ABCF->ab-angle b

Percentage Accurate: 18.3% → 62.6%

Time: 10.3s

Alternatives: 11

Speedup: 10.8×

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: 18.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 62.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_4 := t\_3 \cdot \left(F + F\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(C + A, t\_4, \left(-\sqrt{\mathsf{fma}\left(C - A, C - A, B\_m \cdot B\_m\right)}\right) \cdot t\_4\right)}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(t\_3, F \cdot t\_0, t\_0 \cdot \left(t\_3 \cdot F\right)\right)}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (fma (/ (* B_m B_m) C) -0.5 A) A))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3 (fma -4.0 (* C A) (* B_m B_m)))
        (t_4 (* t_3 (+ F F))))
   (if (<= t_2 (- INFINITY))
     (* -0.25 (/ (* (sqrt (* -16.0 F)) (sqrt C)) C))
     (if (<= t_2 -4e-186)
       (/
        (-
         (sqrt
          (fma
           (+ C A)
           t_4
           (* (- (sqrt (fma (- C A) (- C A) (* B_m B_m)))) t_4))))
        t_1)
       (if (<= t_2 0.0)
         (/ (- (sqrt (fma t_3 (* F t_0) (* t_0 (* t_3 F))))) t_1)
         (if (<= t_2 INFINITY)
           (* -0.25 (/ (* (sqrt (* -16.0 C)) (sqrt F)) C))
           (/ (sqrt (* -2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(((B_m * B_m) / C), -0.5, A) + A;
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = fma(-4.0, (C * A), (B_m * B_m));
	double t_4 = t_3 * (F + F);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * F)) * sqrt(C)) / C);
	} else if (t_2 <= -4e-186) {
		tmp = -sqrt(fma((C + A), t_4, (-sqrt(fma((C - A), (C - A), (B_m * B_m))) * t_4))) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = -sqrt(fma(t_3, (F * t_0), (t_0 * (t_3 * F)))) / t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * C)) * sqrt(F)) / C);
	} else {
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A)
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_4 = Float64(t_3 * Float64(F + F))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * F)) * sqrt(C)) / C));
	elseif (t_2 <= -4e-186)
		tmp = Float64(Float64(-sqrt(fma(Float64(C + A), t_4, Float64(Float64(-sqrt(fma(Float64(C - A), Float64(C - A), Float64(B_m * B_m)))) * t_4)))) / t_1);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-sqrt(fma(t_3, Float64(F * t_0), Float64(t_0 * Float64(t_3 * F))))) / t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * C)) * sqrt(F)) / C));
	else
		tmp = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * 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$1), $MachinePrecision]}, Block[{t$95$3 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(F + F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-186], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] * t$95$4 + N[((-N[Sqrt[N[(N[(C - A), $MachinePrecision] * N[(C - A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[((-N[Sqrt[N[(t$95$3 * N[(F * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(t$95$3 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 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 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot C\right)}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot C}}{C} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      7. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      12. lower-sqrt.f64 35.4

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   6. Applied rewrites 35.4%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   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))) < -3.9999999999999996e-186

   1. Initial program 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 18.2%

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

   if -3.9999999999999996e-186 < (/.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))) < 0.0

   1. Initial program 18.3%

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

   2. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 26.9

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

   4. Applied rewrites 26.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. count-2-rev N/A

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

   6. Applied rewrites 26.1%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(\left(-16 \cdot C\right) \cdot F\right)}^{\frac{1}{2}}}{C} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

