ABCF->ab-angle b

Percentage Accurate: 18.9% → 50.1%

Time: 16.4s

Alternatives: 9

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% 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: 50.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot 2}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-203}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-1}{B\_m} \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* A 4.0)))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))
            (* (* F (- (pow B_m 2.0) t_0)) 2.0)))
          t_1)))
   (if (<= t_2 (- INFINITY))
     (/
      (* (sqrt (* (+ A A) F)) (sqrt (* (fma -4.0 (* C A) (* B_m B_m)) 2.0)))
      t_1)
     (if (<= t_2 -1e-203)
       t_2
       (if (<= t_2 INFINITY)
         (/
          1.0
          (/
           (fma (* C 4.0) A (* (- B_m) B_m))
           (sqrt
            (*
             (fma (* -4.0 C) A (* B_m B_m))
             (* (* F 2.0) (+ (fma (/ (* B_m B_m) C) -0.5 A) A))))))
         (* (- (sqrt 2.0)) (sqrt (* (/ -1.0 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 t_0 = C * (A * 4.0);
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))) * ((F * (pow(B_m, 2.0) - t_0)) * 2.0))) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (sqrt(((A + A) * F)) * sqrt((fma(-4.0, (C * A), (B_m * B_m)) * 2.0))) / t_1;
	} else if (t_2 <= -1e-203) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 1.0 / (fma((C * 4.0), A, (-B_m * B_m)) / sqrt((fma((-4.0 * C), A, (B_m * B_m)) * ((F * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A)))));
	} else {
		tmp = -sqrt(2.0) * sqrt(((-1.0 / 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)
	t_0 = Float64(C * Float64(A * 4.0))
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0))) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * F)) * sqrt(Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * 2.0))) / t_1);
	elseif (t_2 <= -1e-203)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = Float64(1.0 / Float64(fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m)) / sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))))));
	else
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(Float64(-1.0 / B_m) * F)));
	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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -1e-203], t$95$2, If[LessEqual[t$95$2, Infinity], N[(1.0 / N[(N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(N[(-1.0 / B$95$m), $MachinePrecision] * F), $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 3.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 8.2%

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

   5. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 23.4

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

   7. Applied rewrites 23.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-203

   1. Initial program 99.0%

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

   if -1e-203 < (/.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 18.8%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 42.0

