ABCF->ab-angle b

Percentage Accurate: 18.9% → 54.3%

Time: 27.5s

Alternatives: 12

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% 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: 54.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(A - C, B\_m\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left(A \cdot -4, C, {B\_m}^{2}\right)}\right)}\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{-t\_0}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_1) F)))
        (t_3 (- t_1 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (*
      (sqrt 2.0)
      (-
       (sqrt
        (*
         F
         (*
          (+ A (- C (hypot (- A C) B_m)))
          (/ 1.0 (fma (* A -4.0) C (pow B_m 2.0))))))))
     (if (<= t_4 -5e-205)
       (/ (sqrt (* t_0 (* F (* 2.0 (- (+ A C) (hypot B_m (- A C))))))) (- t_0))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_2 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))) t_3)
         (* (sqrt 2.0) (* (sqrt (* F (- A (hypot A B_m)))) (/ -1.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B_m, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = sqrt(2.0) * -sqrt((F * ((A + (C - hypot((A - C), B_m))) * (1.0 / fma((A * -4.0), C, pow(B_m, 2.0))))));
	} else if (t_4 <= -5e-205) {
		tmp = sqrt((t_0 * (F * (2.0 * ((A + C) - hypot(B_m, (A - C))))))) / -t_0;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C)))))) / t_3;
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(A + Float64(C - hypot(Float64(A - C), B_m))) * Float64(1.0 / fma(Float64(A * -4.0), C, (B_m ^ 2.0))))))));
	elseif (t_4 <= -5e-205)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * Float64(2.0 * Float64(Float64(A + C) - hypot(B_m, Float64(A - C))))))) / Float64(-t_0));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_2 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C)))))) / t_3);
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - hypot(A, B_m)))) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(A + N[(C - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(A * -4.0), $MachinePrecision] * C + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -5e-205], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(2.0 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$2 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $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 3.2%

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

   3. Simplified 3.2%

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

   5. Taylor expanded in F around 0 29.7%

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

   7. Simplified 71.7%

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

      1. div-inv 71.7%

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

      2. associate--r- 71.7%

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

      3. fma-undefine 71.7%

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

      4. associate-*r* 71.7%

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

      5. *-commutative 71.7%

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

      6. fma-define 71.7%

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

   9. Applied egg-rr 71.7%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000001e-205

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-frac-neg2 99.5%

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

      2. associate-*l* 99.6%

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

      3. associate-+r- 99.6%

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

   6. Applied egg-rr 99.6%

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

   if -5.00000000000000001e-205 < (/.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 22.7%

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

   3. Taylor expanded in C around inf 27.2%

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

      1. mul-1-neg 27.2%

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

      2. associate--l+ 27.2%

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

   5. Simplified 27.2%

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

   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%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in F around 0 0.2%

