ABCF->ab-angle b

Percentage Accurate: 18.7% → 46.0%

Time: 11.0s

Alternatives: 11

Speedup: 5.8×

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: 18.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 46.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \mathsf{fma}\left(F \cdot B\_m, B\_m, \left(-4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(\left(-0.5 \cdot \frac{B\_m \cdot B\_m}{C} - -1 \cdot A\right) + A\right)}}{{B\_m}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.2e+37)
   (/
    (-
     (sqrt
      (*
       (* 2.0 (fma (* F B_m) B_m (* (* -4.0 (* A C)) F)))
       (+ (- (* -0.5 (/ (* B_m B_m) C)) (* -1.0 A)) A))))
    (- (pow B_m 2.0) (* (* 4.0 A) C)))
   (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (- A (hypot A B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.2e+37) {
		tmp = -sqrt(((2.0 * fma((F * B_m), B_m, ((-4.0 * (A * C)) * F))) * (((-0.5 * ((B_m * B_m) / C)) - (-1.0 * A)) + A))) / (pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - hypot(A, B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.2e+37)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * fma(Float64(F * B_m), B_m, Float64(Float64(-4.0 * Float64(A * C)) * F))) * Float64(Float64(Float64(-0.5 * Float64(Float64(B_m * B_m) / C)) - Float64(-1.0 * A)) + A)))) / Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.2e+37], N[((-N[Sqrt[N[(N[(2.0 * N[(N[(F * B$95$m), $MachinePrecision] * B$95$m + N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.2e37

   1. Initial program 24.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 26.4

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 35.4

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

   3. Applied rewrites 35.4%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. sqr-neg-rev N/A

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

      15. fp-cancel-sub-sign N/A

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

      16. unpow1 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-pow1 N/A

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

      19. pow2 N/A

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

      20. sqr-neg-rev N/A

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

      21. pow2 N/A

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

      22. sqrt-pow1 N/A

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

      23. pow-plus-rev N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

      26. pow2 N/A

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

      27. sqr-neg-rev N/A

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

      28. lift-*.f64 N/A

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

   5. Applied rewrites 35.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. pow2 N/A

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

      18. sqrt-pow1 N/A

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

      19. metadata-eval N/A

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

      20. unpow1 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 35.4%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 43.1

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

   10. Applied rewrites 43.1%

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

   if 1.2e37 < B

   1. Initial program 10.7%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 7.3%

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

   4. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. pow2 N/A

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

      10. lower-hypot.f64 49.8

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

   6. Applied rewrites 49.8%

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

Alternative 2: 44.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+41}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{{B\_m}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e+41)
   (/
    (- (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))))
    (- (pow B_m 2.0) (* (* 4.0 A) C)))
   (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (- A (hypot A B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e+41) {
		tmp = -sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / (pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - hypot(A, B_m)))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+41) {
		tmp = -Math.sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / (Math.pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -1.0 * ((Math.sqrt(2.0) / B_m) * Math.sqrt((F * (A - Math.hypot(A, B_m)))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+41:
		tmp = -math.sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / (math.pow(B_m, 2.0) - ((4.0 * A) * C))
	else:
		tmp = -1.0 * ((math.sqrt(2.0) / B_m) * math.sqrt((F * (A - math.hypot(A, B_m)))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+41)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A)))))))) / Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+41)
		tmp = -sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / ((B_m ^ 2.0) - ((4.0 * A) * C));
	else
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - hypot(A, B_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+41], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000001e41

   1. Initial program 23.9%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 28.1%

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

   4. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 39.3

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

   6. Applied rewrites 39.3%

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

   if 1.00000000000000001e41 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 9.5%