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

   5. Applied rewrites 42.0%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 42.0

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

   7. Applied rewrites 42.0%

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

   9. Applied rewrites 42.0%

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

   11. Applied rewrites 43.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 2.9

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

   5. Applied rewrites 2.9%

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 4.0%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 10.1%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -1 \cdot 10^{-203}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-1}{B} \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := \frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_1}\\ t_3 := -\sqrt{2}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot 2}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-158}:\\ \;\;\;\;t\_3 \cdot \sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)} \cdot F}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C \cdot 4, A, \left(-B\_m\right) \cdot B\_m\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \sqrt{\frac{-1}{B\_m} \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* A 4.0)))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2
         (/
          (sqrt
           (*
            (- (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))
            (* (* F (- (pow B_m 2.0) t_0)) 2.0)))
          t_1))
        (t_3 (- (sqrt 2.0))))
   (if (<= t_2 -4e+165)
     (/
      (* (sqrt (* (+ A A) F)) (sqrt (* (fma -4.0 (* C A) (* B_m B_m)) 2.0)))
      t_1)
     (if (<= t_2 -2e-158)
       (*
        t_3
        (sqrt
         (*
          (/
           (- (+ C A) (sqrt (fma (- A C) (- A C) (* B_m B_m))))
           (fma (* -4.0 A) C (* B_m B_m)))
          F)))
       (if (<= t_2 INFINITY)
         (/
          1.0
          (/
           (fma (* C 4.0) A (* (- B_m) B_m))
           (sqrt
            (*
             (fma (* -4.0 C) A (* B_m B_m))
             (* (* F 2.0) (+ (fma (/ (* B_m B_m) C) -0.5 A) A))))))
         (* t_3 (sqrt (* (/ -1.0 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 t_0 = C * (A * 4.0);
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = sqrt((((C + A) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))) * ((F * (pow(B_m, 2.0) - t_0)) * 2.0))) / t_1;
	double t_3 = -sqrt(2.0);
	double tmp;
	if (t_2 <= -4e+165) {
		tmp = (sqrt(((A + A) * F)) * sqrt((fma(-4.0, (C * A), (B_m * B_m)) * 2.0))) / t_1;
	} else if (t_2 <= -2e-158) {
		tmp = t_3 * sqrt(((((C + A) - sqrt(fma((A - C), (A - C), (B_m * B_m)))) / fma((-4.0 * A), C, (B_m * B_m))) * F));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 1.0 / (fma((C * 4.0), A, (-B_m * B_m)) / sqrt((fma((-4.0 * C), A, (B_m * B_m)) * ((F * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A)))));
	} else {
		tmp = t_3 * sqrt(((-1.0 / 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)
	t_0 = Float64(C * Float64(A * 4.0))
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(sqrt(Float64(Float64(Float64(C + A) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0))) / t_1)
	t_3 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (t_2 <= -4e+165)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + A) * F)) * sqrt(Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * 2.0))) / t_1);
	elseif (t_2 <= -2e-158)
		tmp = Float64(t_3 * sqrt(Float64(Float64(Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m)))) / fma(Float64(-4.0 * A), C, Float64(B_m * B_m))) * F)));
	elseif (t_2 <= Inf)
		tmp = Float64(1.0 / Float64(fma(Float64(C * 4.0), A, Float64(Float64(-B_m) * B_m)) / sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * Float64(Float64(F * 2.0) * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))))));
	else
		tmp = Float64(t_3 * sqrt(Float64(Float64(-1.0 / B_m) * F)));
	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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[t$95$2, -4e+165], N[(N[(N[Sqrt[N[(N[(A + A), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, -2e-158], N[(t$95$3 * N[Sqrt[N[(N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(1.0 / N[(N[(N[(C * 4.0), $MachinePrecision] * A + N[((-B$95$m) * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(F * 2.0), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[Sqrt[N[(N[(-1.0 / B$95$m), $MachinePrecision] * F), $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))) < -3.9999999999999996e165

   1. Initial program 11.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*l* N/A

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

      7. sqrt-prod N/A

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

      8. pow1/2 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 8.7%

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

   5. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 23.6

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

   7. Applied rewrites 23.6%

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

   if -3.9999999999999996e165 < (/.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))) < -2.00000000000000013e-158

   1. Initial program 99.0%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 1.1

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

   5. Applied rewrites 1.1%

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 11.3%

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

   12. Applied rewrites 96.6%

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

   if -2.00000000000000013e-158 < (/.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 22.5%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 41.7

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

   5. Applied rewrites 41.7%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 41.7

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

   7. Applied rewrites 41.7%

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

   9. Applied rewrites 41.7%

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

   11. Applied rewrites 42.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 2.9

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

   5. Applied rewrites 2.9%

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 4.0%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 10.1%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -4 \cdot 10^{+165}:\\ \;\;\;\;\frac{\sqrt{\left(A + A\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -2 \cdot 10^{-158}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}}{\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)} \cdot F}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(C \cdot 4, A, \left(-B\right) \cdot B\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot \left(\left(F \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-1}{B} \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+128}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{C}, A\right) + A\right) \cdot \left(t\_0 \cdot \left(F \cdot 2\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(\frac{C}{B\_m} - 1\right) + \frac{A}{B\_m}}{B\_m} \cdot F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e+128)
     (/
      (sqrt (* (+ (fma -0.5 (/ (* B_m B_m) C) A) A) (* t_0 (* F 2.0))))
      (- t_0))
     (* (- (sqrt 2.0)) (sqrt (* (/ (+ (- (/ C B_m) 1.0) (/ A B_m)) 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 t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+128) {
		tmp = sqrt(((fma(-0.5, ((B_m * B_m) / C), A) + A) * (t_0 * (F * 2.0)))) / -t_0;
	} else {
		tmp = -sqrt(2.0) * sqrt((((((C / B_m) - 1.0) + (A / B_m)) / 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)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+128)
		tmp = Float64(sqrt(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / C), A) + A) * Float64(t_0 * Float64(F * 2.0)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(Float64(Float64(Float64(Float64(C / B_m) - 1.0) + Float64(A / B_m)) / B_m) * F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+128], N[(N[Sqrt[N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] + A), $MachinePrecision] + A), $MachinePrecision] * N[(t$95$0 * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(N[(N[(N[(N[(C / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] + N[(A / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e128

