ABCF->ab-angle b

Percentage Accurate: 18.6% → 37.2%

Time: 10.4s

Alternatives: 13

Speedup: 7.2×

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: 18.6% 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: 37.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;C \leq -1.32 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(4 \cdot A, C, B \cdot B\right)} \cdot \sqrt{\left(\left(\left(\mathsf{hypot}\left(B, C - A\right) - C\right) - A\right) \cdot 2\right) \cdot F}}{t\_0}\\ \mathbf{elif}\;C \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot 2\right)} \cdot \sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_1\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))) (t_1 (fma -4.0 (* C A) (* B B))))
   (if (<= C -1.32e-47)
     (/
      (-
       (*
        (sqrt (fma (* 4.0 A) C (* B B)))
        (sqrt (* (* (- (- (hypot B (- C A)) C) A) 2.0) F))))
      t_0)
     (if (<= C 4.6e-17)
       (/
        (-
         (*
          (sqrt (* (- (+ A C) (hypot (- A C) B)) (* F 2.0)))
          (sqrt (fma (* A C) -4.0 (* B B)))))
        t_0)
       (/
        (-
         (sqrt
          (* (- A (fma -1.0 A (* 0.5 (/ (pow B 2.0) C)))) (* (* 2.0 F) t_1))))
        t_1)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = fma(-4.0, (C * A), (B * B));
	double tmp;
	if (C <= -1.32e-47) {
		tmp = -(sqrt(fma((4.0 * A), C, (B * B))) * sqrt(((((hypot(B, (C - A)) - C) - A) * 2.0) * F))) / t_0;
	} else if (C <= 4.6e-17) {
		tmp = -(sqrt((((A + C) - hypot((A - C), B)) * (F * 2.0))) * sqrt(fma((A * C), -4.0, (B * B)))) / t_0;
	} else {
		tmp = -sqrt(((A - fma(-1.0, A, (0.5 * (pow(B, 2.0) / C)))) * ((2.0 * F) * t_1))) / t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = fma(-4.0, Float64(C * A), Float64(B * B))
	tmp = 0.0
	if (C <= -1.32e-47)
		tmp = Float64(Float64(-Float64(sqrt(fma(Float64(4.0 * A), C, Float64(B * B))) * sqrt(Float64(Float64(Float64(Float64(hypot(B, Float64(C - A)) - C) - A) * 2.0) * F)))) / t_0);
	elseif (C <= 4.6e-17)
		tmp = Float64(Float64(-Float64(sqrt(Float64(Float64(Float64(A + C) - hypot(Float64(A - C), B)) * Float64(F * 2.0))) * sqrt(fma(Float64(A * C), -4.0, Float64(B * B))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(A - fma(-1.0, A, Float64(0.5 * Float64((B ^ 2.0) / C)))) * Float64(Float64(2.0 * F) * t_1)))) / t_1);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
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]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.32e-47], N[((-N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * C + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(N[(N[Sqrt[B ^ 2 + N[(C - A), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision] - A), $MachinePrecision] * 2.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 4.6e-17], N[((-N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(N[(A - N[(-1.0 * A + N[(0.5 * N[(N[Power[B, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.32e-47

