ABCF->ab-angle b

Percentage Accurate: 19.7% → 47.8%

Time: 30.6s

Alternatives: 10

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: 19.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 47.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(t_0 \cdot F\right) \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (pow B 2.0))))
   (if (<= (pow B 2.0) 2e-30)
     (/ (- (sqrt (* 2.0 (* (* t_0 F) (+ A A))))) t_0)
     (/ (sqrt (* (- A (hypot B A)) (* 2.0 F))) (- B)))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(A, (C * -4.0), pow(B, 2.0));
	double tmp;
	if (pow(B, 2.0) <= 2e-30) {
		tmp = -sqrt((2.0 * ((t_0 * F) * (A + A)))) / t_0;
	} else {
		tmp = sqrt(((A - hypot(B, A)) * (2.0 * F))) / -B;
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(A, Float64(C * -4.0), (B ^ 2.0))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-30)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(Float64(t_0 * F) * Float64(A + A))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * F))) / Float64(-B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-30], N[((-N[Sqrt[N[(2.0 * N[(N[(t$95$0 * F), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2e-30

   1. Initial program 19.9%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in C around inf 22.9%

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

      1. cancel-sign-sub-inv 22.9%

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

      2. metadata-eval 22.9%

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

      3. *-lft-identity 22.9%

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

   6. Simplified 22.9%

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

   if 2e-30 < (pow.f64 B 2)

   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 22.7%

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

   4. Taylor expanded in C around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. distribute-rgt-neg-in 11.3%

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

      3. +-commutative 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-def 25.2%

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

   6. Simplified 25.2%

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

      1. associate-*l/ 25.2%

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

      2. frac-2neg 25.2%

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

   8. Applied egg-rr 25.4%

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

      1. associate-*r* 25.4%

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

      2. *-commutative 25.4%

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

   10. Simplified 25.4%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right) \cdot F\right) \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]

Alternative 2: 46.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-t_1}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-59}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;t_1 \cdot \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0))))
        (t_1 (sqrt (* -8.0 (* (* F (* A C)) (* 2.0 A))))))
   (if (<= (pow B 2.0) 5e-149)
     (/ (- t_1) t_0)
     (if (<= (pow B 2.0) 1e-59)
       (- (/ (sqrt (* 2.0 (* t_0 (* F (- (+ A C) (hypot B (- A C))))))) t_0))
       (if (<= (pow B 2.0) 2e-30)
         (* t_1 (/ -1.0 t_0))
         (/ (sqrt (* (- A (hypot B A)) (* 2.0 F))) (- B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = sqrt((-8.0 * ((F * (A * C)) * (2.0 * A))));
	double tmp;
	if (pow(B, 2.0) <= 5e-149) {
		tmp = -t_1 / t_0;
	} else if (pow(B, 2.0) <= 1e-59) {
		tmp = -(sqrt((2.0 * (t_0 * (F * ((A + C) - hypot(B, (A - C))))))) / t_0);
	} else if (pow(B, 2.0) <= 2e-30) {
		tmp = t_1 * (-1.0 / t_0);
	} else {
		tmp = sqrt(((A - hypot(B, A)) * (2.0 * F))) / -B;
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = sqrt(Float64(-8.0 * Float64(Float64(F * Float64(A * C)) * Float64(2.0 * A))))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-149)
		tmp = Float64(Float64(-t_1) / t_0);
	elseif ((B ^ 2.0) <= 1e-59)
		tmp = Float64(-Float64(sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(Float64(A + C) - hypot(B, Float64(A - C))))))) / t_0));
	elseif ((B ^ 2.0) <= 2e-30)
		tmp = Float64(t_1 * Float64(-1.0 / t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * F))) / Float64(-B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(-8.0 * N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-149], N[((-t$95$1) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-59], (-N[(N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-30], N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B)), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 4.99999999999999968e-149