   1. Initial program 24.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 26.3

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

   5. Applied rewrites 26.3%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 26.3

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

   7. Applied rewrites 26.3%

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

   9. Applied rewrites 26.3%

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

   if 1.0000000000000001e128 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 7.8%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 2.8

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

   5. Applied rewrites 2.8%

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 18.8%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+128}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{C}, A\right) + A\right) \cdot \left(\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right) \cdot \left(F \cdot 2\right)\right)}}{-\mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(\frac{C}{B} - 1\right) + \frac{A}{B}}{B} \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+128}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{-0.5}{C} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{-1}{B\_m} \cdot F}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 1e+128)
     (* t_0 (sqrt (* (/ -0.5 C) F)))
     (* t_0 (sqrt (* (/ -1.0 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 t_0 = -sqrt(2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 1e+128) {
		tmp = t_0 * sqrt(((-0.5 / C) * F));
	} else {
		tmp = t_0 * sqrt(((-1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if ((b_m ** 2.0d0) <= 1d+128) then
        tmp = t_0 * sqrt((((-0.5d0) / c) * f))
    else
        tmp = t_0 * sqrt((((-1.0d0) / 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 t_0 = -Math.sqrt(2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+128) {
		tmp = t_0 * Math.sqrt(((-0.5 / C) * F));
	} else {
		tmp = t_0 * Math.sqrt(((-1.0 / 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):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+128:
		tmp = t_0 * math.sqrt(((-0.5 / C) * F))
	else:
		tmp = t_0 * math.sqrt(((-1.0 / 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)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+128)
		tmp = Float64(t_0 * sqrt(Float64(Float64(-0.5 / C) * F)));
	else
		tmp = Float64(t_0 * sqrt(Float64(Float64(-1.0 / 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)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+128)
		tmp = t_0 * sqrt(((-0.5 / C) * F));
	else
		tmp = t_0 * sqrt(((-1.0 / 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_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+128], N[(t$95$0 * N[Sqrt[N[(N[(-0.5 / C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(N[(-1.0 / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.0000000000000001e128

   1. Initial program 24.2%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 1.8

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

   5. Applied rewrites 1.8%

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

   7. Applied rewrites 1.8%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 5.4%

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

   11. Taylor expanded in A around -inf

       \[\leadsto \left(-\sqrt{F \cdot \frac{\frac{-1}{2}}{C}}\right) \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.5%

       \[\leadsto \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right) \cdot \sqrt{2} \]

   if 1.0000000000000001e128 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 7.8%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 2.8

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

   5. Applied rewrites 2.8%

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

   7. Applied rewrites 2.8%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 5.8%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 17.5%

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

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+128}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-0.5}{C} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-1}{B} \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 4.3 \cdot 10^{-234}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot F\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)}}{-\left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;B\_m \leq 4.2 \cdot 10^{+64}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{-0.5}{C} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{\left(\frac{C}{B\_m} - 1\right) + \frac{A}{B\_m}}{B\_m} \cdot F}\\ \end{array} \end{array} \]

