ABCF->ab-angle a

Percentage Accurate: 20.0% → 53.0%

Time: 8.5s

Alternatives: 10

Speedup: 9.0×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, f)
use fmin_fmax_functions
    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: 20.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, f)
use fmin_fmax_functions
    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: 53.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{B\_m \cdot B\_m}{A}\\ t_2 := {B\_m}^{2} - t\_0\\ t_3 := 2 \cdot \left(t\_2 \cdot F\right)\\ t_4 := \frac{-\sqrt{t\_3 \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_2}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{-1 \cdot \sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(-0.5 \cdot t\_1\right)}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-162}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\left(B\_m \cdot B\_m - t\_0\right) \cdot F\right)} \cdot \sqrt{\left(A + C\right) + \sqrt{\mathsf{fma}\left(A - C, A - C, B\_m \cdot B\_m\right)}}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{-\sqrt{t\_3 \cdot \mathsf{fma}\left(-0.5, t\_1, 2 \cdot C\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (/ (* B_m B_m) A))
        (t_2 (- (pow B_m 2.0) t_0))
        (t_3 (* 2.0 (* t_2 F)))
        (t_4
         (/
          (-
           (sqrt
            (* t_3 (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_2)))
   (if (<= t_4 (- INFINITY))
     (* (/ (* -1.0 (sqrt 2.0)) B_m) (sqrt (* F (* -0.5 t_1))))
     (if (<= t_4 -1e-162)
       (/
        (-
         (*
          (sqrt (* 2.0 (* (- (* B_m B_m) t_0) F)))
          (sqrt (+ (+ A C) (sqrt (fma (- A C) (- A C) (* B_m B_m)))))))
        t_2)
       (if (<= t_4 INFINITY)
         (/ (- (sqrt (* t_3 (fma -0.5 t_1 (* 2.0 C))))) t_2)
         (* -1.0 (* (/ (sqrt F) (sqrt B_m)) (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = (B_m * B_m) / A;
	double t_2 = pow(B_m, 2.0) - t_0;
	double t_3 = 2.0 * (t_2 * F);
	double t_4 = -sqrt((t_3 * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_2;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((-1.0 * sqrt(2.0)) / B_m) * sqrt((F * (-0.5 * t_1)));
	} else if (t_4 <= -1e-162) {
		tmp = -(sqrt((2.0 * (((B_m * B_m) - t_0) * F))) * sqrt(((A + C) + sqrt(fma((A - C), (A - C), (B_m * B_m)))))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = -sqrt((t_3 * fma(-0.5, t_1, (2.0 * C)))) / t_2;
	} else {
		tmp = -1.0 * ((sqrt(F) / sqrt(B_m)) * sqrt(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)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(Float64(B_m * B_m) / A)
	t_2 = Float64((B_m ^ 2.0) - t_0)
	t_3 = Float64(2.0 * Float64(t_2 * F))
	t_4 = Float64(Float64(-sqrt(Float64(t_3 * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-1.0 * sqrt(2.0)) / B_m) * sqrt(Float64(F * Float64(-0.5 * t_1))));
	elseif (t_4 <= -1e-162)
		tmp = Float64(Float64(-Float64(sqrt(Float64(2.0 * Float64(Float64(Float64(B_m * B_m) - t_0) * F))) * sqrt(Float64(Float64(A + C) + sqrt(fma(Float64(A - C), Float64(A - C), Float64(B_m * B_m))))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * fma(-0.5, t_1, Float64(2.0 * C))))) / t_2);
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(F) / sqrt(B_m)) * sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(t$95$3 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(-1.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-162], N[((-N[(N[Sqrt[N[(2.0 * N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[((-N[Sqrt[N[(t$95$3 * N[(-0.5 * t$95$1 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 20.0%

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

   2. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 14.3

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

   4. Applied rewrites 14.3%

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

   5. Taylor expanded in F around -inf

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

      1. sqrt-prod N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 14.3%

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

   8. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 14.6

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

   10. Applied rewrites 14.6%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -9.99999999999999954e-163

   1. Initial program 20.0%

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

   3. Applied rewrites 22.7%

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

   if -9.99999999999999954e-163 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 20.0%

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

   2. Taylor expanded in A around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 27.4

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

   4. Applied rewrites 27.4%

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

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

   1. Initial program 20.0%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.6

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

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 27.5

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

   6. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 2: 52.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 7.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{-\left(B\_m \cdot \sqrt{2}\right) \cdot \left(\sqrt{F} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