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

   4. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. pow2 N/A

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

      10. lower-hypot.f64 49.4

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

   6. Applied rewrites 49.4%

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

Alternative 3: 43.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{\left(A - 0.5 \cdot \mathsf{fma}\left(-1, A + B\_m, -1 \cdot \left(A - B\_m\right)\right)\right) \cdot \left(\left(F + F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 5e-198)
     (/
      (sqrt
       (*
        (- A (* 0.5 (fma -1.0 (+ A B_m) (* -1.0 (- A B_m)))))
        (* (+ F F) t_0)))
      (- t_0))
     (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (- A (hypot A B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-198) {
		tmp = sqrt(((A - (0.5 * fma(-1.0, (A + B_m), (-1.0 * (A - B_m))))) * ((F + F) * t_0))) / -t_0;
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - hypot(A, B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-198)
		tmp = Float64(sqrt(Float64(Float64(A - Float64(0.5 * fma(-1.0, Float64(A + B_m), Float64(-1.0 * Float64(A - B_m))))) * Float64(Float64(F + F) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-198], N[(N[Sqrt[N[(N[(A - N[(0.5 * N[(-1.0 * N[(A + B$95$m), $MachinePrecision] + N[(-1.0 * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-198

   1. Initial program 19.3%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 26.6%

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

      1. lift-hypot.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. fp-cancel-sub-sign N/A

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. sqr-neg-rev N/A

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

      9. pow2 N/A

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

      10. sqrt-pow1 N/A

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

      11. pow-plus-rev N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

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

      15. sqr-neg-rev N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. sqrt-prod N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 0.0

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 42.8

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

   10. Applied rewrites 42.8%

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

   if 4.9999999999999999e-198 < (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. 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. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 17.8%

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

   4. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. pow2 N/A

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

      10. lower-hypot.f64 43.1

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

   6. Applied rewrites 43.1%

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

Alternative 4: 45.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.26 \cdot 10^{-93}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \mathsf{fma}\left(F \cdot B\_m, B\_m, \left(-4 \cdot \left(A \cdot C\right)\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{{B\_m}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{+75}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.26e-93)
   (/
    (- (sqrt (* (* 2.0 (fma (* F B_m) B_m (* (* -4.0 (* A C)) F))) (+ A A))))
    (- (pow B_m 2.0) (* (* 4.0 A) C)))
   (if (<= B_m 6e+75)
     (*
      -1.0
      (*
       (sqrt
        (/
         (* F (- (+ A C) (hypot B_m (- A C))))
         (- (* B_m B_m) (* 4.0 (* A C)))))
       (sqrt 2.0)))
     (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (- A (hypot A B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.26e-93) {
		tmp = -sqrt(((2.0 * fma((F * B_m), B_m, ((-4.0 * (A * C)) * F))) * (A + A))) / (pow(B_m, 2.0) - ((4.0 * A) * C));
	} else if (B_m <= 6e+75) {
		tmp = -1.0 * (sqrt(((F * ((A + C) - hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C))))) * sqrt(2.0));
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - hypot(A, B_m)))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.26e-93)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * fma(Float64(F * B_m), B_m, Float64(Float64(-4.0 * Float64(A * C)) * F))) * Float64(A + A)))) / Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	elseif (B_m <= 6e+75)
		tmp = Float64(-1.0 * Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - hypot(B_m, Float64(A - C)))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C))))) * sqrt(2.0)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.26e-93], N[((-N[Sqrt[N[(N[(2.0 * N[(N[(F * B$95$m), $MachinePrecision] * B$95$m + N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6e+75], N[(-1.0 * N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.2600000000000001e-93

   1. Initial program 19.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 21.6

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 30.4

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

   3. Applied rewrites 30.4%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. sqr-neg-rev N/A

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

      15. fp-cancel-sub-sign N/A

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

      16. unpow1 N/A

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

      17. metadata-eval N/A

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

      18. sqrt-pow1 N/A

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

      19. pow2 N/A

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

      20. sqr-neg-rev N/A

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

      21. pow2 N/A

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

      22. sqrt-pow1 N/A

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

      23. pow-plus-rev N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

      26. pow2 N/A

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

      27. sqr-neg-rev N/A

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

      28. lift-*.f64 N/A

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

   5. Applied rewrites 30.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. rem-sqrt-square-rev N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. pow2 N/A

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

      18. sqrt-pow1 N/A

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

      19. metadata-eval N/A

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

      20. unpow1 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 N/A

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

   7. Applied rewrites 30.4%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 45.6%

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

   if 1.2600000000000001e-93 < B < 6e75

   1. Initial program 33.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 33.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 42.7

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

   3. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 42.7

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. metadata-eval 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 39.1%