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

   7. Simplified 4.3%

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

   8. Taylor expanded in C around 0 2.1%

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

      1. unpow2 2.1%

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

      2. unpow2 2.1%

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

      3. hypot-undefine 18.0%

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

   10. Simplified 18.0%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \left(\left(A + \left(C - \mathsf{hypot}\left(A - C, B\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left(A \cdot -4, C, {B}^{2}\right)}\right)}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-205}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 47.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -\sqrt{2}\\ t_2 := C - \left(\mathsf{hypot}\left(A - C, B\_m\right) - A\right)\\ t_3 := t\_0 - {B\_m}^{2}\\ t_4 := \frac{{B\_m}^{2}}{C}\\ t_5 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_6 := \frac{\sqrt{t\_5 \cdot \left(A + \left(A + -0.5 \cdot t\_4\right)\right)}}{t\_3}\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot A\right)}}{t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_2}{C \cdot \left(A \cdot -4 + t\_4\right)}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_2}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+129}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (sqrt 2.0)))
        (t_2 (- C (- (hypot (- A C) B_m) A)))
        (t_3 (- t_0 (pow B_m 2.0)))
        (t_4 (/ (pow B_m 2.0) C))
        (t_5 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_6 (/ (sqrt (* t_5 (+ A (+ A (* -0.5 t_4))))) t_3)))
   (if (<= (pow B_m 2.0) 4e-163)
     (/ (sqrt (* t_5 (* 2.0 A))) t_3)
     (if (<= (pow B_m 2.0) 4e-12)
       (* (sqrt (* F (/ t_2 (* C (+ (* A -4.0) t_4))))) t_1)
       (if (<= (pow B_m 2.0) 4e+40)
         t_6
         (if (<= (pow B_m 2.0) 4e+80)
           (* (sqrt (* F (/ t_2 (fma -4.0 (* A C) (pow B_m 2.0))))) t_1)
           (if (<= (pow B_m 2.0) 2e+129)
             t_6
             (*
              (sqrt 2.0)
              (* (sqrt (* F (- A (hypot A B_m)))) (/ -1.0 B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -sqrt(2.0);
	double t_2 = C - (hypot((A - C), B_m) - A);
	double t_3 = t_0 - pow(B_m, 2.0);
	double t_4 = pow(B_m, 2.0) / C;
	double t_5 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_6 = sqrt((t_5 * (A + (A + (-0.5 * t_4))))) / t_3;
	double tmp;
	if (pow(B_m, 2.0) <= 4e-163) {
		tmp = sqrt((t_5 * (2.0 * A))) / t_3;
	} else if (pow(B_m, 2.0) <= 4e-12) {
		tmp = sqrt((F * (t_2 / (C * ((A * -4.0) + t_4))))) * t_1;
	} else if (pow(B_m, 2.0) <= 4e+40) {
		tmp = t_6;
	} else if (pow(B_m, 2.0) <= 4e+80) {
		tmp = sqrt((F * (t_2 / fma(-4.0, (A * C), pow(B_m, 2.0))))) * t_1;
	} else if (pow(B_m, 2.0) <= 2e+129) {
		tmp = t_6;
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(C - Float64(hypot(Float64(A - C), B_m) - A))
	t_3 = Float64(t_0 - (B_m ^ 2.0))
	t_4 = Float64((B_m ^ 2.0) / C)
	t_5 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_6 = Float64(sqrt(Float64(t_5 * Float64(A + Float64(A + Float64(-0.5 * t_4))))) / t_3)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * A))) / t_3);
	elseif ((B_m ^ 2.0) <= 4e-12)
		tmp = Float64(sqrt(Float64(F * Float64(t_2 / Float64(C * Float64(Float64(A * -4.0) + t_4))))) * t_1);
	elseif ((B_m ^ 2.0) <= 4e+40)
		tmp = t_6;
	elseif ((B_m ^ 2.0) <= 4e+80)
		tmp = Float64(sqrt(Float64(F * Float64(t_2 / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * t_1);
	elseif ((B_m ^ 2.0) <= 2e+129)
		tmp = t_6;
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - hypot(A, B_m)))) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(C - N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(t$95$5 * N[(A + N[(A + N[(-0.5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-163], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-12], N[(N[Sqrt[N[(F * N[(t$95$2 / N[(C * N[(N[(A * -4.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+40], t$95$6, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+80], N[(N[Sqrt[N[(F * N[(t$95$2 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+129], t$95$6, N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999969e-163

   1. Initial program 23.3%

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

   3. Taylor expanded in A around -inf 19.7%

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

      1. *-commutative 19.7%

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

   5. Simplified 19.7%

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

   if 3.99999999999999969e-163 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999992e-12

   1. Initial program 33.0%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in F around 0 32.7%

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

   7. Simplified 43.5%

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

   8. Taylor expanded in C around inf 43.5%

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

   if 3.99999999999999992e-12 < (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000012e40 or 4e80 < (pow.f64 B #s(literal 2 binary64)) < 2e129

