ABCF->ab-angle a

Percentage Accurate: 19.2% → 50.7%

Time: 15.5s

Alternatives: 10

Speedup: 16.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 50.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_0\right)\right) \cdot 2\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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 (* C (* A 4.0))))
   (if (<= (pow B_m 2.0) 2e-95)
     (/
      (sqrt
       (*
        (fma (/ (* B_m B_m) A) -0.5 (* C 2.0))
        (* (* F (- (pow B_m 2.0) t_0)) 2.0)))
      (- t_0 (pow B_m 2.0)))
     (/ (- (sqrt F)) (sqrt (* 0.5 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 = C * (A * 4.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-95) {
		tmp = sqrt((fma(((B_m * B_m) / A), -0.5, (C * 2.0)) * ((F * (pow(B_m, 2.0) - t_0)) * 2.0))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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(C * Float64(A * 4.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-95)
		tmp = Float64(sqrt(Float64(fma(Float64(Float64(B_m * B_m) / A), -0.5, Float64(C * 2.0)) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_0)) * 2.0))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-95], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5 + N[(C * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999998e-95

   1. Initial program 21.1%

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

   3. Taylor expanded in A around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 28.3

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

   5. Applied rewrites 28.3%

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

   if 1.99999999999999998e-95 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.7%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 26.2

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

   5. Applied rewrites 26.2%

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

   7. Applied rewrites 26.3%

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

   9. Applied rewrites 26.3%

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

   11. Applied rewrites 32.5%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\frac{B \cdot B}{A}, -0.5, C \cdot 2\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\left(C \cdot 2\right) \cdot \left(\left(t\_0 \cdot F\right) \cdot 2\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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 (* C A) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e-147)
     (* (sqrt (* (* C 2.0) (* (* t_0 F) 2.0))) (/ -1.0 t_0))
     (/ (- (sqrt F)) (sqrt (* 0.5 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((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-147) {
		tmp = sqrt(((C * 2.0) * ((t_0 * F) * 2.0))) * (-1.0 / t_0);
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-147)
		tmp = Float64(sqrt(Float64(Float64(C * 2.0) * Float64(Float64(t_0 * F) * 2.0))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-147], N[(N[Sqrt[N[(N[(C * 2.0), $MachinePrecision] * N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000013e-147

   1. Initial program 19.1%

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

   3. Taylor expanded in C around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.3

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

   5. Applied rewrites 4.3%

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

   7. Applied rewrites 4.3%

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

   8. Taylor expanded in C around inf

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

      1. lower-*.f64 29.4

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

   10. Applied rewrites 29.4%

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

   if 5.00000000000000013e-147 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 25.4

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

   5. Applied rewrites 25.4%

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

   7. Applied rewrites 25.5%

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

   9. Applied rewrites 25.5%

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

   11. Applied rewrites 31.1%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\left(C \cdot 2\right) \cdot \left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right)} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)}{\sqrt{\left(\left(\left(C \cdot A\right) \cdot F\right) \cdot -8\right) \cdot \left(C \cdot 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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-147)
   (/
    -1.0
    (/
     (fma -4.0 (* C A) (* B_m B_m))
     (sqrt (* (* (* (* C A) F) -8.0) (* C 2.0)))))
   (/ (- (sqrt F)) (sqrt (* 0.5 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-147) {
		tmp = -1.0 / (fma(-4.0, (C * A), (B_m * B_m)) / sqrt(((((C * A) * F) * -8.0) * (C * 2.0))));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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-147)
		tmp = Float64(-1.0 / Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) / sqrt(Float64(Float64(Float64(Float64(C * A) * F) * -8.0) * Float64(C * 2.0)))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-147], N[(-1.0 / N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(C * A), $MachinePrecision] * F), $MachinePrecision] * -8.0), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000013e-147

   1. Initial program 19.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 29.5

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in C around inf

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

      1. rem-square-sqrt N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. rem-square-sqrt N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 28.4

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

   10. Applied rewrites 28.4%

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

   if 5.00000000000000013e-147 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 25.4