   1. Initial program 16.7%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in C around inf 21.0%

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

      1. associate-*r* 22.6%

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

      2. *-commutative 22.6%

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

      3. cancel-sign-sub-inv 22.6%

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

      4. metadata-eval 22.6%

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

      5. *-lft-identity 22.6%

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

   6. Simplified 22.6%

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

      1. distribute-frac-neg 22.6%

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

      2. associate-*r* 22.6%

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

      3. metadata-eval 22.6%

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

      4. associate-*r* 23.8%

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

      5. *-commutative 23.8%

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

      6. count-2 23.8%

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

   8. Applied egg-rr 23.8%

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

   if 4.99999999999999968e-149 < (pow.f64 B 2) < 1e-59

   1. Initial program 36.2%

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

   3. Simplified 43.9%

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

   if 1e-59 < (pow.f64 B 2) < 2e-30

   1. Initial program 3.0%

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

   3. Simplified 28.1%

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

   4. Taylor expanded in C around inf 25.8%

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

      1. associate-*r* 25.8%

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

      2. *-commutative 25.8%

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

      3. cancel-sign-sub-inv 25.8%

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

      4. metadata-eval 25.8%

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

      5. *-lft-identity 25.8%

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

   6. Simplified 25.8%

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

      1. div-inv 25.8%

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

      2. associate-*r* 25.8%

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

      3. metadata-eval 25.8%

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

      4. associate-*r* 25.8%

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

      5. *-commutative 25.8%

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

      6. count-2 25.8%

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

   8. Applied egg-rr 25.8%

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

   if 2e-30 < (pow.f64 B 2)

   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 22.7%

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

   4. Taylor expanded in C around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. distribute-rgt-neg-in 11.3%

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

      3. +-commutative 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-def 25.2%

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

   6. Simplified 25.2%

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

      1. associate-*l/ 25.2%

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

      2. frac-2neg 25.2%

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

   8. Applied egg-rr 25.4%

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

      1. associate-*r* 25.4%

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

      2. *-commutative 25.4%

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

   10. Simplified 25.4%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-59}:\\ \;\;\;\;-\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]

Alternative 3: 46.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := -\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{t_2}{t_1}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{\frac{B}{t_0}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;{\left(\frac{t_1}{t_2}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* (- A (hypot B A)) (* 2.0 F))))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2 (- (sqrt (* -8.0 (* (* F (* A C)) (* 2.0 A)))))))
   (if (<= (pow B 2.0) 5e-149)
     (/ t_2 t_1)
     (if (<= (pow B 2.0) 4e-96)
       (/ -1.0 (/ B t_0))
       (if (<= (pow B 2.0) 2e-30) (pow (/ t_1 t_2) -1.0) (/ t_0 (- B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(((A - hypot(B, A)) * (2.0 * F)));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = -sqrt((-8.0 * ((F * (A * C)) * (2.0 * A))));
	double tmp;
	if (pow(B, 2.0) <= 5e-149) {
		tmp = t_2 / t_1;
	} else if (pow(B, 2.0) <= 4e-96) {
		tmp = -1.0 / (B / t_0);
	} else if (pow(B, 2.0) <= 2e-30) {
		tmp = pow((t_1 / t_2), -1.0);
	} else {
		tmp = t_0 / -B;
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * F)))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(-sqrt(Float64(-8.0 * Float64(Float64(F * Float64(A * C)) * Float64(2.0 * A)))))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-149)
		tmp = Float64(t_2 / t_1);
	elseif ((B ^ 2.0) <= 4e-96)
		tmp = Float64(-1.0 / Float64(B / t_0));
	elseif ((B ^ 2.0) <= 2e-30)
		tmp = Float64(t_1 / t_2) ^ -1.0;
	else
		tmp = Float64(t_0 / Float64(-B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[N[(-8.0 * N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-149], N[(t$95$2 / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 4e-96], N[(-1.0 / N[(B / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-30], N[Power[N[(t$95$1 / t$95$2), $MachinePrecision], -1.0], $MachinePrecision], N[(t$95$0 / (-B)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 4.99999999999999968e-149

   1. Initial program 16.7%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in C around inf 21.0%

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

      1. associate-*r* 22.6%

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

      2. *-commutative 22.6%

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

      3. cancel-sign-sub-inv 22.6%

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

      4. metadata-eval 22.6%

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

      5. *-lft-identity 22.6%

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

   6. Simplified 22.6%

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

      1. distribute-frac-neg 22.6%

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

      2. associate-*r* 22.6%

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

      3. metadata-eval 22.6%

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

      4. associate-*r* 23.8%

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

      5. *-commutative 23.8%

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

      6. count-2 23.8%

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

   8. Applied egg-rr 23.8%

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

   if 4.99999999999999968e-149 < (pow.f64 B 2) < 3.9999999999999996e-96

   1. Initial program 37.6%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in C around 0 23.4%

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

      1. associate-*l* 23.4%

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

      2. +-commutative 23.4%

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

      3. unpow2 23.4%

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

      4. unpow2 23.4%

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

      5. hypot-def 23.6%

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

   6. Simplified 23.6%

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

      1. clear-num 23.6%

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

      2. inv-pow 23.6%

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

      3. sqrt-unprod 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. unpow-1 23.5%