   1. Initial program 14.8%

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

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

      4. sub-negate2 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 21.8%

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

   5. Applied rewrites 30.3%

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

   if -1.32e-47 < C < 4.60000000000000018e-17

   1. Initial program 31.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 40.1%

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

   5. Applied rewrites 42.3%

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

   if 4.60000000000000018e-17 < C

   1. Initial program 6.0%

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

   3. Applied rewrites 7.5%

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

   5. Applied rewrites 1.3%

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

   7. Applied rewrites 11.3%

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

   8. Taylor expanded in C around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 33.3

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

   10. Applied rewrites 33.3%

       \[\leadsto \frac{-\sqrt{\left(A - \color{blue}{\mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)}\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 37.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;C \leq -1.32 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(4 \cdot A, C, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(B, C - A\right) - C\right) - A\right)}}{t\_0}\\ \mathbf{elif}\;C \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(F \cdot 2\right)} \cdot \sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_1\right)}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))) (t_1 (fma -4.0 (* C A) (* B B))))
   (if (<= C -1.32e-47)
     (/
      (-
       (*
        (sqrt (* (fma (* 4.0 A) C (* B B)) 2.0))
        (sqrt (* F (- (- (hypot B (- C A)) C) A)))))
      t_0)
     (if (<= C 4.6e-17)
       (/
        (-
         (*
          (sqrt (* (- (+ A C) (hypot (- A C) B)) (* F 2.0)))
          (sqrt (fma (* A C) -4.0 (* B B)))))
        t_0)
       (/
        (-
         (sqrt
          (* (- A (fma -1.0 A (* 0.5 (/ (pow B 2.0) C)))) (* (* 2.0 F) t_1))))
        t_1)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = fma(-4.0, (C * A), (B * B));
	double tmp;
	if (C <= -1.32e-47) {
		tmp = -(sqrt((fma((4.0 * A), C, (B * B)) * 2.0)) * sqrt((F * ((hypot(B, (C - A)) - C) - A)))) / t_0;
	} else if (C <= 4.6e-17) {
		tmp = -(sqrt((((A + C) - hypot((A - C), B)) * (F * 2.0))) * sqrt(fma((A * C), -4.0, (B * B)))) / t_0;
	} else {
		tmp = -sqrt(((A - fma(-1.0, A, (0.5 * (pow(B, 2.0) / C)))) * ((2.0 * F) * t_1))) / t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = fma(-4.0, Float64(C * A), Float64(B * B))
	tmp = 0.0
	if (C <= -1.32e-47)
		tmp = Float64(Float64(-Float64(sqrt(Float64(fma(Float64(4.0 * A), C, Float64(B * B)) * 2.0)) * sqrt(Float64(F * Float64(Float64(hypot(B, Float64(C - A)) - C) - A))))) / t_0);
	elseif (C <= 4.6e-17)
		tmp = Float64(Float64(-Float64(sqrt(Float64(Float64(Float64(A + C) - hypot(Float64(A - C), B)) * Float64(F * 2.0))) * sqrt(fma(Float64(A * C), -4.0, Float64(B * B))))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(A - fma(-1.0, A, Float64(0.5 * Float64((B ^ 2.0) / C)))) * Float64(Float64(2.0 * F) * t_1)))) / t_1);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
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]}, Block[{t$95$1 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, -1.32e-47], N[((-N[(N[Sqrt[N[(N[(N[(4.0 * A), $MachinePrecision] * C + N[(B * B), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(N[(N[Sqrt[B ^ 2 + N[(C - A), $MachinePrecision] ^ 2], $MachinePrecision] - C), $MachinePrecision] - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 4.6e-17], N[((-N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(N[(A - N[(-1.0 * A + N[(0.5 * N[(N[Power[B, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.32e-47