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 4.3e-234)
     (/
      (sqrt
       (*
        (* (* (fma (* -4.0 C) A (* B_m B_m)) F) 2.0)
        (+ (fma (/ (* B_m B_m) C) -0.5 A) A)))
      (- (* (* C A) -4.0)))
     (if (<= B_m 4.2e+64)
       (* t_0 (sqrt (* (/ -0.5 C) F)))
       (* t_0 (sqrt (* (/ (+ (- (/ C B_m) 1.0) (/ A B_m)) 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 t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 4.3e-234) {
		tmp = sqrt((((fma((-4.0 * C), A, (B_m * B_m)) * F) * 2.0) * (fma(((B_m * B_m) / C), -0.5, A) + A))) / -((C * A) * -4.0);
	} else if (B_m <= 4.2e+64) {
		tmp = t_0 * sqrt(((-0.5 / C) * F));
	} else {
		tmp = t_0 * sqrt((((((C / B_m) - 1.0) + (A / B_m)) / 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)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 4.3e-234)
		tmp = Float64(sqrt(Float64(Float64(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * F) * 2.0) * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))) / Float64(-Float64(Float64(C * A) * -4.0)));
	elseif (B_m <= 4.2e+64)
		tmp = Float64(t_0 * sqrt(Float64(Float64(-0.5 / C) * F)));
	else
		tmp = Float64(t_0 * sqrt(Float64(Float64(Float64(Float64(Float64(C / B_m) - 1.0) + Float64(A / B_m)) / B_m) * F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 4.3e-234], N[(N[Sqrt[N[(N[(N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 4.2e+64], N[(t$95$0 * N[Sqrt[N[(N[(-0.5 / C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(N[(N[(N[(N[(C / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] + N[(A / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.3000000000000001e-234

   1. Initial program 16.2%

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

   3. Taylor expanded in C around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 15.1

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

   5. Applied rewrites 15.1%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 15.1

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

   7. Applied rewrites 15.1%

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

   8. Taylor expanded in C around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 13.1

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

   10. Applied rewrites 13.1%

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

   if 4.3000000000000001e-234 < B < 4.2000000000000001e64

   1. Initial program 26.4%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 1.4

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

   5. Applied rewrites 1.4%

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

   7. Applied rewrites 1.4%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 5.7%

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

   11. Taylor expanded in A around -inf

       \[\leadsto \left(-\sqrt{F \cdot \frac{\frac{-1}{2}}{C}}\right) \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 21.4%

       \[\leadsto \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right) \cdot \sqrt{2} \]

   if 4.2000000000000001e64 < B

   1. Initial program 7.6%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 2.0

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

   5. Applied rewrites 2.0%

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

   7. Applied rewrites 2.0%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 6.1%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 43.1%

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

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.3 \cdot 10^{-234}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \left(\mathsf{fma}\left(\frac{B \cdot B}{C}, -0.5, A\right) + A\right)}}{-\left(C \cdot A\right) \cdot -4}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+64}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-0.5}{C} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(\frac{C}{B} - 1\right) + \frac{A}{B}}{B} \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 4.2 \cdot 10^{+64}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{-0.5}{C} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\frac{\left(\frac{C}{B\_m} - 1\right) + \frac{A}{B\_m}}{B\_m} \cdot F}\\ \end{array} \end{array} \]