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

   if 6e75 < B

   1. Initial program 7.6%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 3.5%

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

   4. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. pow2 N/A

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

      10. lower-hypot.f64 50.7

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

   6. Applied rewrites 50.7%

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

Alternative 5: 44.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{{B\_m}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{+75}:\\ \;\;\;\;-1 \cdot \left(\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)\right)}{B\_m \cdot B\_m - 4 \cdot \left(A \cdot C\right)}} \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.4e-95)
   (/
    (- (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))))
    (- (pow B_m 2.0) (* (* 4.0 A) C)))
   (if (<= B_m 6e+75)
     (*
      -1.0
      (*
       (sqrt
        (/
         (* F (- (+ A C) (hypot B_m (- A C))))
         (- (* B_m B_m) (* 4.0 (* A C)))))
       (sqrt 2.0)))
     (* -1.0 (* (/ (sqrt 2.0) B_m) (sqrt (* F (- A (hypot A B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-95) {
		tmp = -sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / (pow(B_m, 2.0) - ((4.0 * A) * C));
	} else if (B_m <= 6e+75) {
		tmp = -1.0 * (sqrt(((F * ((A + C) - hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C))))) * sqrt(2.0));
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - hypot(A, B_m)))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-95) {
		tmp = -Math.sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / (Math.pow(B_m, 2.0) - ((4.0 * A) * C));
	} else if (B_m <= 6e+75) {
		tmp = -1.0 * (Math.sqrt(((F * ((A + C) - Math.hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C))))) * Math.sqrt(2.0));
	} else {
		tmp = -1.0 * ((Math.sqrt(2.0) / B_m) * Math.sqrt((F * (A - Math.hypot(A, B_m)))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.4e-95:
		tmp = -math.sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / (math.pow(B_m, 2.0) - ((4.0 * A) * C))
	elif B_m <= 6e+75:
		tmp = -1.0 * (math.sqrt(((F * ((A + C) - math.hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C))))) * math.sqrt(2.0))
	else:
		tmp = -1.0 * ((math.sqrt(2.0) / B_m) * math.sqrt((F * (A - math.hypot(A, B_m)))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.4e-95)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A)))))))) / Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	elseif (B_m <= 6e+75)
		tmp = Float64(-1.0 * Float64(sqrt(Float64(Float64(F * Float64(Float64(A + C) - hypot(B_m, Float64(A - C)))) / Float64(Float64(B_m * B_m) - Float64(4.0 * Float64(A * C))))) * sqrt(2.0)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.4e-95)
		tmp = -sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / ((B_m ^ 2.0) - ((4.0 * A) * C));
	elseif (B_m <= 6e+75)
		tmp = -1.0 * (sqrt(((F * ((A + C) - hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C))))) * sqrt(2.0));
	else
		tmp = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - hypot(A, B_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.4e-95], N[((-N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(-1.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6e+75], N[(-1.0 * N[(N[Sqrt[N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(B$95$m * B$95$m), $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.3999999999999998e-95

   1. Initial program 19.5%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 23.0%

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

   4. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 43.3

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

   6. Applied rewrites 43.3%

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

   if 4.3999999999999998e-95 < B < 6e75

   1. Initial program 33.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 34.0

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 42.7

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

   3. Applied rewrites 42.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 42.7

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

      4. unpow1 N/A

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

      5. metadata-eval N/A

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

      6. pow-to-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. metadata-eval 6.5

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

   5. Applied rewrites 6.5%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 39.1%

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

   if 6e75 < B

   1. Initial program 7.6%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 3.5%

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

   4. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. pow2 N/A

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

      10. lower-hypot.f64 50.7

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

   6. Applied rewrites 50.7%

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

Alternative 6: 24.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \frac{\sqrt{\left(A - 0.5 \cdot \mathsf{fma}\left(-1, A + B\_m, -1 \cdot \left(A - B\_m\right)\right)\right) \cdot \left(\left(F + F\right) \cdot t\_0\right)}}{-t\_0} \end{array} \end{array} \]

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 (fma -4.0 (* C A) (* B_m B_m))))
   (/
    (sqrt
     (* (- A (* 0.5 (fma -1.0 (+ A B_m) (* -1.0 (- A B_m))))) (* (+ F F) t_0)))
    (- 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 = fma(-4.0, (C * A), (B_m * B_m));
	return sqrt(((A - (0.5 * fma(-1.0, (A + B_m), (-1.0 * (A - B_m))))) * ((F + F) * t_0))) / -t_0;
}