   1. Initial program 24.8%

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

   3. Taylor expanded in C around inf 32.2%

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

      1. mul-1-neg 32.2%

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

      2. associate--l+ 32.2%

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

   5. Simplified 32.2%

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

   if 4.00000000000000012e40 < (pow.f64 B #s(literal 2 binary64)) < 4e80

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in F around 0 52.2%

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

   7. Simplified 63.4%

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

   if 2e129 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.3%

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

   3. Simplified 10.3%

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

   5. Taylor expanded in F around 0 15.1%

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

   7. Simplified 26.4%

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

   8. Taylor expanded in C around 0 11.5%

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

      1. unpow2 11.5%

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

      2. unpow2 11.5%

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

      3. hypot-undefine 29.1%

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

   10. Simplified 29.1%

       \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{F \cdot \frac{C - \left(\mathsf{hypot}\left(A - C, B\right) - A\right)}{C \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{F \cdot \frac{C - \left(\mathsf{hypot}\left(A - C, B\right) - A\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 46.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -\sqrt{2}\\ t_2 := C - \left(\mathsf{hypot}\left(A - C, B\_m\right) - A\right)\\ t_3 := t\_0 - {B\_m}^{2}\\ t_4 := \frac{{B\_m}^{2}}{C}\\ t_5 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_6 := \frac{\sqrt{t\_5 \cdot \left(A + \left(A + -0.5 \cdot t\_4\right)\right)}}{t\_3}\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot A\right)}}{t\_3}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_2}{C \cdot \left(A \cdot -4 + t\_4\right)}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_2}{{B\_m}^{2}}} \cdot t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+129}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (sqrt 2.0)))
        (t_2 (- C (- (hypot (- A C) B_m) A)))
        (t_3 (- t_0 (pow B_m 2.0)))
        (t_4 (/ (pow B_m 2.0) C))
        (t_5 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_6 (/ (sqrt (* t_5 (+ A (+ A (* -0.5 t_4))))) t_3)))
   (if (<= (pow B_m 2.0) 4e-163)
     (/ (sqrt (* t_5 (* 2.0 A))) t_3)
     (if (<= (pow B_m 2.0) 4e-12)
       (* (sqrt (* F (/ t_2 (* C (+ (* A -4.0) t_4))))) t_1)
       (if (<= (pow B_m 2.0) 4e+40)
         t_6
         (if (<= (pow B_m 2.0) 4e+80)
           (* (sqrt (* F (/ t_2 (pow B_m 2.0)))) t_1)
           (if (<= (pow B_m 2.0) 2e+129)
             t_6
             (*
              (sqrt 2.0)
              (* (sqrt (* F (- A (hypot A B_m)))) (/ -1.0 B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -sqrt(2.0);
	double t_2 = C - (hypot((A - C), B_m) - A);
	double t_3 = t_0 - pow(B_m, 2.0);
	double t_4 = pow(B_m, 2.0) / C;
	double t_5 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_6 = sqrt((t_5 * (A + (A + (-0.5 * t_4))))) / t_3;
	double tmp;
	if (pow(B_m, 2.0) <= 4e-163) {
		tmp = sqrt((t_5 * (2.0 * A))) / t_3;
	} else if (pow(B_m, 2.0) <= 4e-12) {
		tmp = sqrt((F * (t_2 / (C * ((A * -4.0) + t_4))))) * t_1;
	} else if (pow(B_m, 2.0) <= 4e+40) {
		tmp = t_6;