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

   5. Applied rewrites 25.4%

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

   7. Applied rewrites 25.5%

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

   9. Applied rewrites 25.5%

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

   11. Applied rewrites 31.1%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\left(C \cdot A\right) \cdot F\right) \cdot -8\right) \cdot \left(C \cdot 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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-147)
   (*
    (sqrt (* (* (* (* C C) F) A) -16.0))
    (/ -1.0 (fma (* C A) -4.0 (* B_m B_m))))
   (/ (- (sqrt F)) (sqrt (* 0.5 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-147) {
		tmp = sqrt(((((C * C) * F) * A) * -16.0)) * (-1.0 / fma((C * A), -4.0, (B_m * B_m)));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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-147)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -16.0)) * Float64(-1.0 / fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-147], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000013e-147

   1. Initial program 19.1%

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

   3. Taylor expanded in C around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 4.3

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

   5. Applied rewrites 4.3%

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

   7. Applied rewrites 4.3%

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

   8. Taylor expanded in C around inf

      \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 18.6

         \[\leadsto \sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   10. Applied rewrites 18.6%

       \[\leadsto \sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 5.00000000000000013e-147 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 15.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 25.4

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

   5. Applied rewrites 25.4%

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

   7. Applied rewrites 25.5%

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

   9. Applied rewrites 25.5%

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

   11. Applied rewrites 31.1%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16} \cdot \frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot \sqrt{\left(\left(t\_0 \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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 (* C A) -4.0 (* B_m B_m))))
   (if (<= B_m 6e-45)
     (*
      (/ -1.0 t_0)
      (sqrt (* (* (* t_0 F) 2.0) (fma -0.5 (/ (* B_m B_m) A) (* C 2.0)))))
     (/ (- (sqrt F)) (sqrt (* 0.5 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((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (B_m <= 6e-45) {
		tmp = (-1.0 / t_0) * sqrt((((t_0 * F) * 2.0) * fma(-0.5, ((B_m * B_m) / A), (C * 2.0))));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 6e-45)
		tmp = Float64(Float64(-1.0 / t_0) * sqrt(Float64(Float64(Float64(t_0 * F) * 2.0) * fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(C * 2.0)))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6e-45], N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Sqrt[N[(N[(N[(t$95$0 * F), $MachinePrecision] * 2.0), $MachinePrecision] * N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(C * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 6.00000000000000022e-45

   1. Initial program 18.3%

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

   3. Taylor expanded in C around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 3.3

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

   5. Applied rewrites 3.3%

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 19.3

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

   10. Applied rewrites 19.3%

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

   if 6.00000000000000022e-45 < B

   1. Initial program 13.4%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 47.3

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

   5. Applied rewrites 47.3%

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

   7. Applied rewrites 47.4%

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

   9. Applied rewrites 47.4%

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

   11. Applied rewrites 60.1%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \cdot \sqrt{\left(\left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot F\right) \cdot 2\right) \cdot \mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 8 \cdot 10^{-71}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(C \cdot 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

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 -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 8e-71)
     (/ -1.0 (/ t_0 (sqrt (* (* (* F 2.0) t_0) (* C 2.0)))))
     (/ (- (sqrt F)) (sqrt (* 0.5 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(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (B_m <= 8e-71) {
		tmp = -1.0 / (t_0 / sqrt((((F * 2.0) * t_0) * (C * 2.0))));
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 8e-71)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(C * 2.0)))));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8e-71], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(C * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 7.9999999999999993e-71

   1. Initial program 18.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. neg-mul-1 N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in C around inf