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))
	return Float64(sqrt(Float64(Float64(A - Float64(0.5 * fma(-1.0, Float64(A + B_m), Float64(-1.0 * Float64(A - B_m))))) * Float64(Float64(F + F) * t_0))) / Float64(-t_0))
end

Wolfram

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

Derivation

1. Initial program 18.7%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 23.8%

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

   1. lift-hypot.f64 N/A

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

   2. sqr-neg-rev N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. sqrt-pow1 N/A

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

   7. pow2 N/A

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

   8. sqr-neg-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. pow-plus-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-squares N/A

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

   19. sqrt-prod N/A

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

   20. lower-*.f64 N/A

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

5. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

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

8. Taylor expanded in C around inf

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 24.2

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

10. Applied rewrites 24.2%

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

Alternative 7: 21.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - 0.5 \cdot \mathsf{fma}\left(-1, A + B\_m, -1 \cdot \left(A - B\_m\right)\right)\right)\right)\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/
  (sqrt
   (*
    -8.0
    (* A (* C (* F (- A (* 0.5 (fma -1.0 (+ A B_m) (* -1.0 (- A B_m))))))))))
  (- (fma -4.0 (* C A) (* B_m 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((-8.0 * (A * (C * (F * (A - (0.5 * fma(-1.0, (A + B_m), (-1.0 * (A - B_m)))))))))) / -fma(-4.0, (C * A), (B_m * 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(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(0.5 * fma(-1.0, Float64(A + B_m), Float64(-1.0 * Float64(A - B_m)))))))))) / Float64(-fma(-4.0, Float64(C * A), Float64(B_m * 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[(-8.0 * N[(A * N[(C * N[(F * N[(A - N[(0.5 * N[(-1.0 * N[(A + B$95$m), $MachinePrecision] + N[(-1.0 * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.7%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 23.8%

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

   1. lift-hypot.f64 N/A

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

   2. sqr-neg-rev N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. sqrt-pow1 N/A

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

   7. pow2 N/A

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

   8. sqr-neg-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. pow-plus-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-squares N/A

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

   19. sqrt-prod N/A

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

   20. lower-*.f64 N/A

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

5. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

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

8. Taylor expanded in C around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 21.2

       \[\leadsto \frac{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - 0.5 \cdot \mathsf{fma}\left(-1, A + B, -1 \cdot \left(A - B\right)\right)\right)\right)\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]

10. Applied rewrites 21.2%

    \[\leadsto \frac{\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - 0.5 \cdot \mathsf{fma}\left(-1, A + B, -1 \cdot \left(A - B\right)\right)\right)\right)\right)\right)}}}{-\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
11. Add Preprocessing

Alternative 8: 4.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \frac{\sqrt{\left(\left(C + A\right) - C\right) \cdot \left(\left(F + F\right) \cdot t\_0\right)}}{-t\_0} \end{array} \end{array} \]

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 (fma -4.0 (* C A) (* B_m B_m))))
   (/ (sqrt (* (- (+ C A) C) (* (+ F F) t_0))) (- 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 = fma(-4.0, (C * A), (B_m * B_m));
	return sqrt((((C + A) - C) * ((F + F) * t_0))) / -t_0;
}

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))
	return Float64(sqrt(Float64(Float64(Float64(C + A) - C) * Float64(Float64(F + F) * t_0))) / Float64(-t_0))
end

Wolfram

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

Derivation

1. Initial program 18.7%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 23.8%

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

   1. lift-hypot.f64 N/A

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

   2. sqr-neg-rev N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. sqrt-pow1 N/A

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

   7. pow2 N/A

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

   8. sqr-neg-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. pow-plus-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-squares N/A

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

   19. sqrt-prod N/A

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

   20. lower-*.f64 N/A

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

5. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

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

8. Taylor expanded in C around inf

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

10. Applied rewrites 4.9%

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

Alternative 9: 4.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -1 \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \sqrt{F \cdot \left(A - \sqrt{\left(A + B\_m\right) \cdot \left(A - B\_m\right)}\right)}\right) \end{array} \]