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

      2. associate-/r* 23.7%

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

      3. *-commutative 23.7%

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

      4. associate-*r* 23.7%

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

   10. Simplified 23.7%

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

   11. Taylor expanded in A around 0 23.7%

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

   if 3.9999999999999996e-96 < (pow.f64 B 2) < 2e-30

   1. Initial program 26.7%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in C around inf 21.2%

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

      1. associate-*r* 21.2%

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

      2. *-commutative 21.2%

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

      3. cancel-sign-sub-inv 21.2%

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

      4. metadata-eval 21.2%

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

      5. *-lft-identity 21.2%

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

   6. Simplified 21.2%

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

      1. clear-num 21.4%

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

      2. inv-pow 21.4%

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

      3. associate-*r* 21.4%

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

      4. metadata-eval 21.4%

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

      5. associate-*r* 21.4%

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

      6. *-commutative 21.4%

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

      7. count-2 21.4%

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

   8. Applied egg-rr 21.4%

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

   if 2e-30 < (pow.f64 B 2)

   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 22.7%

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

   4. Taylor expanded in C around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. distribute-rgt-neg-in 11.3%

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

      3. +-commutative 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-def 25.2%

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

   6. Simplified 25.2%

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

      1. associate-*l/ 25.2%

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

      2. frac-2neg 25.2%

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

   8. Applied egg-rr 25.4%

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

      1. associate-*r* 25.4%

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

      2. *-commutative 25.4%

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

   10. Simplified 25.4%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]

Alternative 4: 46.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := -\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{t_2}{t_1}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{\frac{\frac{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}{B}}{t_0}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;{\left(\frac{t_1}{t_2}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* (- A (hypot B A)) (* 2.0 F))))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2 (- (sqrt (* -8.0 (* (* F (* A C)) (* 2.0 A)))))))
   (if (<= (pow B 2.0) 5e-149)
     (/ t_2 t_1)
     (if (<= (pow B 2.0) 4e-96)
       (/ -1.0 (/ (/ (fma A (* C -4.0) (pow B 2.0)) B) t_0))
       (if (<= (pow B 2.0) 2e-30) (pow (/ t_1 t_2) -1.0) (/ t_0 (- B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(((A - hypot(B, A)) * (2.0 * F)));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = -sqrt((-8.0 * ((F * (A * C)) * (2.0 * A))));
	double tmp;
	if (pow(B, 2.0) <= 5e-149) {
		tmp = t_2 / t_1;
	} else if (pow(B, 2.0) <= 4e-96) {
		tmp = -1.0 / ((fma(A, (C * -4.0), pow(B, 2.0)) / B) / t_0);
	} else if (pow(B, 2.0) <= 2e-30) {
		tmp = pow((t_1 / t_2), -1.0);
	} else {
		tmp = t_0 / -B;
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * F)))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(-sqrt(Float64(-8.0 * Float64(Float64(F * Float64(A * C)) * Float64(2.0 * A)))))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-149)
		tmp = Float64(t_2 / t_1);
	elseif ((B ^ 2.0) <= 4e-96)
		tmp = Float64(-1.0 / Float64(Float64(fma(A, Float64(C * -4.0), (B ^ 2.0)) / B) / t_0));
	elseif ((B ^ 2.0) <= 2e-30)
		tmp = Float64(t_1 / t_2) ^ -1.0;
	else
		tmp = Float64(t_0 / Float64(-B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[N[(-8.0 * N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-149], N[(t$95$2 / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 4e-96], N[(-1.0 / N[(N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-30], N[Power[N[(t$95$1 / t$95$2), $MachinePrecision], -1.0], $MachinePrecision], N[(t$95$0 / (-B)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 4.99999999999999968e-149