      12. lower-sqrt.f64 12.0

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   6. Applied rewrites 12.0%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   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 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

      \[\leadsto \frac{\sqrt{-2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 62.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_0}\\ t_2 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_3 := t\_2 \cdot \left(F + F\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(C + A, t\_3, \left(-\sqrt{\mathsf{fma}\left(C - A, C - A, B\_m \cdot B\_m\right)}\right) \cdot t\_3\right)}}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{-\sqrt{\left(t\_2 \cdot 2\right) \cdot \left(F \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)\right)}}{t\_0}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_0))
        (t_2 (fma -4.0 (* C A) (* B_m B_m)))
        (t_3 (* t_2 (+ F F))))
   (if (<= t_1 (- INFINITY))
     (* -0.25 (/ (* (sqrt (* -16.0 F)) (sqrt C)) C))
     (if (<= t_1 -4e-186)
       (/
        (-
         (sqrt
          (fma
           (+ C A)
           t_3
           (* (- (sqrt (fma (- C A) (- C A) (* B_m B_m)))) t_3))))
        t_0)
       (if (<= t_1 0.0)
         (/
          (- (sqrt (* (* t_2 2.0) (* F (+ (fma (/ (* B_m B_m) C) -0.5 A) A)))))
          t_0)
         (if (<= t_1 INFINITY)
           (* -0.25 (/ (* (sqrt (* -16.0 C)) (sqrt F)) C))
           (/ (sqrt (* -2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_0;
	double t_2 = fma(-4.0, (C * A), (B_m * B_m));
	double t_3 = t_2 * (F + F);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * F)) * sqrt(C)) / C);
	} else if (t_1 <= -4e-186) {
		tmp = -sqrt(fma((C + A), t_3, (-sqrt(fma((C - A), (C - A), (B_m * B_m))) * t_3))) / t_0;
	} else if (t_1 <= 0.0) {
		tmp = -sqrt(((t_2 * 2.0) * (F * (fma(((B_m * B_m) / C), -0.5, A) + A)))) / t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * C)) * sqrt(F)) / C);
	} else {
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_0)
	t_2 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_3 = Float64(t_2 * Float64(F + F))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * F)) * sqrt(C)) / C));
	elseif (t_1 <= -4e-186)
		tmp = Float64(Float64(-sqrt(fma(Float64(C + A), t_3, Float64(Float64(-sqrt(fma(Float64(C - A), Float64(C - A), Float64(B_m * B_m)))) * t_3)))) / t_0);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_2 * 2.0) * Float64(F * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))))) / t_0);
	elseif (t_1 <= Inf)
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * C)) * sqrt(F)) / C));
	else
		tmp = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(F + F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-186], N[((-N[Sqrt[N[(N[(C + A), $MachinePrecision] * t$95$3 + N[((-N[Sqrt[N[(N[(C - A), $MachinePrecision] * N[(C - A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[((-N[Sqrt[N[(N[(t$95$2 * 2.0), $MachinePrecision] * N[(F * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 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 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot C\right)}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot C}}{C} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      7. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      12. lower-sqrt.f64 35.4

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   6. Applied rewrites 35.4%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   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))) < -3.9999999999999996e-186

   1. Initial program 18.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-flip N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

   3. Applied rewrites 18.2%

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

   if -3.9999999999999996e-186 < (/.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))) < 0.0

   1. Initial program 18.3%

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

   2. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 26.9

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

   4. Applied rewrites 26.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 26.9%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(\left(-16 \cdot C\right) \cdot F\right)}^{\frac{1}{2}}}{C} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

      12. lower-sqrt.f64 12.0

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   6. Applied rewrites 12.0%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   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 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