   1. Initial program 14.8%

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

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

      4. sub-negate2 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

   3. Applied rewrites 21.8%

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

   5. Applied rewrites 30.3%

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

   if -1.32e-47 < C < 4.60000000000000018e-17

   1. Initial program 31.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 40.1%

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

   5. Applied rewrites 42.3%

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

   if 4.60000000000000018e-17 < C

   1. Initial program 6.0%

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

   3. Applied rewrites 7.5%

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

   5. Applied rewrites 1.3%

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

   7. Applied rewrites 11.3%

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

   8. Taylor expanded in C around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 33.3

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

   10. Applied rewrites 33.3%

       \[\leadsto \frac{-\sqrt{\left(A - \color{blue}{\mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)}\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 38.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := \mathsf{hypot}\left(A - C, B\right)\\ \mathbf{if}\;C \leq -5.2 \cdot 10^{-263}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(C + A\right) - t\_1\right) \cdot \left(\left(F + F\right) \cdot t\_0\right)}}{t\_0}\\ \mathbf{elif}\;C \leq 4.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(A + C\right) - t\_1\right) \cdot \left(F \cdot 2\right)} \cdot \sqrt{\mathsf{fma}\left(A \cdot C, -4, B \cdot B\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B))) (t_1 (hypot (- A C) B)))
   (if (<= C -5.2e-263)
     (/ (- (sqrt (* (- (+ C A) t_1) (* (+ F F) t_0)))) t_0)
     (if (<= C 4.6e-17)
       (/
        (-
         (*
          (sqrt (* (- (+ A C) t_1) (* F 2.0)))
          (sqrt (fma (* A C) -4.0 (* B B)))))
        (- (pow B 2.0) (* (* 4.0 A) C)))
       (/
        (-
         (sqrt
          (* (- A (fma -1.0 A (* 0.5 (/ (pow B 2.0) C)))) (* (* 2.0 F) t_0))))
        t_0)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double t_1 = hypot((A - C), B);
	double tmp;
	if (C <= -5.2e-263) {
		tmp = -sqrt((((C + A) - t_1) * ((F + F) * t_0))) / t_0;
	} else if (C <= 4.6e-17) {
		tmp = -(sqrt((((A + C) - t_1) * (F * 2.0))) * sqrt(fma((A * C), -4.0, (B * B)))) / (pow(B, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = -sqrt(((A - fma(-1.0, A, (0.5 * (pow(B, 2.0) / C)))) * ((2.0 * F) * t_0))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_1 = hypot(Float64(A - C), B)
	tmp = 0.0
	if (C <= -5.2e-263)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(C + A) - t_1) * Float64(Float64(F + F) * t_0)))) / t_0);
	elseif (C <= 4.6e-17)
		tmp = Float64(Float64(-Float64(sqrt(Float64(Float64(Float64(A + C) - t_1) * Float64(F * 2.0))) * sqrt(fma(Float64(A * C), -4.0, Float64(B * B))))) / Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(A - fma(-1.0, A, Float64(0.5 * Float64((B ^ 2.0) / C)))) * Float64(Float64(2.0 * F) * t_0)))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]}, If[LessEqual[C, -5.2e-263], N[((-N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - t$95$1), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[C, 4.6e-17], N[((-N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] - t$95$1), $MachinePrecision] * N[(F * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(A * C), $MachinePrecision] * -4.0 + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(A - N[(-1.0 * A + N[(0.5 * N[(N[Power[B, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if C < -5.2000000000000001e-263

