neg 11.2%

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

   2. +-commutative 11.2%

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

   3. unpow2 11.2%

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

   4. unpow2 11.2%

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

   5. hypot-define 18.6%

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

5. Simplified 18.6%

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

   1. pow1/2 18.6%

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

   2. pow-to-exp 18.5%

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

7. Applied egg-rr 18.5%

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

8. Taylor expanded in C around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 3.3%

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

   5. unpow2 3.3%

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

   6. rem-square-sqrt 3.3%

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

   7. metadata-eval 3.3%

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

10. Simplified 3.3%

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

11. Final simplification 3.3%

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

Reproduce

herbie shell --seed 2024172 
(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))))