	} else if (pow(B_m, 2.0) <= 4e+80) {
		tmp = sqrt((F * (t_2 / pow(B_m, 2.0)))) * t_1;
	} else if (pow(B_m, 2.0) <= 2e+129) {
		tmp = t_6;
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -Math.sqrt(2.0);
	double t_2 = C - (Math.hypot((A - C), B_m) - A);
	double t_3 = t_0 - Math.pow(B_m, 2.0);
	double t_4 = Math.pow(B_m, 2.0) / C;
	double t_5 = 2.0 * ((Math.pow(B_m, 2.0) - t_0) * F);
	double t_6 = Math.sqrt((t_5 * (A + (A + (-0.5 * t_4))))) / t_3;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e-163) {
		tmp = Math.sqrt((t_5 * (2.0 * A))) / t_3;
	} else if (Math.pow(B_m, 2.0) <= 4e-12) {
		tmp = Math.sqrt((F * (t_2 / (C * ((A * -4.0) + t_4))))) * t_1;
	} else if (Math.pow(B_m, 2.0) <= 4e+40) {
		tmp = t_6;
	} else if (Math.pow(B_m, 2.0) <= 4e+80) {
		tmp = Math.sqrt((F * (t_2 / Math.pow(B_m, 2.0)))) * t_1;
	} else if (Math.pow(B_m, 2.0) <= 2e+129) {
		tmp = t_6;
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt((F * (A - Math.hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = -math.sqrt(2.0)
	t_2 = C - (math.hypot((A - C), B_m) - A)
	t_3 = t_0 - math.pow(B_m, 2.0)
	t_4 = math.pow(B_m, 2.0) / C
	t_5 = 2.0 * ((math.pow(B_m, 2.0) - t_0) * F)
	t_6 = math.sqrt((t_5 * (A + (A + (-0.5 * t_4))))) / t_3
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e-163:
		tmp = math.sqrt((t_5 * (2.0 * A))) / t_3
	elif math.pow(B_m, 2.0) <= 4e-12:
		tmp = math.sqrt((F * (t_2 / (C * ((A * -4.0) + t_4))))) * t_1
	elif math.pow(B_m, 2.0) <= 4e+40:
		tmp = t_6
	elif math.pow(B_m, 2.0) <= 4e+80:
		tmp = math.sqrt((F * (t_2 / math.pow(B_m, 2.0)))) * t_1
	elif math.pow(B_m, 2.0) <= 2e+129:
		tmp = t_6
	else:
		tmp = math.sqrt(2.0) * (math.sqrt((F * (A - math.hypot(A, B_m)))) * (-1.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-sqrt(2.0))
	t_2 = Float64(C - Float64(hypot(Float64(A - C), B_m) - A))
	t_3 = Float64(t_0 - (B_m ^ 2.0))
	t_4 = Float64((B_m ^ 2.0) / C)
	t_5 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_6 = Float64(sqrt(Float64(t_5 * Float64(A + Float64(A + Float64(-0.5 * t_4))))) / t_3)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * A))) / t_3);
	elseif ((B_m ^ 2.0) <= 4e-12)
		tmp = Float64(sqrt(Float64(F * Float64(t_2 / Float64(C * Float64(Float64(A * -4.0) + t_4))))) * t_1);
	elseif ((B_m ^ 2.0) <= 4e+40)
		tmp = t_6;
	elseif ((B_m ^ 2.0) <= 4e+80)
		tmp = Float64(sqrt(Float64(F * Float64(t_2 / (B_m ^ 2.0)))) * t_1);
	elseif ((B_m ^ 2.0) <= 2e+129)
		tmp = t_6;
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - hypot(A, B_m)))) * Float64(-1.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = -sqrt(2.0);
	t_2 = C - (hypot((A - C), B_m) - A);
	t_3 = t_0 - (B_m ^ 2.0);
	t_4 = (B_m ^ 2.0) / C;
	t_5 = 2.0 * (((B_m ^ 2.0) - t_0) * F);
	t_6 = sqrt((t_5 * (A + (A + (-0.5 * t_4))))) / t_3;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = sqrt((t_5 * (2.0 * A))) / t_3;
	elseif ((B_m ^ 2.0) <= 4e-12)
		tmp = sqrt((F * (t_2 / (C * ((A * -4.0) + t_4))))) * t_1;
	elseif ((B_m ^ 2.0) <= 4e+40)
		tmp = t_6;
	elseif ((B_m ^ 2.0) <= 4e+80)
		tmp = sqrt((F * (t_2 / (B_m ^ 2.0)))) * t_1;
	elseif ((B_m ^ 2.0) <= 2e+129)
		tmp = t_6;
	else
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$2 = N[(C - N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Sqrt[N[(t$95$5 * N[(A + N[(A + N[(-0.5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-163], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-12], N[(N[Sqrt[N[(F * N[(t$95$2 / N[(C * N[(N[(A * -4.0), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+40], t$95$6, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+80], N[(N[Sqrt[N[(F * N[(t$95$2 / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+129], t$95$6, N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999969e-163