   1. Initial program 25.7%

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

   3. Applied rewrites 35.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 35.0

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

   5. Applied rewrites 35.0%

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

   if -5.2000000000000001e-263 < C < 4.60000000000000018e-17

   1. Initial program 31.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

   3. Applied rewrites 38.8%

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

   5. Applied rewrites 48.7%

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

   if 4.60000000000000018e-17 < C

   1. Initial program 6.0%

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

   3. Applied rewrites 7.5%

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

   5. Applied rewrites 1.3%

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

   7. Applied rewrites 11.3%

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

   8. Taylor expanded in C around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 33.3

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

   10. Applied rewrites 33.3%

       \[\leadsto \frac{-\sqrt{\left(A - \color{blue}{\mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)}\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 35.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := \left(2 \cdot F\right) \cdot t\_0\\ \mathbf{if}\;C \leq 2.7 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \left(\mathsf{hypot}\left(A - C, B\right) - C\right)\right) \cdot t\_1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)\right) \cdot t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B))) (t_1 (* (* 2.0 F) t_0)))
   (if (<= C 2.7e-61)
     (/ (- (sqrt (* (- A (- (hypot (- A C) B) C)) t_1))) t_0)
     (/ (- (sqrt (* (- A (fma -1.0 A (* 0.5 (/ (pow B 2.0) C)))) t_1))) t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double t_1 = (2.0 * F) * t_0;
	double tmp;
	if (C <= 2.7e-61) {
		tmp = -sqrt(((A - (hypot((A - C), B) - C)) * t_1)) / t_0;
	} else {
		tmp = -sqrt(((A - fma(-1.0, A, (0.5 * (pow(B, 2.0) / C)))) * t_1)) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	tmp = 0.0
	if (C <= 2.7e-61)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A - Float64(hypot(Float64(A - C), B) - C)) * t_1))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(A - fma(-1.0, A, Float64(0.5 * Float64((B ^ 2.0) / C)))) * t_1))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[C, 2.7e-61], N[((-N[Sqrt[N[(N[(A - N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(N[(A - N[(-1.0 * A + N[(0.5 * N[(N[Power[B, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if C < 2.69999999999999993e-61

   1. Initial program 29.4%

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

   3. Applied rewrites 37.8%

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

   5. Applied rewrites 6.6%

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

   7. Applied rewrites 37.9%

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

   if 2.69999999999999993e-61 < C

   1. Initial program 8.0%

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

   3. Applied rewrites 9.8%

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

   5. Applied rewrites 2.0%

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

   7. Applied rewrites 13.4%

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

   8. Taylor expanded in C around inf

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 32.8

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

   10. Applied rewrites 32.8%

       \[\leadsto \frac{-\sqrt{\left(A - \color{blue}{\mathsf{fma}\left(-1, A, 0.5 \cdot \frac{{B}^{2}}{C}\right)}\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 31.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := \left(2 \cdot F\right) \cdot t\_0\\ \mathbf{if}\;C \leq 1.75 \cdot 10^{-64}:\\ \;\;\;\;\frac{-\sqrt{\left(A - \left(\mathsf{hypot}\left(A - C, B\right) - C\right)\right) \cdot t\_1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot A\right) \cdot t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B))) (t_1 (* (* 2.0 F) t_0)))
   (if (<= C 1.75e-64)
     (/ (- (sqrt (* (- A (- (hypot (- A C) B) C)) t_1))) t_0)
     (/ (- (sqrt (* (* 2.0 A) t_1))) t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double t_1 = (2.0 * F) * t_0;
	double tmp;
	if (C <= 1.75e-64) {
		tmp = -sqrt(((A - (hypot((A - C), B) - C)) * t_1)) / t_0;
	} else {
		tmp = -sqrt(((2.0 * A) * t_1)) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	tmp = 0.0
	if (C <= 1.75e-64)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A - Float64(hypot(Float64(A - C), B) - C)) * t_1))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * A) * t_1))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[C, 1.75e-64], N[((-N[Sqrt[N[(N[(A - N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision] - C), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if C < 1.7500000000000002e-64

   1. Initial program 29.4%

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

   3. Applied rewrites 37.8%

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

   5. Applied rewrites 6.7%

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

   7. Applied rewrites 37.9%

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

   if 1.7500000000000002e-64 < C

   1. Initial program 8.2%

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

   3. Applied rewrites 10.1%

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 26.0

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

   8. Applied rewrites 26.0%

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

Alternative 6: 31.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;C \leq 2.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(C + A\right) - \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\left(F + F\right) \cdot t\_0\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot A\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B))))
   (if (<= C 2.2e-99)
     (/ (- (sqrt (* (- (+ C A) (hypot (- A C) B)) (* (+ F F) t_0)))) t_0)
     (/ (- (sqrt (* (* 2.0 A) (* (* 2.0 F) t_0)))) t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double tmp;
	if (C <= 2.2e-99) {
		tmp = -sqrt((((C + A) - hypot((A - C), B)) * ((F + F) * t_0))) / t_0;
	} else {
		tmp = -sqrt(((2.0 * A) * ((2.0 * F) * t_0))) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	tmp = 0.0
	if (C <= 2.2e-99)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(C + A) - hypot(Float64(A - C), B)) * Float64(Float64(F + F) * t_0)))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * A) * Float64(Float64(2.0 * F) * t_0)))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[C, 2.2e-99], N[((-N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(F + F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 2.20000000000000004e-99