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 4.2e+64)
     (* t_0 (sqrt (* (/ -0.5 C) F)))
     (* t_0 (sqrt (* (/ (+ (- (/ C B_m) 1.0) (/ A B_m)) 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 t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 4.2e+64) {
		tmp = t_0 * sqrt(((-0.5 / C) * F));
	} else {
		tmp = t_0 * sqrt((((((C / B_m) - 1.0) + (A / B_m)) / 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) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 4.2d+64) then
        tmp = t_0 * sqrt((((-0.5d0) / c) * f))
    else
        tmp = t_0 * sqrt((((((c / b_m) - 1.0d0) + (a / b_m)) / 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 t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 4.2e+64) {
		tmp = t_0 * Math.sqrt(((-0.5 / C) * F));
	} else {
		tmp = t_0 * Math.sqrt((((((C / B_m) - 1.0) + (A / B_m)) / 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):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 4.2e+64:
		tmp = t_0 * math.sqrt(((-0.5 / C) * F))
	else:
		tmp = t_0 * math.sqrt((((((C / B_m) - 1.0) + (A / B_m)) / 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)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 4.2e+64)
		tmp = Float64(t_0 * sqrt(Float64(Float64(-0.5 / C) * F)));
	else
		tmp = Float64(t_0 * sqrt(Float64(Float64(Float64(Float64(Float64(C / B_m) - 1.0) + Float64(A / B_m)) / 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)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 4.2e+64)
		tmp = t_0 * sqrt(((-0.5 / C) * F));
	else
		tmp = t_0 * sqrt((((((C / B_m) - 1.0) + (A / B_m)) / 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_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 4.2e+64], N[(t$95$0 * N[Sqrt[N[(N[(-0.5 / C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(N[(N[(N[(N[(C / B$95$m), $MachinePrecision] - 1.0), $MachinePrecision] + N[(A / B$95$m), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4.2000000000000001e64

   1. Initial program 19.1%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 2.3

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

   5. Applied rewrites 2.3%

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

   7. Applied rewrites 2.3%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 5.5%

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

   11. Taylor expanded in A around -inf

       \[\leadsto \left(-\sqrt{F \cdot \frac{\frac{-1}{2}}{C}}\right) \cdot \sqrt{2} \]
   12. Step-by-step derivation

   13. Applied rewrites 17.1%

       \[\leadsto \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right) \cdot \sqrt{2} \]

   if 4.2000000000000001e64 < B

   1. Initial program 7.6%

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

   3. Taylor expanded in B around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. rem-square-sqrt N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 2.0

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

   5. Applied rewrites 2.0%

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

   7. Applied rewrites 2.0%

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

   8. Taylor expanded in F around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 6.1%

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

   11. Taylor expanded in B around inf

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

   13. Applied rewrites 43.1%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.2 \cdot 10^{+64}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{-0.5}{C} \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\left(\frac{C}{B} - 1\right) + \frac{A}{B}}{B} \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.7% accurate, 11.2× speedup

Math

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

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)) (sqrt (* (/ -1.0 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 -sqrt(2.0) * sqrt(((-1.0 / 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 = -sqrt(2.0d0) * sqrt((((-1.0d0) / 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 -Math.sqrt(2.0) * Math.sqrt(((-1.0 / 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 -math.sqrt(2.0) * math.sqrt(((-1.0 / 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(Float64(-sqrt(2.0)) * sqrt(Float64(Float64(-1.0 / 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 = -sqrt(2.0) * sqrt(((-1.0 / 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[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(N[(-1.0 / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.2

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

5. Applied rewrites 2.2%

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

7. Applied rewrites 2.2%

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

8. Taylor expanded in F around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 5.6%

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

11. Taylor expanded in B around inf

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

13. Applied rewrites 12.1%

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

14. Final simplification 12.1%

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

Alternative 8: 1.6% accurate, 14.9× speedup

Math

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

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))))

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)));
}

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)))
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)));
}

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)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(2.0 / Float64(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 = 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[Sqrt[N[(2.0 / N[(B$95$m / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.2

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

5. Applied rewrites 2.2%

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

7. Applied rewrites 2.2%

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

9. Applied rewrites 2.2%

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

11. Applied rewrites 2.3%

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

Alternative 9: 1.5% accurate, 18.2× speedup

Math

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

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)))

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));
}

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))
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));
}

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))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return sqrt(Float64(Float64(2.0 / 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 = 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[Sqrt[N[(N[(2.0 / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. rem-square-sqrt N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 2.2

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

5. Applied rewrites 2.2%

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

7. Applied rewrites 2.2%

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

9. Applied rewrites 2.2%

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

11. Applied rewrites 2.2%

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

12. Final simplification 2.2%

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

Reproduce

herbie shell --seed 2024249 
(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))))