   1. Initial program 16.7%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in C around inf 21.0%

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

      1. associate-*r* 22.6%

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

      2. *-commutative 22.6%

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

      3. cancel-sign-sub-inv 22.6%

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

      4. metadata-eval 22.6%

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

      5. *-lft-identity 22.6%

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

   6. Simplified 22.6%

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

      1. distribute-frac-neg 22.6%

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

      2. associate-*r* 22.6%

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

      3. metadata-eval 22.6%

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

      4. associate-*r* 23.8%

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

      5. *-commutative 23.8%

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

      6. count-2 23.8%

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

   8. Applied egg-rr 23.8%

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

   if 4.99999999999999968e-149 < (pow.f64 B 2) < 3.9999999999999996e-96

   1. Initial program 37.6%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in C around 0 23.4%

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

      1. associate-*l* 23.4%

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

      2. +-commutative 23.4%

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

      3. unpow2 23.4%

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

      4. unpow2 23.4%

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

      5. hypot-def 23.6%

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

   6. Simplified 23.6%

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

      1. clear-num 23.6%

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

      2. inv-pow 23.6%

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

      3. sqrt-unprod 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. unpow-1 23.5%

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

      2. associate-/r* 23.7%

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

      3. *-commutative 23.7%

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

      4. associate-*r* 23.7%

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

   10. Simplified 23.7%

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

   if 3.9999999999999996e-96 < (pow.f64 B 2) < 2e-30

   1. Initial program 26.7%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in C around inf 21.2%

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

      1. associate-*r* 21.2%

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

      2. *-commutative 21.2%

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

      3. cancel-sign-sub-inv 21.2%

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

      4. metadata-eval 21.2%

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

      5. *-lft-identity 21.2%

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

   6. Simplified 21.2%

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

      1. clear-num 21.4%

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

      2. inv-pow 21.4%

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

      3. associate-*r* 21.4%

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

      4. metadata-eval 21.4%

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

      5. associate-*r* 21.4%

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

      6. *-commutative 21.4%

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

      7. count-2 21.4%

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

   8. Applied egg-rr 21.4%

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

   if 2e-30 < (pow.f64 B 2)

   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 22.7%

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

   4. Taylor expanded in C around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. distribute-rgt-neg-in 11.3%

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

      3. +-commutative 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-def 25.2%

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

   6. Simplified 25.2%

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

      1. associate-*l/ 25.2%

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

      2. frac-2neg 25.2%

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

   8. Applied egg-rr 25.4%

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

      1. associate-*r* 25.4%

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

      2. *-commutative 25.4%

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

   10. Simplified 25.4%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{\frac{\frac{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}{B}}{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]

Alternative 5: 46.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-t_2}{t_1}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{\frac{B}{t_0}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;t_2 \cdot \frac{-1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* (- A (hypot B A)) (* 2.0 F))))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2 (sqrt (* -8.0 (* (* F (* A C)) (* 2.0 A))))))
   (if (<= (pow B 2.0) 5e-149)
     (/ (- t_2) t_1)
     (if (<= (pow B 2.0) 4e-96)
       (/ -1.0 (/ B t_0))
       (if (<= (pow B 2.0) 2e-30) (* t_2 (/ -1.0 t_1)) (/ t_0 (- B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(((A - hypot(B, A)) * (2.0 * F)));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = sqrt((-8.0 * ((F * (A * C)) * (2.0 * A))));
	double tmp;
	if (pow(B, 2.0) <= 5e-149) {
		tmp = -t_2 / t_1;
	} else if (pow(B, 2.0) <= 4e-96) {
		tmp = -1.0 / (B / t_0);
	} else if (pow(B, 2.0) <= 2e-30) {
		tmp = t_2 * (-1.0 / t_1);
	} else {
		tmp = t_0 / -B;
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * F)))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = sqrt(Float64(-8.0 * Float64(Float64(F * Float64(A * C)) * Float64(2.0 * A))))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-149)
		tmp = Float64(Float64(-t_2) / t_1);
	elseif ((B ^ 2.0) <= 4e-96)
		tmp = Float64(-1.0 / Float64(B / t_0));
	elseif ((B ^ 2.0) <= 2e-30)
		tmp = Float64(t_2 * Float64(-1.0 / t_1));
	else
		tmp = Float64(t_0 / Float64(-B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(-8.0 * N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-149], N[((-t$95$2) / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 4e-96], N[(-1.0 / N[(B / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-30], N[(t$95$2 * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / (-B)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 4.99999999999999968e-149