      \[\leadsto \frac{\sqrt{-2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 62.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C - \sqrt{\mathsf{fma}\left(C - A, C - A, B\_m \cdot B\_m\right)}\right) + A\right) \cdot \left(t\_0 \cdot \left(F + F\right)\right)}}{C \cdot \left(A \cdot 4\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{-\sqrt{\left(t\_0 \cdot 2\right) \cdot \left(F \cdot \left(\mathsf{fma}\left(\frac{B\_m \cdot B\_m}{C}, -0.5, A\right) + A\right)\right)}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m)))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1)))
   (if (<= t_2 (- INFINITY))
     (* -0.25 (/ (* (sqrt (* -16.0 F)) (sqrt C)) C))
     (if (<= t_2 -4e-186)
       (/
        (sqrt
         (*
          (+ (- C (sqrt (fma (- C A) (- C A) (* B_m B_m)))) A)
          (* t_0 (+ F F))))
        (- (* C (* A 4.0)) (* B_m B_m)))
       (if (<= t_2 0.0)
         (/
          (- (sqrt (* (* t_0 2.0) (* F (+ (fma (/ (* B_m B_m) C) -0.5 A) A)))))
          t_1)
         (if (<= t_2 INFINITY)
           (* -0.25 (/ (* (sqrt (* -16.0 C)) (sqrt F)) C))
           (/ (sqrt (* -2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * F)) * sqrt(C)) / C);
	} else if (t_2 <= -4e-186) {
		tmp = sqrt((((C - sqrt(fma((C - A), (C - A), (B_m * B_m)))) + A) * (t_0 * (F + F)))) / ((C * (A * 4.0)) - (B_m * B_m));
	} else if (t_2 <= 0.0) {
		tmp = -sqrt(((t_0 * 2.0) * (F * (fma(((B_m * B_m) / C), -0.5, A) + A)))) / t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * C)) * sqrt(F)) / C);
	} else {
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * F)) * sqrt(C)) / C));
	elseif (t_2 <= -4e-186)
		tmp = Float64(sqrt(Float64(Float64(Float64(C - sqrt(fma(Float64(C - A), Float64(C - A), Float64(B_m * B_m)))) + A) * Float64(t_0 * Float64(F + F)))) / Float64(Float64(C * Float64(A * 4.0)) - Float64(B_m * B_m)));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * 2.0) * Float64(F * Float64(fma(Float64(Float64(B_m * B_m) / C), -0.5, A) + A))))) / t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * C)) * sqrt(F)) / C));
	else
		tmp = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * 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$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-186], N[(N[Sqrt[N[(N[(N[(C - N[Sqrt[N[(N[(C - A), $MachinePrecision] * N[(C - A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * N[(t$95$0 * N[(F + F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[((-N[Sqrt[N[(N[(t$95$0 * 2.0), $MachinePrecision] * N[(F * N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision] * -0.5 + A), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 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 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot C\right)}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot C}}{C} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      7. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      12. lower-sqrt.f64 35.4

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   6. Applied rewrites 35.4%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   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))) < -3.9999999999999996e-186

   1. Initial program 18.3%

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

   3. Applied rewrites 19.3%

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

   if -3.9999999999999996e-186 < (/.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))) < 0.0

   1. Initial program 18.3%

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

   2. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 26.9

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

   4. Applied rewrites 26.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 26.9%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(\left(-16 \cdot C\right) \cdot F\right)}^{\frac{1}{2}}}{C} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

      12. lower-sqrt.f64 12.0

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   6. Applied rewrites 12.0%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   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 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

      \[\leadsto \frac{\sqrt{-2 \cdot F}}{\color{blue}{-\sqrt{B}}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_0}\\ t_2 := -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C - \sqrt{\mathsf{fma}\left(C - A, C - A, B\_m \cdot B\_m\right)}\right) + A\right) \cdot \left(\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot \left(F + F\right)\right)}}{C \cdot \left(A \cdot 4\right) - B\_m \cdot B\_m}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_0))
        (t_2 (* -0.25 (/ (* (sqrt (* -16.0 F)) (sqrt C)) C))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -4e-186)
       (/
        (sqrt
         (*
          (+ (- C (sqrt (fma (- C A) (- C A) (* B_m B_m)))) A)
          (* (fma -4.0 (* C A) (* B_m B_m)) (+ F F))))
        (- (* C (* A 4.0)) (* B_m B_m)))
       (if (<= t_1 0.0)
         t_2
         (if (<= t_1 INFINITY)
           (* -0.25 (/ (* (sqrt (* -16.0 C)) (sqrt F)) C))
           (/ (sqrt (* -2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_0;
	double t_2 = -0.25 * ((sqrt((-16.0 * F)) * sqrt(C)) / C);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -4e-186) {
		tmp = sqrt((((C - sqrt(fma((C - A), (C - A), (B_m * B_m)))) + A) * (fma(-4.0, (C * A), (B_m * B_m)) * (F + F)))) / ((C * (A * 4.0)) - (B_m * B_m));
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * C)) * sqrt(F)) / C);
	} else {
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_0)
	t_2 = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * F)) * sqrt(C)) / C))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -4e-186)
		tmp = Float64(sqrt(Float64(Float64(Float64(C - sqrt(fma(Float64(C - A), Float64(C - A), Float64(B_m * B_m)))) + A) * Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * Float64(F + F)))) / Float64(Float64(C * Float64(A * 4.0)) - Float64(B_m * B_m)));
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * C)) * sqrt(F)) / C));
	else
		tmp = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -4e-186], N[(N[Sqrt[N[(N[(N[(C - N[Sqrt[N[(N[(C - A), $MachinePrecision] * N[(C - A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision] * N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * N[(F + F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, Infinity], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $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 or -3.9999999999999996e-186 < (/.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))) < 0.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot C\right)}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot C}}{C} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      7. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      12. lower-sqrt.f64 35.4