   1. Initial program 28.7%

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

   3. Applied rewrites 37.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if 2.20000000000000004e-99 < C

   1. Initial program 10.3%

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

   3. Applied rewrites 12.4%

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 26.2

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

   8. Applied rewrites 26.2%

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

Alternative 7: 28.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ t_1 := \left(2 \cdot F\right) \cdot t\_0\\ t_2 := \sqrt{B - \left(A - C\right)}\\ \mathbf{if}\;C \leq 2.9 \cdot 10^{-66}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(t\_2, -t\_2, A + C\right) \cdot t\_1}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot A\right) \cdot t\_1}}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B)))
        (t_1 (* (* 2.0 F) t_0))
        (t_2 (sqrt (- B (- A C)))))
   (if (<= C 2.9e-66)
     (/ (- (sqrt (* (fma t_2 (- t_2) (+ A C)) t_1))) t_0)
     (/ (- (sqrt (* (* 2.0 A) t_1))) t_0))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double t_1 = (2.0 * F) * t_0;
	double t_2 = sqrt((B - (A - C)));
	double tmp;
	if (C <= 2.9e-66) {
		tmp = -sqrt((fma(t_2, -t_2, (A + C)) * t_1)) / t_0;
	} else {
		tmp = -sqrt(((2.0 * A) * t_1)) / t_0;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	t_1 = Float64(Float64(2.0 * F) * t_0)
	t_2 = sqrt(Float64(B - Float64(A - C)))
	tmp = 0.0
	if (C <= 2.9e-66)
		tmp = Float64(Float64(-sqrt(Float64(fma(t_2, Float64(-t_2), Float64(A + C)) * t_1))) / t_0);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * A) * t_1))) / t_0);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(B - N[(A - C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[C, 2.9e-66], N[((-N[Sqrt[N[(N[(t$95$2 * (-t$95$2) + N[(A + C), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[((-N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if C < 2.90000000000000011e-66

   1. Initial program 29.4%

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

   3. Applied rewrites 37.8%

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

   5. Applied rewrites 6.6%

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

   7. Applied rewrites 30.6%

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

   if 2.90000000000000011e-66 < C

   1. Initial program 8.3%

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

   3. Applied rewrites 10.2%

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 26.1

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

   8. Applied rewrites 26.1%

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

Alternative 8: 27.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\\ \mathbf{if}\;B \leq 9 \cdot 10^{+87}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot A\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{\left(B + C\right) \cdot \left(B - C\right)}\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B B))))
   (if (<= B 9e+87)
     (/ (- (sqrt (* (* 2.0 A) (* (* 2.0 F) t_0)))) t_0)
     (*
      -1.0
      (* (/ (sqrt 2.0) B) (sqrt (* F (- C (sqrt (* (+ B C) (- B C)))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B * B));
	double tmp;
	if (B <= 9e+87) {
		tmp = -sqrt(((2.0 * A) * ((2.0 * F) * t_0))) / t_0;
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B) * sqrt((F * (C - sqrt(((B + C) * (B - C)))))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B * B))
	tmp = 0.0
	if (B <= 9e+87)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * A) * Float64(Float64(2.0 * F) * t_0)))) / t_0);
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(C - sqrt(Float64(Float64(B + C) * Float64(B - C))))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 9e+87], N[((-N[Sqrt[N[(N[(2.0 * A), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(C - N[Sqrt[N[(N[(B + C), $MachinePrecision] * N[(B - C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 9.0000000000000005e87