   1. Initial program 16.7%

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

   3. Simplified 27.0%

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

   4. Taylor expanded in C around inf 21.0%

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

      1. associate-*r* 22.6%

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

      2. *-commutative 22.6%

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

      3. cancel-sign-sub-inv 22.6%

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

      4. metadata-eval 22.6%

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

      5. *-lft-identity 22.6%

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

   6. Simplified 22.6%

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

      1. distribute-frac-neg 22.6%

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

      2. associate-*r* 22.6%

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

      3. metadata-eval 22.6%

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

      4. associate-*r* 23.8%

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

      5. *-commutative 23.8%

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

      6. count-2 23.8%

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

   8. Applied egg-rr 23.8%

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

   if 4.99999999999999968e-149 < (pow.f64 B 2) < 3.9999999999999996e-96

   1. Initial program 37.6%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in C around 0 23.4%

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

      1. associate-*l* 23.4%

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

      2. +-commutative 23.4%

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

      3. unpow2 23.4%

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

      4. unpow2 23.4%

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

      5. hypot-def 23.6%

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

   6. Simplified 23.6%

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

      1. clear-num 23.6%

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

      2. inv-pow 23.6%

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

      3. sqrt-unprod 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. unpow-1 23.5%

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

      2. associate-/r* 23.7%

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

      3. *-commutative 23.7%

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

      4. associate-*r* 23.7%

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

   10. Simplified 23.7%

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

   11. Taylor expanded in A around 0 23.7%

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

   if 3.9999999999999996e-96 < (pow.f64 B 2) < 2e-30

   1. Initial program 26.7%

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

   3. Simplified 39.2%

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

   4. Taylor expanded in C around inf 21.2%

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

      1. associate-*r* 21.2%

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

      2. *-commutative 21.2%

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

      3. cancel-sign-sub-inv 21.2%

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

      4. metadata-eval 21.2%

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

      5. *-lft-identity 21.2%

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

   6. Simplified 21.2%

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

      1. div-inv 21.2%

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

      2. associate-*r* 21.2%

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

      3. metadata-eval 21.2%

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

      4. associate-*r* 21.2%

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

      5. *-commutative 21.2%

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

      6. count-2 21.2%

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

   8. Applied egg-rr 21.2%

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

   if 2e-30 < (pow.f64 B 2)

   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 22.7%

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

   4. Taylor expanded in C around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. distribute-rgt-neg-in 11.3%

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

      3. +-commutative 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-def 25.2%

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

   6. Simplified 25.2%

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

      1. associate-*l/ 25.2%

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

      2. frac-2neg 25.2%

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

   8. Applied egg-rr 25.4%

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

      1. associate-*r* 25.4%

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

      2. *-commutative 25.4%

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

   10. Simplified 25.4%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]

Alternative 6: 46.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} t_0 := \sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}\\ t_1 := \frac{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{\frac{B}{t_0}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (sqrt (* (- A (hypot B A)) (* 2.0 F))))
        (t_1
         (/
          (- (sqrt (* -8.0 (* (* F (* A C)) (* 2.0 A)))))
          (fma B B (* A (* C -4.0))))))
   (if (<= (pow B 2.0) 5e-149)
     t_1
     (if (<= (pow B 2.0) 4e-96)
       (/ -1.0 (/ B t_0))
       (if (<= (pow B 2.0) 2e-30) t_1 (/ t_0 (- B)))))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt(((A - hypot(B, A)) * (2.0 * F)));
	double t_1 = -sqrt((-8.0 * ((F * (A * C)) * (2.0 * A)))) / fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (pow(B, 2.0) <= 5e-149) {
		tmp = t_1;
	} else if (pow(B, 2.0) <= 4e-96) {
		tmp = -1.0 / (B / t_0);
	} else if (pow(B, 2.0) <= 2e-30) {
		tmp = t_1;
	} else {
		tmp = t_0 / -B;
	}
	return tmp;
}