FPCore

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

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - sqrt(((A + B_m) * (A - 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 = (-1.0d0) * ((sqrt(2.0d0) / b_m) * sqrt((f * (a - sqrt(((a + b_m) * (a - 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 -1.0 * ((Math.sqrt(2.0) / B_m) * Math.sqrt((F * (A - Math.sqrt(((A + B_m) * (A - B_m)))))));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -1.0 * ((math.sqrt(2.0) / B_m) * math.sqrt((F * (A - math.sqrt(((A + B_m) * (A - B_m)))))))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-1.0 * Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(F * Float64(A - sqrt(Float64(Float64(A + B_m) * Float64(A - 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 = -1.0 * ((sqrt(2.0) / B_m) * sqrt((F * (A - sqrt(((A + B_m) * (A - B_m)))))));
end

Wolfram

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

Derivation

1. Initial program 18.7%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 23.8%

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

   1. lift-hypot.f64 N/A

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

   2. sqr-neg-rev N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. sqrt-pow1 N/A

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

   7. pow2 N/A

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

   8. sqr-neg-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. pow-plus-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-squares N/A

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

   19. sqrt-prod N/A

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

   20. lower-*.f64 N/A

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

5. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

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

8. Taylor expanded in C around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 4.1

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

10. Applied rewrites 4.1%

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

Alternative 10: 2.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}{-\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* -16.0 (* A (* (* C C) F)))) (- (fma -4.0 (* C A) (* B_m 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((-16.0 * (A * ((C * C) * F)))) / -fma(-4.0, (C * A), (B_m * 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(-16.0 * Float64(A * Float64(Float64(C * C) * F)))) / Float64(-fma(-4.0, Float64(C * A), Float64(B_m * 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[(-16.0 * N[(A * N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.7%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 23.8%

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

   1. lift-hypot.f64 N/A

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

   2. sqr-neg-rev N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. sqrt-pow1 N/A

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

   7. pow2 N/A

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

   8. sqr-neg-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. pow-plus-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-squares N/A

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

   19. sqrt-prod N/A

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

   20. lower-*.f64 N/A

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

5. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

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

8. Taylor expanded in B around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 2.1

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

10. Applied rewrites 2.1%

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

Alternative 11: 0.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -1 \cdot \left(\sqrt{\frac{F}{A}} \cdot \left(\sqrt{-0.5} \cdot \sqrt{2}\right)\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* -1.0 (* (sqrt (/ F A)) (* (sqrt -0.5) (sqrt 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -1.0 * (sqrt((F / A)) * (sqrt(-0.5) * sqrt(2.0)));
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-1.0d0) * (sqrt((f / a)) * (sqrt((-0.5d0)) * sqrt(2.0d0)))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -1.0 * (Math.sqrt((F / A)) * (Math.sqrt(-0.5) * Math.sqrt(2.0)));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -1.0 * (math.sqrt((F / A)) * (math.sqrt(-0.5) * math.sqrt(2.0)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-1.0 * Float64(sqrt(Float64(F / A)) * Float64(sqrt(-0.5) * sqrt(2.0))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -1.0 * (sqrt((F / A)) * (sqrt(-0.5) * sqrt(2.0)));
end

Wolfram

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

Derivation

1. Initial program 18.7%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 23.8%

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

   1. lift-hypot.f64 N/A

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

   2. sqr-neg-rev N/A

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

   3. fp-cancel-sub-sign N/A

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

   4. unpow1 N/A

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

   5. metadata-eval N/A

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

   6. sqrt-pow1 N/A

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

   7. pow2 N/A

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

   8. sqr-neg-rev N/A

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

   9. pow2 N/A

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

   10. sqrt-pow1 N/A

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

   11. pow-plus-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. pow2 N/A

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

   15. sqr-neg-rev N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-squares N/A

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

   19. sqrt-prod N/A

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

   20. lower-*.f64 N/A

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

5. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 0.0

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

7. Applied rewrites 0.0%

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

8. Taylor expanded in B around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 0.0

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

10. Applied rewrites 0.0%

    \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{A}} \cdot \left(\sqrt{-0.5} \cdot \sqrt{2}\right)\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(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))))