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   6. Applied rewrites 35.4%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   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))) < -3.9999999999999996e-186

   1. Initial program 18.3%

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

   3. Applied rewrites 19.3%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(\left(-16 \cdot C\right) \cdot F\right)}^{\frac{1}{2}}}{C} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

      12. lower-sqrt.f64 12.0

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   6. Applied rewrites 12.0%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   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 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

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

Alternative 5: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_0}\\ t_2 := -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(-2 \cdot \left(F - \frac{\left(C + A\right) \cdot F}{B\_m}\right)\right) \cdot \left(B\_m \cdot B\_m\right)\right) \cdot B\_m}}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_0))
        (t_2 (* -0.25 (/ (* (sqrt (* -16.0 F)) (sqrt C)) C))))
   (if (<= t_1 -2e+153)
     t_2
     (if (<= t_1 -4e-186)
       (/
        (- (sqrt (* (* (* -2.0 (- F (/ (* (+ C A) F) B_m))) (* B_m B_m)) B_m)))
        t_0)
       (if (<= t_1 0.0)
         t_2
         (if (<= t_1 INFINITY)
           (* -0.25 (/ (* (sqrt (* -16.0 C)) (sqrt F)) C))
           (/ (sqrt (* -2.0 F)) (- (sqrt B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_0;
	double t_2 = -0.25 * ((sqrt((-16.0 * F)) * sqrt(C)) / C);
	double tmp;
	if (t_1 <= -2e+153) {
		tmp = t_2;
	} else if (t_1 <= -4e-186) {
		tmp = -sqrt((((-2.0 * (F - (((C + A) * F) / B_m))) * (B_m * B_m)) * B_m)) / t_0;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * C)) * sqrt(F)) / C);
	} else {
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / t_0;
	double t_2 = -0.25 * ((Math.sqrt((-16.0 * F)) * Math.sqrt(C)) / C);
	double tmp;
	if (t_1 <= -2e+153) {
		tmp = t_2;
	} else if (t_1 <= -4e-186) {
		tmp = -Math.sqrt((((-2.0 * (F - (((C + A) * F) / B_m))) * (B_m * B_m)) * B_m)) / t_0;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = -0.25 * ((Math.sqrt((-16.0 * C)) * Math.sqrt(F)) / C);
	} else {
		tmp = Math.sqrt((-2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / t_0
	t_2 = -0.25 * ((math.sqrt((-16.0 * F)) * math.sqrt(C)) / C)
	tmp = 0
	if t_1 <= -2e+153:
		tmp = t_2
	elif t_1 <= -4e-186:
		tmp = -math.sqrt((((-2.0 * (F - (((C + A) * F) / B_m))) * (B_m * B_m)) * B_m)) / t_0
	elif t_1 <= 0.0:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = -0.25 * ((math.sqrt((-16.0 * C)) * math.sqrt(F)) / C)
	else:
		tmp = math.sqrt((-2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_0)
	t_2 = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * F)) * sqrt(C)) / C))
	tmp = 0.0
	if (t_1 <= -2e+153)
		tmp = t_2;
	elseif (t_1 <= -4e-186)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(-2.0 * Float64(F - Float64(Float64(Float64(C + A) * F) / B_m))) * Float64(B_m * B_m)) * B_m))) / t_0);
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * C)) * sqrt(F)) / C));
	else
		tmp = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_0;
	t_2 = -0.25 * ((sqrt((-16.0 * F)) * sqrt(C)) / C);
	tmp = 0.0;
	if (t_1 <= -2e+153)
		tmp = t_2;
	elseif (t_1 <= -4e-186)
		tmp = -sqrt((((-2.0 * (F - (((C + A) * F) / B_m))) * (B_m * B_m)) * B_m)) / t_0;
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = -0.25 * ((sqrt((-16.0 * C)) * sqrt(F)) / C);
	else
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+153], t$95$2, If[LessEqual[t$95$1, -4e-186], N[((-N[Sqrt[N[(N[(N[(-2.0 * N[(F - N[(N[(N[(C + A), $MachinePrecision] * F), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * B$95$m), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, Infinity], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $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))) < -2e153 or -3.9999999999999996e-186 < (/.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))) < 0.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot C\right)}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot C}}{C} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      7. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      12. lower-sqrt.f64 35.4