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	t_0 = sqrt(Float64(Float64(A - hypot(B, A)) * Float64(2.0 * F)))
	t_1 = Float64(Float64(-sqrt(Float64(-8.0 * Float64(Float64(F * Float64(A * C)) * Float64(2.0 * A))))) / fma(B, B, Float64(A * Float64(C * -4.0))))
	tmp = 0.0
	if ((B ^ 2.0) <= 5e-149)
		tmp = t_1;
	elseif ((B ^ 2.0) <= 4e-96)
		tmp = Float64(-1.0 / Float64(B / t_0));
	elseif ((B ^ 2.0) <= 2e-30)
		tmp = t_1;
	else
		tmp = Float64(t_0 / Float64(-B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(A - N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-N[Sqrt[N[(-8.0 * N[(N[(F * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 5e-149], t$95$1, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 4e-96], N[(-1.0 / N[(B / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-30], t$95$1, N[(t$95$0 / (-B)), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 4.99999999999999968e-149 or 3.9999999999999996e-96 < (pow.f64 B 2) < 2e-30

   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 28.5%

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

   4. Taylor expanded in C around inf 21.0%

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

      1. associate-*r* 22.4%

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

      2. *-commutative 22.4%

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

      3. cancel-sign-sub-inv 22.4%

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

      4. metadata-eval 22.4%

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

      5. *-lft-identity 22.4%

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

   6. Simplified 22.4%

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

      1. distribute-frac-neg 22.4%

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

      2. associate-*r* 22.4%

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

      3. metadata-eval 22.4%

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

      4. associate-*r* 23.5%

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

      5. *-commutative 23.5%

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

      6. count-2 23.5%

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

   8. Applied egg-rr 23.5%

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

   if 4.99999999999999968e-149 < (pow.f64 B 2) < 3.9999999999999996e-96

   1. Initial program 37.6%

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

   3. Simplified 38.9%

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

   4. Taylor expanded in C around 0 23.4%

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

      1. associate-*l* 23.4%

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

      2. +-commutative 23.4%

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

      3. unpow2 23.4%

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

      4. unpow2 23.4%

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

      5. hypot-def 23.6%

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

   6. Simplified 23.6%

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

      1. clear-num 23.6%

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

      2. inv-pow 23.6%

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

      3. sqrt-unprod 23.5%

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

   8. Applied egg-rr 23.5%

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

      1. unpow-1 23.5%

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

      2. associate-/r* 23.7%

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

      3. *-commutative 23.7%

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

      4. associate-*r* 23.7%

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

   10. Simplified 23.7%

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

   11. Taylor expanded in A around 0 23.7%

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

   if 2e-30 < (pow.f64 B 2)

   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 22.7%

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

   4. Taylor expanded in C around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. distribute-rgt-neg-in 11.3%

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

      3. +-commutative 11.3%

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

      4. unpow2 11.3%

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

      5. unpow2 11.3%

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

      6. hypot-def 25.2%

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

   6. Simplified 25.2%

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

      1. associate-*l/ 25.2%

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

      2. frac-2neg 25.2%

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

   8. Applied egg-rr 25.4%

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

      1. associate-*r* 25.4%

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

      2. *-commutative 25.4%

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

   10. Simplified 25.4%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(\left(F \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \end{array} \]

Alternative 7: 26.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ [A, C] = \mathsf{sort}([A, C])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -4.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{{\left(A \cdot F\right)}^{0.5}}{B} \cdot \left(-2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(-B\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= A -4.6e+60)
   (* (/ (pow (* A F) 0.5) B) (- 2.0))
   (* (sqrt (* F (- B))) (/ (- (sqrt 2.0)) B))))

C

B = abs(B);
assert(A < C);
double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -4.6e+60) {
		tmp = (pow((A * F), 0.5) / B) * -2.0;
	} else {
		tmp = sqrt((F * -B)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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) :: tmp
    if (a <= (-4.6d+60)) then
        tmp = (((a * f) ** 0.5d0) / b) * -2.0d0
    else
        tmp = sqrt((f * -b)) * (-sqrt(2.0d0) / b)
    end if
    code = tmp
end function

Java

B = Math.abs(B);
assert A < C;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (A <= -4.6e+60) {
		tmp = (Math.pow((A * F), 0.5) / B) * -2.0;
	} else {
		tmp = Math.sqrt((F * -B)) * (-Math.sqrt(2.0) / B);
	}
	return tmp;
}

Python

B = abs(B)
[A, C] = sort([A, C])
def code(A, B, C, F):
	tmp = 0
	if A <= -4.6e+60:
		tmp = (math.pow((A * F), 0.5) / B) * -2.0
	else:
		tmp = math.sqrt((F * -B)) * (-math.sqrt(2.0) / B)
	return tmp

