ABCF->ab-angle b

Percentage Accurate: 18.3% → 47.6%

Time: 23.0s

Alternatives: 9

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.3% 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: 47.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot F\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-39)
     (/ (sqrt (* (* t_0 F) (* 2.0 (+ A A)))) (- t_0))
     (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-39) {
		tmp = sqrt(((t_0 * F) * (2.0 * (A + A)))) / -t_0;
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-39)
		tmp = Float64(sqrt(Float64(Float64(t_0 * F) * Float64(2.0 * Float64(A + A)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-39], N[(N[Sqrt[N[(N[(t$95$0 * F), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 1.99999999999999986e-39

   1. Initial program 22.6%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in C around inf 22.2%

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

      1. sub-neg 22.2%

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

      2. mul-1-neg 22.2%

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

      3. remove-double-neg 22.2%

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

   7. Simplified 22.2%

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

   if 1.99999999999999986e-39 < (pow.f64 B 2)

   1. Initial program 17.9%

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

   3. Simplified 15.5%

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

   5. Taylor expanded in C around 0 12.2%

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

      1. associate-*r* 12.2%

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

      2. mul-1-neg 12.2%

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

      3. +-commutative 12.2%

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

      4. unpow2 12.2%

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

      5. unpow2 12.2%

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

      6. hypot-define 27.3%

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

   7. Simplified 27.3%

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

   9. Applied egg-rr 27.4%

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

       1. unpow1 27.4%

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

       2. distribute-neg-frac2 27.4%

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

       3. unpow1/2 27.3%

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

       4. hypot-undefine 12.2%

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

       5. unpow2 12.2%

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

       6. unpow2 12.2%

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

       7. +-commutative 12.2%

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

       8. unpow2 12.2%

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

       9. unpow2 12.2%

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

       10. hypot-undefine 27.3%

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

   11. Simplified 27.3%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 42.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-192)
   (/
    (sqrt (* 2.0 (* (* A -4.0) (* (+ A A) (* C F)))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-192) {
		tmp = sqrt((2.0 * ((A * -4.0) * ((A + A) * (C * F))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-192) {
		tmp = Math.sqrt((2.0 * ((A * -4.0) * ((A + A) * (C * F))))) / ((4.0 * (A * C)) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-192:
		tmp = math.sqrt((2.0 * ((A * -4.0) * ((A + A) * (C * F))))) / ((4.0 * (A * C)) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-192)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(A * -4.0) * Float64(Float64(A + A) * Float64(C * F))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-192)
		tmp = sqrt((2.0 * ((A * -4.0) * ((A + A) * (C * F))))) / ((4.0 * (A * C)) - (B_m ^ 2.0));
	else
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-192], N[(N[Sqrt[N[(2.0 * N[(N[(A * -4.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.0000000000000001e-192

   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 25.3%

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

   5. Taylor expanded in C around inf 21.3%

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

      1. associate-*r* 21.3%

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

      2. associate-*r* 18.2%

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

      3. *-commutative 18.2%

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

      4. sub-neg 18.2%

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

      5. mul-1-neg 18.2%

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

      6. remove-double-neg 18.2%

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

   7. Simplified 18.2%

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

   if 5.0000000000000001e-192 < (pow.f64 B 2)

   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 18.9%

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

   5. Taylor expanded in C around 0 11.8%

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

      1. associate-*r* 11.8%

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

      2. mul-1-neg 11.8%

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

      3. +-commutative 11.8%

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

      4. unpow2 11.8%

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

      5. unpow2 11.8%

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

      6. hypot-define 23.6%

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

   7. Simplified 23.6%

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

   9. Applied egg-rr 23.7%

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

       1. unpow1 23.7%

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

       2. distribute-neg-frac2 23.7%

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

       3. unpow1/2 23.7%

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

       4. hypot-undefine 11.8%

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

       5. unpow2 11.8%

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

       6. unpow2 11.8%

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

       7. +-commutative 11.8%

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

       8. unpow2 11.8%

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

       9. unpow2 11.8%

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

       10. hypot-undefine 23.7%

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

   11. Simplified 23.7%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A \cdot -4\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 44.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-184)
   (/
    (sqrt (* 2.0 (* -4.0 (* A (* C (* F (+ A A)))))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-184) {
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-184) {
		tmp = Math.sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / ((4.0 * (A * C)) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-184:
		tmp = math.sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / ((4.0 * (A * C)) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-184)
		tmp = Float64(sqrt(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-184)
		tmp = sqrt((2.0 * (-4.0 * (A * (C * (F * (A + A))))))) / ((4.0 * (A * C)) - (B_m ^ 2.0));
	else
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-184], N[(N[Sqrt[N[(2.0 * N[(-4.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.00000000000000003e-184

   1. Initial program 20.2%

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

   3. Simplified 25.6%

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

   5. Taylor expanded in C around inf 22.6%

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

   if 5.00000000000000003e-184 < (pow.f64 B 2)