   1. Initial program 23.3%

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

   3. Taylor expanded in A around -inf 19.7%

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

      1. *-commutative 19.7%

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

   5. Simplified 19.7%

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

   if 3.99999999999999969e-163 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999992e-12

   1. Initial program 33.0%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in F around 0 32.7%

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

   7. Simplified 43.5%

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

   8. Taylor expanded in C around inf 43.5%

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

   if 3.99999999999999992e-12 < (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000012e40 or 4e80 < (pow.f64 B #s(literal 2 binary64)) < 2e129

   1. Initial program 24.8%

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

   3. Taylor expanded in C around inf 32.2%

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

      1. mul-1-neg 32.2%

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

      2. associate--l+ 32.2%

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

   5. Simplified 32.2%

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

   if 4.00000000000000012e40 < (pow.f64 B #s(literal 2 binary64)) < 4e80

   1. Initial program 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in F around 0 52.2%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in A around 0 62.2%

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

   if 2e129 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.3%

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

   3. Simplified 10.3%

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

   5. Taylor expanded in F around 0 15.1%

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

   7. Simplified 26.4%

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

   8. Taylor expanded in C around 0 11.5%

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

      1. unpow2 11.5%

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

      2. unpow2 11.5%

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

      3. hypot-undefine 29.1%

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

   10. Simplified 29.1%

       \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{F \cdot \frac{C - \left(\mathsf{hypot}\left(A - C, B\right) - A\right)}{C \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\sqrt{F \cdot \frac{C - \left(\mathsf{hypot}\left(A - C, B\right) - A\right)}{{B}^{2}}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+129}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{F \cdot \frac{C - \left(\mathsf{hypot}\left(A - C, B\_m\right) - A\right)}{C \cdot \left(A \cdot -4 + \frac{{B\_m}^{2}}{C}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
          (- t_0 (pow B_m 2.0)))))
   (if (<= (pow B_m 2.0) 4e-163)
     t_1
     (if (<= (pow B_m 2.0) 4e-12)
       (*
        (sqrt
         (*
          F
          (/
           (- C (- (hypot (- A C) B_m) A))
           (* C (+ (* A -4.0) (/ (pow B_m 2.0) C))))))
        (- (sqrt 2.0)))
       (if (<= (pow B_m 2.0) 4e+40)
         t_1
         (* (sqrt 2.0) (* (sqrt (* F (- A (hypot A B_m)))) (/ -1.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 4e-163) {
		tmp = t_1;
	} else if (pow(B_m, 2.0) <= 4e-12) {
		tmp = sqrt((F * ((C - (hypot((A - C), B_m) - A)) / (C * ((A * -4.0) + (pow(B_m, 2.0) / C)))))) * -sqrt(2.0);
	} else if (pow(B_m, 2.0) <= 4e+40) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e-163) {
		tmp = t_1;
	} else if (Math.pow(B_m, 2.0) <= 4e-12) {
		tmp = Math.sqrt((F * ((C - (Math.hypot((A - C), B_m) - A)) / (C * ((A * -4.0) + (Math.pow(B_m, 2.0) / C)))))) * -Math.sqrt(2.0);
	} else if (Math.pow(B_m, 2.0) <= 4e+40) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt((F * (A - Math.hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e-163:
		tmp = t_1
	elif math.pow(B_m, 2.0) <= 4e-12:
		tmp = math.sqrt((F * ((C - (math.hypot((A - C), B_m) - A)) / (C * ((A * -4.0) + (math.pow(B_m, 2.0) / C)))))) * -math.sqrt(2.0)
	elif math.pow(B_m, 2.0) <= 4e+40:
		tmp = t_1
	else:
		tmp = math.sqrt(2.0) * (math.sqrt((F * (A - math.hypot(A, B_m)))) * (-1.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 4e-12)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(C - Float64(hypot(Float64(A - C), B_m) - A)) / Float64(C * Float64(Float64(A * -4.0) + Float64((B_m ^ 2.0) / C)))))) * Float64(-sqrt(2.0)));
	elseif ((B_m ^ 2.0) <= 4e+40)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - hypot(A, B_m)))) * Float64(-1.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = t_1;
	elseif ((B_m ^ 2.0) <= 4e-12)
		tmp = sqrt((F * ((C - (hypot((A - C), B_m) - A)) / (C * ((A * -4.0) + ((B_m ^ 2.0) / C)))))) * -sqrt(2.0);
	elseif ((B_m ^ 2.0) <= 4e+40)
		tmp = t_1;
	else
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-163], t$95$1, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-12], N[(N[Sqrt[N[(F * N[(N[(C - N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision] / N[(C * N[(N[(A * -4.0), $MachinePrecision] + N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+40], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999969e-163 or 3.99999999999999992e-12 < (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000012e40