Julia

B = abs(B)
A, C = sort([A, C])
function code(A, B, C, F)
	tmp = 0.0
	if (A <= -4.6e+60)
		tmp = Float64(Float64((Float64(A * F) ^ 0.5) / B) * Float64(-2.0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(-B))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

MATLAB

B = abs(B)
A, C = num2cell(sort([A, C])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (A <= -4.6e+60)
		tmp = (((A * F) ^ 0.5) / B) * -2.0;
	else
		tmp = sqrt((F * -B)) * (-sqrt(2.0) / B);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[A, -4.6e+60], N[(N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B), $MachinePrecision] * (-2.0)), $MachinePrecision], N[(N[Sqrt[N[(F * (-B)), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.60000000000000034e60

   1. Initial program 18.9%

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

   3. Simplified 38.7%

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

   4. Taylor expanded in C around inf 34.6%

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

      1. cancel-sign-sub-inv 34.6%

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

      2. metadata-eval 34.6%

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

      3. *-lft-identity 34.6%

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

   6. Simplified 34.6%

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

      1. pow1/2 34.6%

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

      2. pow-to-exp 32.4%

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

      3. associate-*l* 35.0%

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

      4. count-2 35.0%

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

   8. Applied egg-rr 35.0%

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

   9. Taylor expanded in C around 0 15.7%

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

       1. associate-*r/ 15.8%

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

       2. *-rgt-identity 15.8%

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

   11. Simplified 15.8%

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

       1. add-sqr-sqrt 15.6%

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

       2. *-un-lft-identity 15.6%

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

       3. times-frac 15.7%

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

       4. pow1/2 15.7%

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

       5. sqrt-pow1 15.7%

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

       6. *-commutative 15.7%

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

       7. metadata-eval 15.7%

          \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{A \cdot F}}}{B}\right) \]

       8. pow1/2 16.0%

          \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}}{B}\right) \]

       9. sqrt-pow1 16.0%

          \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{\left(\frac{0.5}{2}\right)}}}{B}\right) \]

       10. *-commutative 16.0%

           \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{{\color{blue}{\left(F \cdot A\right)}}^{\left(\frac{0.5}{2}\right)}}{B}\right) \]

       11. metadata-eval 16.0%

           \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{{\left(F \cdot A\right)}^{\color{blue}{0.25}}}{B}\right) \]

   13. Applied egg-rr 16.0%

       \[\leadsto -2 \cdot \color{blue}{\left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{{\left(F \cdot A\right)}^{0.25}}{B}\right)} \]
   14. Step-by-step derivation

       1. /-rgt-identity 16.0%

          \[\leadsto -2 \cdot \left(\color{blue}{{\left(F \cdot A\right)}^{0.25}} \cdot \frac{{\left(F \cdot A\right)}^{0.25}}{B}\right) \]

       2. associate-*r/ 15.9%

          \[\leadsto -2 \cdot \color{blue}{\frac{{\left(F \cdot A\right)}^{0.25} \cdot {\left(F \cdot A\right)}^{0.25}}{B}} \]

       3. pow-sqr 16.0%

          \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{\left(2 \cdot 0.25\right)}}}{B} \]

       4. metadata-eval 16.0%

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

   15. Simplified 16.0%

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

   if -4.60000000000000034e60 < A

   1. Initial program 20.3%

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

   3. Simplified 23.6%

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

   4. Taylor expanded in C around 0 7.8%

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

      1. mul-1-neg 7.8%

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

      2. distribute-rgt-neg-in 7.8%

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

      3. +-commutative 7.8%

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

      4. unpow2 7.8%

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

      5. unpow2 7.8%

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

      6. hypot-def 15.1%

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

   6. Simplified 15.1%

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

   7. Taylor expanded in A around 0 13.9%

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

      1. associate-*r* 13.9%

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

      2. mul-1-neg 13.9%

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

   9. Simplified 13.9%

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

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -4.6 \cdot 10^{+60}:\\ \;\;\;\;\frac{{\left(A \cdot F\right)}^{0.5}}{B} \cdot \left(-2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(-B\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8: 31.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.1%