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 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= B_m 7.2e-81)
     (/ (- (sqrt (* (* 2.0 (* t_0 F)) (* 2.0 C)))) t_0)
     (if (<= B_m 5.5e+103)
       (/
        (-
         (*
          (* B_m (sqrt 2.0))
          (* (sqrt F) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))))
        t_0)
       (* -1.0 (* (/ (sqrt F) (sqrt B_m)) (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 7.2e-81) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0;
	} else if (B_m <= 5.5e+103) {
		tmp = -((B_m * sqrt(2.0)) * (sqrt(F) * sqrt((C + sqrt(fma(B_m, B_m, (C * C))))))) / t_0;
	} else {
		tmp = -1.0 * ((sqrt(F) / sqrt(B_m)) * sqrt(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)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 7.2e-81)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(2.0 * C)))) / t_0);
	elseif (B_m <= 5.5e+103)
		tmp = Float64(Float64(-Float64(Float64(B_m * sqrt(2.0)) * Float64(sqrt(F) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))))) / t_0);
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(F) / sqrt(B_m)) * sqrt(2.0)));
	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[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 7.2e-81], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 5.5e+103], N[((-N[(N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.1999999999999997e-81

   1. Initial program 20.0%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 27.3

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

   4. Applied rewrites 27.3%

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

   if 7.1999999999999997e-81 < B < 5.50000000000000001e103

   1. Initial program 20.0%

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

   2. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 16.7

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

   4. Applied rewrites 16.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-prod N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

   6. Applied rewrites 18.1%

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

   if 5.50000000000000001e103 < B

   1. Initial program 20.0%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.6

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

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 27.5

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

   6. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 3: 49.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.15 \cdot 10^{-73}:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{-\left(B\_m \cdot \sqrt{2}\right) \cdot \left(\sqrt{F} \cdot \sqrt{C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}}\right)}{{B\_m}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

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 (<= B_m 2.15e-73)
   (* -1.0 (sqrt (* -1.0 (/ F A))))
   (if (<= B_m 5.5e+103)
     (/
      (-
       (*
        (* B_m (sqrt 2.0))
        (* (sqrt F) (sqrt (+ C (sqrt (fma B_m B_m (* C C))))))))
      (- (pow B_m 2.0) (* (* 4.0 A) C)))
     (* -1.0 (* (/ (sqrt F) (sqrt B_m)) (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.15e-73) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else if (B_m <= 5.5e+103) {
		tmp = -((B_m * sqrt(2.0)) * (sqrt(F) * sqrt((C + sqrt(fma(B_m, B_m, (C * C))))))) / (pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -1.0 * ((sqrt(F) / sqrt(B_m)) * sqrt(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 (B_m <= 2.15e-73)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	elseif (B_m <= 5.5e+103)
		tmp = Float64(Float64(-Float64(Float64(B_m * sqrt(2.0)) * Float64(sqrt(F) * sqrt(Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))))) / Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(F) / sqrt(B_m)) * sqrt(2.0)));
	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[B$95$m, 2.15e-73], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.5e+103], N[((-N[(N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.1499999999999999e-73

   1. Initial program 20.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 18.2%

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

   5. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 27.1

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   7. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   if 2.1499999999999999e-73 < B < 5.50000000000000001e103

   1. Initial program 20.0%

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

   2. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 16.7

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

   4. Applied rewrites 16.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. sqrt-prod N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

   6. Applied rewrites 18.1%

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

   if 5.50000000000000001e103 < B

   1. Initial program 20.0%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.6

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

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 27.5

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

   6. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 4: 48.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.9 \cdot 10^{-63}:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{+92}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F \cdot \left(A + \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

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 (<= B_m 4.9e-63)
   (* -1.0 (sqrt (* -1.0 (/ F A))))
   (if (<= B_m 6e+92)
     (*
      -1.0
      (sqrt
       (*
        (/
         (* F (+ A (+ C (sqrt (fma B_m B_m (* (- A C) (- A C)))))))
         (fma B_m B_m (* -4.0 (* A C))))
        2.0)))
     (* -1.0 (* (/ (sqrt F) (sqrt B_m)) (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.9e-63) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else if (B_m <= 6e+92) {
		tmp = -1.0 * sqrt((((F * (A + (C + sqrt(fma(B_m, B_m, ((A - C) * (A - C))))))) / fma(B_m, B_m, (-4.0 * (A * C)))) * 2.0));
	} else {
		tmp = -1.0 * ((sqrt(F) / sqrt(B_m)) * sqrt(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 (B_m <= 4.9e-63)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	elseif (B_m <= 6e+92)
		tmp = Float64(-1.0 * sqrt(Float64(Float64(Float64(F * Float64(A + Float64(C + sqrt(fma(B_m, B_m, Float64(Float64(A - C) * Float64(A - C))))))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))) * 2.0)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(F) / sqrt(B_m)) * sqrt(2.0)));
	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[B$95$m, 4.9e-63], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6e+92], N[(-1.0 * N[Sqrt[N[(N[(N[(F * N[(A + N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.90000000000000015e-63