   1. Initial program 23.7%

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

   3. Taylor expanded in A around -inf 19.8%

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

      1. *-commutative 19.8%

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

   5. Simplified 19.8%

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

   if 3.99999999999999969e-163 < (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999992e-12

   1. Initial program 33.0%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in F around 0 32.7%

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

   7. Simplified 43.5%

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

   8. Taylor expanded in C around inf 43.5%

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

   if 4.00000000000000012e40 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.3%

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

   3. Simplified 14.3%

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

   5. Taylor expanded in F around 0 17.3%

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

   7. Simplified 28.2%

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

   8. Taylor expanded in C around 0 10.6%

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

      1. unpow2 10.6%

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

      2. unpow2 10.6%

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

      3. hypot-undefine 26.1%

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

   10. Simplified 26.1%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{F \cdot \frac{C - \left(\mathsf{hypot}\left(A - C, B\right) - A\right)}{C \cdot \left(A \cdot -4 + \frac{{B}^{2}}{C}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 4e+40)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (* (sqrt 2.0) (* (sqrt (* F (- A (hypot A B_m)))) (/ -1.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 4e+40) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e+40) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt((F * (A - Math.hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e+40:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt(2.0) * (math.sqrt((F * (A - math.hypot(A, B_m)))) * (-1.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+40)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - hypot(A, B_m)))) * Float64(-1.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e+40)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+40], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000012e40

   1. Initial program 25.2%

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

   3. Taylor expanded in A around -inf 20.1%

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

      1. *-commutative 20.1%

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

   5. Simplified 20.1%

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

   if 4.00000000000000012e40 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.3%

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

   3. Simplified 14.3%

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

   5. Taylor expanded in F around 0 17.3%

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

   7. Simplified 28.2%

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

   8. Taylor expanded in C around 0 10.6%

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

      1. unpow2 10.6%

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

      2. unpow2 10.6%

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

      3. hypot-undefine 26.1%

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

   10. Simplified 26.1%

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

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e+40)
   (/
    (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
    (- (fma C (* A -4.0) (pow B_m 2.0))))
   (* (sqrt 2.0) (* (sqrt (* F (- A (hypot A B_m)))) (/ -1.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e+40) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+40)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - hypot(A, B_m)))) * Float64(-1.0 / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+40], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000012e40

   1. Initial program 25.2%

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

   3. Simplified 26.7%

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

   5. Taylor expanded in C around inf 18.5%

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

      1. associate-*r* 18.5%

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

      2. rem-square-sqrt 0.0%

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

      3. unpow2 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 18.5%

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

      7. associate-*r* 17.1%

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

      8. mul-1-neg 17.1%

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

   7. Simplified 17.1%

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

      1. pow1 17.1%

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

      2. associate-*l* 18.5%

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

   9. Applied egg-rr 18.5%

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

       1. unpow1 18.5%

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

   11. Simplified 18.5%

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

   if 4.00000000000000012e40 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.3%

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

   3. Simplified 14.3%

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

   5. Taylor expanded in F around 0 17.3%

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

   7. Simplified 28.2%

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

   8. Taylor expanded in C around 0 10.6%

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

      1. unpow2 10.6%

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

      2. unpow2 10.6%

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

      3. hypot-undefine 26.1%

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

   10. Simplified 26.1%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+40}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e-163)
   (/
    (sqrt (* (* A -8.0) (* (+ A A) (* C F))))
    (- (* C (* -4.0 (- A))) (pow B_m 2.0)))
   (* (sqrt 2.0) (* (sqrt (* F (- A (hypot A B_m)))) (/ -1.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e-163) {
		tmp = sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - pow(B_m, 2.0));
	} else {
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e-163) {
		tmp = Math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt((F * (A - Math.hypot(A, B_m)))) * (-1.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e-163:
		tmp = math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt(2.0) * (math.sqrt((F * (A - math.hypot(A, B_m)))) * (-1.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(Float64(A + A) * Float64(C * F)))) / Float64(Float64(C * Float64(-4.0 * Float64(-A))) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(A - hypot(A, B_m)))) * Float64(-1.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - (B_m ^ 2.0));
	else
		tmp = sqrt(2.0) * (sqrt((F * (A - hypot(A, B_m)))) * (-1.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-163], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * N[(-4.0 * (-A)), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999969e-163