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

      1. lower-*.f64 17.8

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

   7. Applied rewrites 17.8%

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

   if 7.9999999999999993e-71 < B

   1. Initial program 14.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 45.7

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

   5. Applied rewrites 45.7%

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

   7. Applied rewrites 45.8%

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

   9. Applied rewrites 45.8%

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

   11. Applied rewrites 57.3%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-71}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(C \cdot 2\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.2% accurate, 9.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{\frac{\left(B\_m \cdot B\_m\right) \cdot F}{-A}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9e-10)
   (/ (sqrt (/ (* (* B_m B_m) F) (- A))) (- B_m))
   (/ (- (sqrt F)) (sqrt (* 0.5 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9e-10) {
		tmp = sqrt((((B_m * B_m) * F) / -A)) / -B_m;
	} else {
		tmp = -sqrt(F) / sqrt((0.5 * 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) :: tmp
    if (b_m <= 9d-10) then
        tmp = sqrt((((b_m * b_m) * f) / -a)) / -b_m
    else
        tmp = -sqrt(f) / sqrt((0.5d0 * 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 tmp;
	if (B_m <= 9e-10) {
		tmp = Math.sqrt((((B_m * B_m) * F) / -A)) / -B_m;
	} else {
		tmp = -Math.sqrt(F) / Math.sqrt((0.5 * B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9e-10:
		tmp = math.sqrt((((B_m * B_m) * F) / -A)) / -B_m
	else:
		tmp = -math.sqrt(F) / math.sqrt((0.5 * 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 <= 9e-10)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) * F) / Float64(-A))) / Float64(-B_m));
	else
		tmp = Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * 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 <= 9e-10)
		tmp = sqrt((((B_m * B_m) * F) / -A)) / -B_m;
	else
		tmp = -sqrt(F) / sqrt((0.5 * B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9e-10], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] * F), $MachinePrecision] / (-A)), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.9999999999999999e-10

   1. Initial program 18.6%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 3.9

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

   5. Applied rewrites 3.9%

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

   7. Applied rewrites 3.9%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 6.0%

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

   if 8.9999999999999999e-10 < B

   1. Initial program 12.5%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 46.4

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

   5. Applied rewrites 46.4%

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

   7. Applied rewrites 46.6%

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

   9. Applied rewrites 46.6%

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

   11. Applied rewrites 59.8%

       \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{\frac{\left(B \cdot B\right) \cdot F}{-A}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.5% accurate, 12.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 8.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot B\_m\right) \cdot 2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 (<= F 8.7e-97)
   (/ (sqrt (* (* F B_m) 2.0)) (- B_m))
   (- (sqrt (* (/ F B_m) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 8.7e-97) {
		tmp = sqrt(((F * B_m) * 2.0)) / -B_m;
	} else {
		tmp = -sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 8.7d-97) then
        tmp = sqrt(((f * b_m) * 2.0d0)) / -b_m
    else
        tmp = -sqrt(((f / b_m) * 2.0d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 8.7e-97) {
		tmp = Math.sqrt(((F * B_m) * 2.0)) / -B_m;
	} else {
		tmp = -Math.sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 8.7e-97:
		tmp = math.sqrt(((F * B_m) * 2.0)) / -B_m
	else:
		tmp = -math.sqrt(((F / B_m) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 8.7e-97)
		tmp = Float64(sqrt(Float64(Float64(F * B_m) * 2.0)) / Float64(-B_m));
	else
		tmp = Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 8.7e-97)
		tmp = sqrt(((F * B_m) * 2.0)) / -B_m;
	else
		tmp = -sqrt(((F / B_m) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 8.7e-97], N[(N[Sqrt[N[(N[(F * B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 8.6999999999999998e-97

   1. Initial program 20.4%

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

   3. Taylor expanded in C around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. +-commutative N/A

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

      14. unpow2 N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 5.8

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

   5. Applied rewrites 5.8%

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

   7. Applied rewrites 5.8%

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

   8. Taylor expanded in B around inf

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

   10. Applied rewrites 17.4%

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

   if 8.6999999999999998e-97 < F

   1. Initial program 14.3%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 23.1

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

   5. Applied rewrites 23.1%

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

   7. Applied rewrites 23.1%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 8.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot B\right) \cdot 2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{F}{B} \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.2% accurate, 12.6× speedup

Math

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

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 F)) (sqrt (* 0.5 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(F) / sqrt((0.5 * B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(f) / sqrt((0.5d0 * b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(F) / Math.sqrt((0.5 * 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(F) / math.sqrt((0.5 * 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(Float64(-sqrt(F)) / sqrt(Float64(0.5 * 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(F) / sqrt((0.5 * 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[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 17.3

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

5. Applied rewrites 17.3%

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

7. Applied rewrites 17.4%

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

9. Applied rewrites 17.4%

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

11. Applied rewrites 20.9%

    \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]

12. Final simplification 20.9%

    \[\leadsto \frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}} \]
13. Add Preprocessing

Alternative 10: 27.8% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.8%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 17.3

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

5. Applied rewrites 17.3%

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

7. Applied rewrites 17.4%

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

9. Applied rewrites 17.4%

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

10. Final simplification 17.4%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))