   1. Initial program 20.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 18.2%

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

   5. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 27.1

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   7. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   if 4.90000000000000015e-63 < B < 6.00000000000000026e92

   1. Initial program 20.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 18.2%

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

   if 6.00000000000000026e92 < B

   1. Initial program 20.0%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.6

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

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 27.5

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

   6. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 5: 48.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-97}:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\frac{-\left(B\_m \cdot \sqrt{2}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}}{B\_m \cdot B\_m - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-97)
   (* -1.0 (sqrt (* -1.0 (/ F A))))
   (if (<= (pow B_m 2.0) 2e+196)
     (/
      (-
       (* (* B_m (sqrt 2.0)) (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))))
      (- (* B_m B_m) (* (* 4.0 A) C)))
     (* -1.0 (* (/ (sqrt F) (sqrt B_m)) (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-97) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else if (pow(B_m, 2.0) <= 2e+196) {
		tmp = -((B_m * sqrt(2.0)) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C))))))) / ((B_m * B_m) - ((4.0 * A) * C));
	} else {
		tmp = -1.0 * ((sqrt(F) / sqrt(B_m)) * sqrt(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 ((B_m ^ 2.0) <= 2e-97)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	elseif ((B_m ^ 2.0) <= 2e+196)
		tmp = Float64(Float64(-Float64(Float64(B_m * sqrt(2.0)) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C)))))))) / Float64(Float64(B_m * B_m) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(F) / sqrt(B_m)) * sqrt(2.0)));
	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], 2e-97], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+196], N[((-N[(N[(B$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000007e-97

   1. Initial program 20.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 18.2%

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

   5. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 27.1

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   7. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   if 2.00000000000000007e-97 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e196

   1. Initial program 20.0%

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

   2. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 16.7

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

   4. Applied rewrites 16.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 16.7

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

   6. Applied rewrites 16.7%

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

   if 1.9999999999999999e196 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.0%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.6

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

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 27.5

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

   6. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 6: 48.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-126}:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+196}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(C + \sqrt{\mathsf{fma}\left(B\_m, B\_m, C \cdot C\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-126)
   (* -1.0 (sqrt (* -1.0 (/ F A))))
   (if (<= (pow B_m 2.0) 2e+196)
     (*
      -1.0
      (* (/ (sqrt 2.0) B_m) (sqrt (* F (+ C (sqrt (fma B_m B_m (* C C))))))))
     (* -1.0 (* (/ (sqrt F) (sqrt B_m)) (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-126) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else if (pow(B_m, 2.0) <= 2e+196) {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (C + sqrt(fma(B_m, B_m, (C * C)))))));
	} else {
		tmp = -1.0 * ((sqrt(F) / sqrt(B_m)) * sqrt(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 ((B_m ^ 2.0) <= 5e-126)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	elseif ((B_m ^ 2.0) <= 2e+196)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(C + sqrt(fma(B_m, B_m, Float64(C * C))))))));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(F) / sqrt(B_m)) * sqrt(2.0)));
	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-126], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+196], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[N[(B$95$m * B$95$m + N[(C * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000006e-126

   1. Initial program 20.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 18.2%

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

   5. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 27.1

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   7. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   if 5.00000000000000006e-126 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e196

   1. Initial program 20.0%

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

   2. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 17.5

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

   4. Applied rewrites 17.5%

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

   if 1.9999999999999999e196 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.0%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.6