   1. Initial program 23.3%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in C around inf 18.7%

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

      1. associate-*r* 18.8%

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

      2. rem-square-sqrt 0.0%

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

      3. unpow2 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 18.8%

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

      7. associate-*r* 18.7%

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

      8. mul-1-neg 18.7%

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

   7. Simplified 18.7%

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

      1. fma-undefine 18.7%

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

   9. Applied egg-rr 18.7%

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

   if 3.99999999999999969e-163 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.0%

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

   3. Simplified 18.2%

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

   5. Taylor expanded in F around 0 19.7%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in C around 0 12.3%

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

      1. unpow2 12.3%

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

      2. unpow2 12.3%

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

      3. hypot-undefine 24.5%

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

   10. Simplified 24.5%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{-1}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e-163)
   (/
    (sqrt (* (* A -8.0) (* (+ A A) (* C F))))
    (- (* C (* -4.0 (- A))) (pow B_m 2.0)))
   (* (sqrt (* F (- A (hypot A B_m)))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e-163) {
		tmp = sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - pow(B_m, 2.0));
	} else {
		tmp = sqrt((F * (A - hypot(A, B_m)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e-163) {
		tmp = Math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt((F * (A - Math.hypot(A, B_m)))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e-163:
		tmp = math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt((F * (A - math.hypot(A, B_m)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(Float64(A + A) * Float64(C * F)))) / Float64(Float64(C * Float64(-4.0 * Float64(-A))) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(A, B_m)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e-163)
		tmp = sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - (B_m ^ 2.0));
	else
		tmp = sqrt((F * (A - hypot(A, B_m)))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-163], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * N[(-4.0 * (-A)), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.99999999999999969e-163

   1. Initial program 23.3%

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

   3. Simplified 24.6%

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

   5. Taylor expanded in C around inf 18.7%

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

      1. associate-*r* 18.8%

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

      2. rem-square-sqrt 0.0%

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

      3. unpow2 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 18.8%

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

      7. associate-*r* 18.7%

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

      8. mul-1-neg 18.7%

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

   7. Simplified 18.7%

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

      1. fma-undefine 18.7%

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

   9. Applied egg-rr 18.7%

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

   if 3.99999999999999969e-163 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.0%

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

   3. Simplified 18.2%

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

   5. Taylor expanded in C around 0 12.3%

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

      1. mul-1-neg 12.3%

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

      2. unpow2 12.3%

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

      3. unpow2 12.3%

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

      4. hypot-define 24.5%

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

   7. Simplified 24.5%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 1.4 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right) - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 1.5 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 1.4e-202)
     (/
      (sqrt (* (* A -8.0) (* (+ A A) (* C F))))
      (- (* C (* -4.0 (- A))) (pow B_m 2.0)))
     (if (<= B_m 1.5e-76)
       (* (sqrt (* F (/ -0.5 C))) t_0)
       (* (sqrt (* F (/ -1.0 B_m))) t_0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 1.4e-202) {
		tmp = sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - pow(B_m, 2.0));
	} else if (B_m <= 1.5e-76) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 1.4d-202) then
        tmp = sqrt(((a * (-8.0d0)) * ((a + a) * (c * f)))) / ((c * ((-4.0d0) * -a)) - (b_m ** 2.0d0))
    else if (b_m <= 1.5d-76) then
        tmp = sqrt((f * ((-0.5d0) / c))) * t_0
    else
        tmp = sqrt((f * ((-1.0d0) / b_m))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 1.4e-202) {
		tmp = Math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - Math.pow(B_m, 2.0));
	} else if (B_m <= 1.5e-76) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = Math.sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 1.4e-202:
		tmp = math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - math.pow(B_m, 2.0))
	elif B_m <= 1.5e-76:
		tmp = math.sqrt((F * (-0.5 / C))) * t_0
	else:
		tmp = math.sqrt((F * (-1.0 / B_m))) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 1.4e-202)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(Float64(A + A) * Float64(C * F)))) / Float64(Float64(C * Float64(-4.0 * Float64(-A))) - (B_m ^ 2.0)));
	elseif (B_m <= 1.5e-76)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(-1.0 / B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 1.4e-202)
		tmp = sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((C * (-4.0 * -A)) - (B_m ^ 2.0));
	elseif (B_m <= 1.5e-76)
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	else
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 1.4e-202], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * N[(-4.0 * (-A)), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.5e-76], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.4000000000000001e-202