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   6. Applied rewrites 35.4%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   if -2e153 < (/.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))) < -3.9999999999999996e-186

   1. Initial program 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{{B}^{3} \cdot \mathsf{fma}\left(-2, \color{blue}{F}, 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{{B}^{3} \cdot \mathsf{fma}\left(-2, F, 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{{B}^{3} \cdot \mathsf{fma}\left(-2, F, 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{{B}^{3} \cdot \mathsf{fma}\left(-2, F, 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. lower-+.f64 7.9

         \[\leadsto \frac{-\sqrt{{B}^{3} \cdot \mathsf{fma}\left(-2, F, 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 7.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{{B}^{3} \cdot \color{blue}{\mathsf{fma}\left(-2, F, 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. *-commutative N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-2, F, 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right) \cdot \color{blue}{{B}^{3}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-2, F, 2 \cdot \frac{F \cdot \left(A + C\right)}{B}\right) \cdot {B}^{\color{blue}{3}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 9.9%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(\left(-16 \cdot C\right) \cdot F\right)}^{\frac{1}{2}}}{C} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

      12. lower-sqrt.f64 12.0

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   6. Applied rewrites 12.0%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   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 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

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

Alternative 6: 57.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ t_1 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt (* -2.0 F)) (- (sqrt B_m))))
        (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0)))))))
          t_1))
        (t_3 (* -0.25 (/ (* (sqrt (* -16.0 F)) (sqrt C)) C))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -4e-186)
       t_0
       (if (<= t_2 0.0)
         t_3
         (if (<= t_2 INFINITY)
           (* -0.25 (/ (* (sqrt (* -16.0 C)) (sqrt F)) C))
           t_0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((-2.0 * F)) / -sqrt(B_m);
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = -0.25 * ((sqrt((-16.0 * F)) * sqrt(C)) / C);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -4e-186) {
		tmp = t_0;
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = -0.25 * ((sqrt((-16.0 * C)) * sqrt(F)) / C);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt((-2.0 * F)) / -Math.sqrt(B_m);
	double t_1 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_2 = -Math.sqrt(((2.0 * (t_1 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / t_1;
	double t_3 = -0.25 * ((Math.sqrt((-16.0 * F)) * Math.sqrt(C)) / C);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else if (t_2 <= -4e-186) {
		tmp = t_0;
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = -0.25 * ((Math.sqrt((-16.0 * C)) * Math.sqrt(F)) / C);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.sqrt((-2.0 * F)) / -math.sqrt(B_m)
	t_1 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_2 = -math.sqrt(((2.0 * (t_1 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / t_1
	t_3 = -0.25 * ((math.sqrt((-16.0 * F)) * math.sqrt(C)) / C)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_3
	elif t_2 <= -4e-186:
		tmp = t_0
	elif t_2 <= 0.0:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = -0.25 * ((math.sqrt((-16.0 * C)) * math.sqrt(F)) / C)
	else:
		tmp = t_0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0))))))) / t_1)
	t_3 = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * F)) * sqrt(C)) / C))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -4e-186)
		tmp = t_0;
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = Float64(-0.25 * Float64(Float64(sqrt(Float64(-16.0 * C)) * sqrt(F)) / C));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt((-2.0 * F)) / -sqrt(B_m);
	t_1 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_1;
	t_3 = -0.25 * ((sqrt((-16.0 * F)) * sqrt(C)) / C);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_3;
	elseif (t_2 <= -4e-186)
		tmp = t_0;
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = -0.25 * ((sqrt((-16.0 * C)) * sqrt(F)) / C);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * 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$1), $MachinePrecision]}, Block[{t$95$3 = N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -4e-186], t$95$0, If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, Infinity], N[(-0.25 * N[(N[(N[Sqrt[N[(-16.0 * C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 3 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 or -3.9999999999999996e-186 < (/.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))) < 0.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(F \cdot C\right)}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{\left(-16 \cdot F\right) \cdot C}}{C} \]