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

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 27.5

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

   6. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 7: 47.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-146}:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{F}}{\sqrt{B\_m}} \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-146)
   (* -1.0 (sqrt (* -1.0 (/ F A))))
   (* -1.0 (* (/ (sqrt F) (sqrt B_m)) (sqrt 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-146) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else {
		tmp = -1.0 * ((sqrt(F) / sqrt(B_m)) * sqrt(2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    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 ** 2.0d0) <= 5d-146) then
        tmp = (-1.0d0) * sqrt(((-1.0d0) * (f / a)))
    else
        tmp = (-1.0d0) * ((sqrt(f) / sqrt(b_m)) * sqrt(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 (Math.pow(B_m, 2.0) <= 5e-146) {
		tmp = -1.0 * Math.sqrt((-1.0 * (F / A)));
	} else {
		tmp = -1.0 * ((Math.sqrt(F) / Math.sqrt(B_m)) * Math.sqrt(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 math.pow(B_m, 2.0) <= 5e-146:
		tmp = -1.0 * math.sqrt((-1.0 * (F / A)))
	else:
		tmp = -1.0 * ((math.sqrt(F) / math.sqrt(B_m)) * math.sqrt(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 ((B_m ^ 2.0) <= 5e-146)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(F) / sqrt(B_m)) * sqrt(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 ((B_m ^ 2.0) <= 5e-146)
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	else
		tmp = -1.0 * ((sqrt(F) / sqrt(B_m)) * sqrt(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[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-146], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999957e-146

   1. Initial program 20.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 18.2%

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

   5. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 27.1

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   7. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   if 4.99999999999999957e-146 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.0%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.6

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

   4. Applied rewrites 27.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{B} \cdot 2}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 27.5

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

   6. Applied rewrites 27.5%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 36.0

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

   8. Applied rewrites 36.0%

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

Alternative 8: 39.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-146}:\\ \;\;\;\;-1 \cdot \sqrt{-1 \cdot \frac{F}{A}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \sqrt{\frac{F}{B\_m} \cdot 2}\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 5e-146)
   (* -1.0 (sqrt (* -1.0 (/ F A))))
   (* -1.0 (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 (pow(B_m, 2.0) <= 5e-146) {
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	} else {
		tmp = -1.0 * sqrt(((F / B_m) * 2.0));
	}
	return tmp;
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    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 ** 2.0d0) <= 5d-146) then
        tmp = (-1.0d0) * sqrt(((-1.0d0) * (f / a)))
    else
        tmp = (-1.0d0) * 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 (Math.pow(B_m, 2.0) <= 5e-146) {
		tmp = -1.0 * Math.sqrt((-1.0 * (F / A)));
	} else {
		tmp = -1.0 * 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 math.pow(B_m, 2.0) <= 5e-146:
		tmp = -1.0 * math.sqrt((-1.0 * (F / A)))
	else:
		tmp = -1.0 * 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 ((B_m ^ 2.0) <= 5e-146)
		tmp = Float64(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))));
	else
		tmp = Float64(-1.0 * 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 ((B_m ^ 2.0) <= 5e-146)
		tmp = -1.0 * sqrt((-1.0 * (F / A)));
	else
		tmp = -1.0 * 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[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-146], N[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.99999999999999957e-146

   1. Initial program 20.0%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 18.2%

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

   5. Taylor expanded in A around -inf

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

      2. lower-/.f64 27.1

         \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   7. Applied rewrites 27.1%

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   if 4.99999999999999957e-146 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.0%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 27.6

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

   4. Applied rewrites 27.6%

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

Alternative 9: 27.1% accurate, 9.0× speedup

Math

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

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 (* -1.0 (sqrt (* -1.0 (/ F A)))))

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 -1.0 * sqrt((-1.0 * (F / A)));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-1.0d0) * sqrt(((-1.0d0) * (f / a)))
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 -1.0 * Math.sqrt((-1.0 * (F / A)));
}

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 -1.0 * math.sqrt((-1.0 * (F / A)))

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(-1.0 * sqrt(Float64(-1.0 * Float64(F / A))))
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 = -1.0 * sqrt((-1.0 * (F / A)));
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[(-1.0 * N[Sqrt[N[(-1.0 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

2. Taylor expanded in F around 0

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

   1. lower-*.f64 N/A

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

   2. sqrt-unprod N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

4. Applied rewrites 18.2%

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

5. Taylor expanded in A around -inf

   \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

   2. lower-/.f64 27.1

      \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]

7. Applied rewrites 27.1%

   \[\leadsto -1 \cdot \sqrt{-1 \cdot \frac{F}{A}} \]
8. Add Preprocessing

Alternative 10: 1.6% accurate, 12.0× speedup

Math

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

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 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) {
	return sqrt(((F / B_m) * -2.0));
}

Fortran

B_m =     private
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(((f / b_m) * (-2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

2. Taylor expanded in C around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 14.3

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

4. Applied rewrites 14.3%

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

5. Taylor expanded in B around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sqrt-unprod N/A

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

   5. metadata-eval N/A

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

   6. lower-sqrt.f64 0.0

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

7. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-unprod N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 1.6

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

9. Applied rewrites 1.6%

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

Reproduce

herbie shell --seed 2025127 
(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))))