   1. Initial program 18.7%

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

   3. Simplified 19.8%

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

   5. Taylor expanded in C around inf 12.9%

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

      1. associate-*r* 13.0%

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

      2. rem-square-sqrt 0.0%

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

      3. unpow2 0.0%

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

      4. *-commutative 0.0%

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

      5. unpow2 0.0%

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

      6. rem-square-sqrt 13.0%

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

      7. associate-*r* 12.3%

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

      8. mul-1-neg 12.3%

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

   7. Simplified 12.3%

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

      1. fma-undefine 12.3%

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

   9. Applied egg-rr 12.3%

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

   if 1.4000000000000001e-202 < B < 1.50000000000000012e-76

   1. Initial program 31.8%

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

   3. Simplified 31.8%

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

   5. Taylor expanded in F around 0 14.8%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in A around -inf 29.3%

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

   if 1.50000000000000012e-76 < B

   1. Initial program 18.1%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in F around 0 21.9%

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

   7. Simplified 35.9%

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

   8. Taylor expanded in B around inf 45.0%

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

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(-4 \cdot \left(-A\right)\right) - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.5 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;B\_m \leq 3.2 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B\_m}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= B_m 3.2e-76)
     (* (sqrt (* F (/ -0.5 C))) t_0)
     (* (sqrt (* F (/ -1.0 B_m))) t_0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (B_m <= 3.2e-76) {
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (b_m <= 3.2d-76) then
        tmp = sqrt((f * ((-0.5d0) / c))) * t_0
    else
        tmp = sqrt((f * ((-1.0d0) / b_m))) * t_0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (B_m <= 3.2e-76) {
		tmp = Math.sqrt((F * (-0.5 / C))) * t_0;
	} else {
		tmp = Math.sqrt((F * (-1.0 / B_m))) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if B_m <= 3.2e-76:
		tmp = math.sqrt((F * (-0.5 / C))) * t_0
	else:
		tmp = math.sqrt((F * (-1.0 / B_m))) * t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (B_m <= 3.2e-76)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 / C))) * t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(-1.0 / B_m))) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (B_m <= 3.2e-76)
		tmp = sqrt((F * (-0.5 / C))) * t_0;
	else
		tmp = sqrt((F * (-1.0 / B_m))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[B$95$m, 3.2e-76], N[(N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.1999999999999998e-76

   1. Initial program 21.0%

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

   3. Simplified 21.5%

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

   5. Taylor expanded in F around 0 14.2%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in A around -inf 15.7%

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

   if 3.1999999999999998e-76 < B

   1. Initial program 18.1%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in F around 0 21.9%

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

   7. Simplified 35.9%

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

   8. Taylor expanded in B around inf 45.0%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.2 \cdot 10^{-76}:\\ \;\;\;\;\sqrt{F \cdot \frac{-0.5}{C}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \frac{-1}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.2%

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

3. Simplified 20.6%

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

5. Taylor expanded in F around 0 16.5%

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

7. Simplified 26.9%

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

8. Taylor expanded in B around inf 15.3%

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

9. Final simplification 15.3%

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

Alternative 12: 0.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.2%

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

3. Simplified 18.8%

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

5. Taylor expanded in C around inf 11.3%

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

   1. associate-*r* 11.3%

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

   2. rem-square-sqrt 0.0%

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

   3. unpow2 0.0%

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

   4. *-commutative 0.0%

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

   5. unpow2 0.0%

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

   6. rem-square-sqrt 11.3%

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

   7. associate-*r* 10.6%

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

   8. mul-1-neg 10.6%

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

7. Simplified 10.6%

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

8. Taylor expanded in C around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

10. Simplified 0.0%

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

11. Final simplification 0.0%

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

Reproduce

herbie shell --seed 2024101 
(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))))