      6. sqrt-prod N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      7. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot {C}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

      12. lower-sqrt.f64 35.4

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   6. Applied rewrites 35.4%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot F} \cdot \sqrt{C}}{C} \]

   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))) < -3.9999999999999996e-186 or +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 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

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

   if 0.0 < (/.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))) < +inf.0

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      2. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot \left(C \cdot F\right)\right)}^{\frac{1}{2}}}{C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(\left(-16 \cdot C\right) \cdot F\right)}^{\frac{1}{2}}}{C} \]

      6. unpow-prod-down N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{{\left(-16 \cdot C\right)}^{\frac{1}{2}} \cdot {F}^{\frac{1}{2}}}{C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot {F}^{\frac{1}{2}}}{C} \]

      11. pow1/2 N/A

          \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

      12. lower-sqrt.f64 12.0

          \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]

   6. Applied rewrites 12.0%

      \[\leadsto -0.25 \cdot \frac{\sqrt{-16 \cdot C} \cdot \sqrt{F}}{C} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-35}:\\ \;\;\;\;-0.25 \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-35)
   (* -0.25 (/ (sqrt (* -16.0 (* C F))) C))
   (/ (sqrt (* -2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-35) {
		tmp = -0.25 * (sqrt((-16.0 * (C * F))) / C);
	} else {
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	}
	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) <= 2d-35) then
        tmp = (-0.25d0) * (sqrt(((-16.0d0) * (c * f))) / c)
    else
        tmp = sqrt(((-2.0d0) * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-35) {
		tmp = -0.25 * (Math.sqrt((-16.0 * (C * F))) / C);
	} else {
		tmp = Math.sqrt((-2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-35:
		tmp = -0.25 * (math.sqrt((-16.0 * (C * F))) / C)
	else:
		tmp = math.sqrt((-2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-35)
		tmp = Float64(-0.25 * Float64(sqrt(Float64(-16.0 * Float64(C * F))) / C));
	else
		tmp = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-35)
		tmp = -0.25 * (sqrt((-16.0 * (C * F))) / C);
	else
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-35], N[(-0.25 * N[(N[Sqrt[N[(-16.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000002e-35