   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 18.6%

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

   5. Taylor expanded in C around 0 12.0%

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

      1. associate-*r* 12.0%

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

      2. mul-1-neg 12.0%

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

      3. +-commutative 12.0%

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

      4. unpow2 12.0%

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

      5. unpow2 12.0%

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

      6. hypot-define 24.1%

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

   7. Simplified 24.1%

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

   9. Applied egg-rr 24.2%

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

       1. unpow1 24.2%

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

       2. distribute-neg-frac2 24.2%

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

       3. unpow1/2 24.2%

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

       4. hypot-undefine 12.0%

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

       5. unpow2 12.0%

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

       6. unpow2 12.0%

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

       7. +-commutative 12.0%

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

       8. unpow2 12.0%

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

       9. unpow2 12.0%

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

       10. hypot-undefine 24.2%

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

   11. Simplified 24.2%

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

4. Final simplification 23.6%

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

Alternative 4: 26.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -8.5 \cdot 10^{+260}:\\ \;\;\;\;\frac{-\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(2 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{B\_m \cdot \left(-F\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -8.5e+260)
   (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (* 2.0 A))))
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* B_m (- F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -8.5e+260) {
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (2.0 * A)));
	} else {
		tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * -F));
	}
	return tmp;
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-8.5d+260)) then
        tmp = (-sqrt(2.0d0) / b_m) * sqrt((f * (2.0d0 * a)))
    else
        tmp = (sqrt(2.0d0) / b_m) * -sqrt((b_m * -f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -8.5e+260) {
		tmp = (-Math.sqrt(2.0) / B_m) * Math.sqrt((F * (2.0 * A)));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((B_m * -F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -8.5e+260:
		tmp = (-math.sqrt(2.0) / B_m) * math.sqrt((F * (2.0 * A)))
	else:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((B_m * -F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -8.5e+260)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(2.0 * A))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(B_m * Float64(-F)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -8.5e+260)
		tmp = (-sqrt(2.0) / B_m) * sqrt((F * (2.0 * A)));
	else
		tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * -F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -8.5e+260], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -8.5e260

   1. Initial program 1.3%

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

   3. Simplified 1.3%

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

   5. Taylor expanded in C around 0 1.0%

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

      1. associate-*r* 1.0%

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

      2. mul-1-neg 1.0%

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

      3. +-commutative 1.0%

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

      4. unpow2 1.0%

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

      5. unpow2 1.0%

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

      6. hypot-define 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in A around -inf 11.7%

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

      1. *-commutative 11.7%

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

   10. Simplified 11.7%

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

   if -8.5e260 < A

   1. Initial program 20.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in C around 0 8.6%

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

      1. associate-*r* 8.6%

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

      2. mul-1-neg 8.6%

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

      3. +-commutative 8.6%

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

      4. unpow2 8.6%

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

      5. unpow2 8.6%

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

      6. hypot-define 16.5%

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

   7. Simplified 16.5%

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

   8. Taylor expanded in A around 0 15.4%

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

      1. associate-*r* 15.4%

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

      2. mul-1-neg 15.4%

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

   10. Simplified 15.4%

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

4. Final simplification 15.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -8.5 \cdot 10^{+260}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(2 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot \left(-F\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 27.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{-\sqrt{2}}{B\_m}\\ \mathbf{if}\;A \leq -1.62 \cdot 10^{+260}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \left(2 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{F \cdot \left(A - B\_m\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt 2.0)) B_m)))
   (if (<= A -1.62e+260)
     (* t_0 (sqrt (* F (* 2.0 A))))
     (* t_0 (sqrt (* F (- A B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt(2.0) / B_m;
	double tmp;
	if (A <= -1.62e+260) {
		tmp = t_0 * sqrt((F * (2.0 * A)));
	} else {
		tmp = t_0 * sqrt((F * (A - B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -sqrt(2.0d0) / b_m
    if (a <= (-1.62d+260)) then
        tmp = t_0 * sqrt((f * (2.0d0 * a)))
    else
        tmp = t_0 * sqrt((f * (a - b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -Math.sqrt(2.0) / B_m;
	double tmp;
	if (A <= -1.62e+260) {
		tmp = t_0 * Math.sqrt((F * (2.0 * A)));
	} else {
		tmp = t_0 * Math.sqrt((F * (A - B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = -math.sqrt(2.0) / B_m
	tmp = 0
	if A <= -1.62e+260:
		tmp = t_0 * math.sqrt((F * (2.0 * A)))
	else:
		tmp = t_0 * math.sqrt((F * (A - B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(-sqrt(2.0)) / B_m)
	tmp = 0.0
	if (A <= -1.62e+260)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(2.0 * A))));
	else
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(A - B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = -sqrt(2.0) / B_m;
	tmp = 0.0;
	if (A <= -1.62e+260)
		tmp = t_0 * sqrt((F * (2.0 * A)));
	else
		tmp = t_0 * sqrt((F * (A - B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]}, If[LessEqual[A, -1.62e+260], N[(t$95$0 * N[Sqrt[N[(F * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < -1.62e260