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

3. Simplified 26.8%

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

4. Taylor expanded in C around 0 8.3%

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

   1. mul-1-neg 8.3%

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

   2. distribute-rgt-neg-in 8.3%

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

   3. +-commutative 8.3%

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

   4. unpow2 8.3%

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

   5. unpow2 8.3%

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

   6. hypot-def 15.2%

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

6. Simplified 15.2%

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

   1. associate-*l/ 15.2%

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

   2. frac-2neg 15.2%

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

8. Applied egg-rr 15.3%

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

   1. associate-*r* 15.3%

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

   2. *-commutative 15.3%

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

10. Simplified 15.3%

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

11. Final simplification 15.3%

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

Alternative 9: 9.1% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = (((a * f) ** 0.5d0) / b) * -2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.1%

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

3. Simplified 27.6%

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

4. Taylor expanded in C around inf 14.5%

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

   1. cancel-sign-sub-inv 14.5%

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

   2. metadata-eval 14.5%

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

   3. *-lft-identity 14.5%

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

6. Simplified 14.5%

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

   1. pow1/2 14.7%

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

   2. pow-to-exp 13.6%

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

   3. associate-*l* 13.9%

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

   4. count-2 13.9%

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

8. Applied egg-rr 13.9%

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

9. Taylor expanded in C around 0 4.4%

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

    1. associate-*r/ 4.4%

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

    2. *-rgt-identity 4.4%

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

11. Simplified 4.4%

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

    1. add-sqr-sqrt 4.4%

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

    2. *-un-lft-identity 4.4%

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

    3. times-frac 4.4%

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

    4. pow1/2 4.4%

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

    5. sqrt-pow1 4.4%

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

    6. *-commutative 4.4%

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

    7. metadata-eval 4.4%

       \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{\color{blue}{0.25}}}{1} \cdot \frac{\sqrt{\sqrt{A \cdot F}}}{B}\right) \]

    8. pow1/2 4.6%

       \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{\sqrt{\color{blue}{{\left(A \cdot F\right)}^{0.5}}}}{B}\right) \]

    9. sqrt-pow1 4.6%

       \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{\color{blue}{{\left(A \cdot F\right)}^{\left(\frac{0.5}{2}\right)}}}{B}\right) \]

    10. *-commutative 4.6%

        \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{{\color{blue}{\left(F \cdot A\right)}}^{\left(\frac{0.5}{2}\right)}}{B}\right) \]

    11. metadata-eval 4.6%

        \[\leadsto -2 \cdot \left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{{\left(F \cdot A\right)}^{\color{blue}{0.25}}}{B}\right) \]

13. Applied egg-rr 4.6%

    \[\leadsto -2 \cdot \color{blue}{\left(\frac{{\left(F \cdot A\right)}^{0.25}}{1} \cdot \frac{{\left(F \cdot A\right)}^{0.25}}{B}\right)} \]
14. Step-by-step derivation

    1. /-rgt-identity 4.6%

       \[\leadsto -2 \cdot \left(\color{blue}{{\left(F \cdot A\right)}^{0.25}} \cdot \frac{{\left(F \cdot A\right)}^{0.25}}{B}\right) \]

    2. associate-*r/ 4.5%

       \[\leadsto -2 \cdot \color{blue}{\frac{{\left(F \cdot A\right)}^{0.25} \cdot {\left(F \cdot A\right)}^{0.25}}{B}} \]

    3. pow-sqr 4.6%

       \[\leadsto -2 \cdot \frac{\color{blue}{{\left(F \cdot A\right)}^{\left(2 \cdot 0.25\right)}}}{B} \]

    4. metadata-eval 4.6%

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

15. Simplified 4.6%

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

16. Final simplification 4.6%

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

Alternative 10: 9.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: B should be positive before calling this function
NOTE: A and C should be sorted in increasing order before calling this function.
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
    code = (sqrt((a * f)) / b) * -2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.1%

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

3. Simplified 27.6%

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

4. Taylor expanded in C around inf 14.5%

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

   1. cancel-sign-sub-inv 14.5%

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

   2. metadata-eval 14.5%

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

   3. *-lft-identity 14.5%

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

6. Simplified 14.5%

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

   1. pow1/2 14.7%

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

   2. pow-to-exp 13.6%

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

   3. associate-*l* 13.9%

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

   4. count-2 13.9%

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

8. Applied egg-rr 13.9%

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

9. Taylor expanded in C around 0 4.4%

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

    1. associate-*r/ 4.4%

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

    2. *-rgt-identity 4.4%

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

11. Simplified 4.4%

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

12. Final simplification 4.4%

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

Reproduce

herbie shell --seed 2023310 
(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))))