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

   if 2.00000000000000002e-35 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

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

Alternative 8: 47.5% accurate, 6.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.8 \cdot 10^{-16}:\\ \;\;\;\;-0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.8e-16)
   (* -0.25 (sqrt (* -16.0 (/ F C))))
   (/ (sqrt (* -2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.8e-16) {
		tmp = -0.25 * sqrt((-16.0 * (F / C)));
	} else {
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	}
	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 <= 3.8d-16) then
        tmp = (-0.25d0) * sqrt(((-16.0d0) * (f / c)))
    else
        tmp = sqrt(((-2.0d0) * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.8e-16) {
		tmp = -0.25 * Math.sqrt((-16.0 * (F / C)));
	} else {
		tmp = Math.sqrt((-2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.8e-16:
		tmp = -0.25 * math.sqrt((-16.0 * (F / C)))
	else:
		tmp = math.sqrt((-2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.8e-16)
		tmp = Float64(-0.25 * sqrt(Float64(-16.0 * Float64(F / C))));
	else
		tmp = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.8e-16)
		tmp = -0.25 * sqrt((-16.0 * (F / C)));
	else
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.8e-16], N[(-0.25 * N[Sqrt[N[(-16.0 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.80000000000000012e-16

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

   5. Taylor expanded in C around inf

      \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{C}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{C}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \sqrt{-16 \cdot \frac{F}{C}} \]

      3. lower-/.f64 27.4

         \[\leadsto -0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}} \]

   7. Applied rewrites 27.4%

      \[\leadsto -0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}} \]

   if 3.80000000000000012e-16 < B

   1. Initial program 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

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

Alternative 9: 46.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.3 \cdot 10^{-48}:\\ \;\;\;\;0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.3e-48)
   (* 0.25 (sqrt (* -16.0 (/ F C))))
   (/ (sqrt (* -2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.3e-48) {
		tmp = 0.25 * sqrt((-16.0 * (F / C)));
	} else {
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	}
	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 <= 1.3d-48) then
        tmp = 0.25d0 * sqrt(((-16.0d0) * (f / c)))
    else
        tmp = sqrt(((-2.0d0) * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.3e-48) {
		tmp = 0.25 * Math.sqrt((-16.0 * (F / C)));
	} else {
		tmp = Math.sqrt((-2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.3e-48:
		tmp = 0.25 * math.sqrt((-16.0 * (F / C)))
	else:
		tmp = math.sqrt((-2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.3e-48)
		tmp = Float64(0.25 * sqrt(Float64(-16.0 * Float64(F / C))));
	else
		tmp = Float64(sqrt(Float64(-2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.3e-48)
		tmp = 0.25 * sqrt((-16.0 * (F / C)));
	else
		tmp = sqrt((-2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.3e-48], N[(0.25 * N[Sqrt[N[(-16.0 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.29999999999999994e-48

   1. Initial program 18.3%

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

   2. Taylor expanded in A around -inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{\sqrt{-16 \cdot \left(C \cdot F\right)}}{C} \]

      5. lower-*.f64 37.1

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

   4. Applied rewrites 37.1%

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

   5. Taylor expanded in C around -inf

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\sqrt{-16 \cdot \frac{F}{C}}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{C}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{C}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \sqrt{-16 \cdot \frac{F}{C}} \]

      4. lower-/.f64 20.6

         \[\leadsto 0.25 \cdot \sqrt{-16 \cdot \frac{F}{C}} \]

   7. Applied rewrites 20.6%

      \[\leadsto 0.25 \cdot \color{blue}{\sqrt{-16 \cdot \frac{F}{C}}} \]

   if 1.29999999999999994e-48 < B

   1. Initial program 18.3%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 26.7

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

   4. Applied rewrites 26.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f64 35.1

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

   6. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 35.1

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

   8. Applied rewrites 35.1%

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

Alternative 10: 35.1% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

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(((-2.0d0) * f)) / -sqrt(b_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

2. Taylor expanded in B around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 26.7

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

4. Applied rewrites 26.7%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. sqrt-div N/A

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

   6. pow1/2 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow1/2 N/A

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

   15. lower-sqrt.f64 35.1

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

6. Applied rewrites 35.1%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 35.1

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

8. Applied rewrites 35.1%

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

Alternative 11: 26.7% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

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 * ((-2.0d0) / b_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

2. Taylor expanded in B around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 26.7

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

4. Applied rewrites 26.7%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 26.7

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 26.7

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

6. Applied rewrites 26.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 26.7

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

8. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2025149 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))