   1. Initial program 1.3%

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

   3. Simplified 1.3%

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

   5. Taylor expanded in C around 0 1.0%

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

      1. associate-*r* 1.0%

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

      2. mul-1-neg 1.0%

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

      3. +-commutative 1.0%

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

      4. unpow2 1.0%

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

      5. unpow2 1.0%

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

      6. hypot-define 12.4%

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

   7. Simplified 12.4%

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

   8. Taylor expanded in A around -inf 11.7%

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

      1. *-commutative 11.7%

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

   10. Simplified 11.7%

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

   if -1.62e260 < A

   1. Initial program 20.8%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in C around 0 8.6%

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

      1. associate-*r* 8.6%

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

      2. mul-1-neg 8.6%

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

      3. +-commutative 8.6%

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

      4. unpow2 8.6%

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

      5. unpow2 8.6%

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

      6. hypot-define 16.5%

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

   7. Simplified 16.5%

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

   8. Taylor expanded in A around 0 14.2%

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

      1. mul-1-neg 14.2%

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

      2. unsub-neg 14.2%

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

   10. Simplified 14.2%

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

4. Final simplification 14.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.62 \cdot 10^{+260}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(2 \cdot A\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.2%

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

3. Simplified 21.1%

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

5. Taylor expanded in C around 0 8.3%

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

   1. associate-*r* 8.3%

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

   2. mul-1-neg 8.3%

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

   3. +-commutative 8.3%

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

   4. unpow2 8.3%

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

   5. unpow2 8.3%

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

   6. hypot-define 16.4%

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

7. Simplified 16.4%

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

9. Applied egg-rr 16.5%

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

    1. unpow1 16.5%

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

    2. distribute-neg-frac2 16.5%

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

    3. unpow1/2 16.4%

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

    4. hypot-undefine 8.3%

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

    5. unpow2 8.3%

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

    6. unpow2 8.3%

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

    7. +-commutative 8.3%

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

    8. unpow2 8.3%

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

    9. unpow2 8.3%

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

    10. hypot-undefine 16.4%

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

11. Simplified 16.4%

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

12. Final simplification 16.4%

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

Alternative 7: 26.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.2%

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

3. Simplified 21.1%

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

5. Taylor expanded in C around 0 8.3%

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

   1. associate-*r* 8.3%

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

   2. mul-1-neg 8.3%

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

   3. +-commutative 8.3%

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

   4. unpow2 8.3%

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

   5. unpow2 8.3%

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

   6. hypot-define 16.4%

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

7. Simplified 16.4%

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

8. Taylor expanded in A around 0 14.9%

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

   1. associate-*r* 14.9%

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

   2. mul-1-neg 14.9%

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

10. Simplified 14.9%

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

11. Final simplification 14.9%

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

Alternative 8: 0.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.2%

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

3. Simplified 21.1%

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

5. Taylor expanded in C around 0 8.3%

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

   1. associate-*r* 8.3%

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

   2. mul-1-neg 8.3%

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

   3. +-commutative 8.3%

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

   4. unpow2 8.3%

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

   5. unpow2 8.3%

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

   6. hypot-define 16.4%

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

7. Simplified 16.4%

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

8. Taylor expanded in B around -inf 1.6%

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

9. Final simplification 1.6%

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

Alternative 9: 0.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(0.25 \cdot \sqrt{\frac{F}{A}}\right) \cdot \sqrt{-16} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (* 0.25 (sqrt (/ F A))) (sqrt -16.0)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return (0.25 * sqrt((F / A))) * sqrt(-16.0);
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (0.25d0 * sqrt((f / a))) * sqrt((-16.0d0))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return (0.25 * Math.sqrt((F / A))) * Math.sqrt(-16.0);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return (0.25 * math.sqrt((F / A))) * math.sqrt(-16.0)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(0.25 * sqrt(Float64(F / A))) * sqrt(-16.0))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (0.25 * sqrt((F / A))) * sqrt(-16.0);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[(0.25 * N[Sqrt[N[(F / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[-16.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.2%

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

3. Simplified 21.1%

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

5. Taylor expanded in A around inf 12.7%

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

   1. associate-*r* 13.7%

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

   2. *-commutative 13.7%

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

   3. sub-neg 13.7%

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

   4. mul-1-neg 13.7%

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

   5. remove-double-neg 13.7%

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

7. Simplified 13.7%

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

8. Taylor expanded in C around inf 0.0%

   \[\leadsto \color{blue}{0.25 \cdot \left(\sqrt{\frac{F}{A}} \cdot \sqrt{-16}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 0.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot \sqrt{\frac{F}{A}}\right) \cdot \sqrt{-16}} \]

10. Simplified 0.0%

    \[\leadsto \color{blue}{\left(0.25 \cdot \sqrt{\frac{F}{A}}\right) \cdot \sqrt{-16}} \]

11. Final simplification 0.0%

    \[\leadsto \left(0.25 \cdot \sqrt{\frac{F}{A}}\right) \cdot \sqrt{-16} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))))