   1. Initial program 21.1%

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

   3. Applied rewrites 27.0%

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

   5. Applied rewrites 5.3%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 31.1

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

   8. Applied rewrites 31.1%

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

   if 9.0000000000000005e87 < 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

   3. Applied rewrites 9.1%

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

   5. Applied rewrites 6.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 6.4

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

   7. Applied rewrites 6.4%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 13.6

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

   10. Applied rewrites 13.6%

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

Alternative 9: 25.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{\left(B + C\right) \cdot \left(B - C\right)}\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 2.3e+79)
   (/
    (- (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))))
    (fma -4.0 (* C A) (* B B)))
   (*
    -1.0
    (* (/ (sqrt 2.0) B) (sqrt (* F (- C (sqrt (* (+ B C) (- B C))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 2.3e+79) {
		tmp = -sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / fma(-4.0, (C * A), (B * B));
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B) * sqrt((F * (C - sqrt(((B + C) * (B - C)))))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 2.3e+79)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A)))))))) / fma(-4.0, Float64(C * A), Float64(B * B)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(C - sqrt(Float64(Float64(B + C) * Float64(B - C))))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 2.3e+79], 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(C - N[Sqrt[N[(N[(B + C), $MachinePrecision] * N[(B - C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.3e79

   1. Initial program 21.0%

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

   3. Applied rewrites 27.0%

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 27.9

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

   8. Applied rewrites 27.9%

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

   if 2.3e79 < B

   1. Initial program 8.5%

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

   3. Applied rewrites 9.9%

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

   5. Applied rewrites 7.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 7.3

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

   7. Applied rewrites 7.3%

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

   8. Taylor expanded in A around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 14.8

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

   10. Applied rewrites 14.8%

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

Alternative 10: 25.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B \leq 2.3 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\left(A + B\right) \cdot \left(B - A\right)}\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= B 2.3e+79)
   (/
    (- (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))))
    (fma -4.0 (* C A) (* B B)))
   (*
    -1.0
    (* (/ (sqrt 2.0) B) (sqrt (* F (- A (sqrt (* (+ A B) (- B A))))))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (B <= 2.3e+79) {
		tmp = -sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / fma(-4.0, (C * A), (B * B));
	} else {
		tmp = -1.0 * ((sqrt(2.0) / B) * sqrt((F * (A - sqrt(((A + B) * (B - A)))))));
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (B <= 2.3e+79)
		tmp = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A)))))))) / fma(-4.0, Float64(C * A), Float64(B * B)));
	else
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(A - sqrt(Float64(Float64(A + B) * Float64(B - A))))))));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[B, 2.3e+79], 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(N[(A + B), $MachinePrecision] * N[(B - A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.3e79

   1. Initial program 21.0%

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

   3. Applied rewrites 27.0%

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 27.9

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

   8. Applied rewrites 27.9%

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

   if 2.3e79 < B

   1. Initial program 8.5%

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

   3. Applied rewrites 9.9%

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

   5. Applied rewrites 7.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 7.3

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

   7. Applied rewrites 7.3%

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

   8. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 14.7

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

   10. Applied rewrites 14.7%

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

Alternative 11: 23.2% accurate, 7.2× speedup

Math

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

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (/
  (- (sqrt (* -8.0 (* A (* C (* F (- A (* -1.0 A))))))))
  (fma -4.0 (* C A) (* B B))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return -sqrt((-8.0 * (A * (C * (F * (A - (-1.0 * A))))))) / fma(-4.0, (C * A), (B * B));
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A - Float64(-1.0 * A)))))))) / fma(-4.0, Float64(C * A), Float64(B * B)))
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := 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[(-4.0 * N[(C * A), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

3. Applied rewrites 23.7%

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

5. Applied rewrites 4.9%

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

6. Taylor expanded in C around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 23.2

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

8. Applied rewrites 23.2%

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

Alternative 12: 3.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ 0.25 \cdot \left(\frac{\sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C, 4 \cdot C\right)}\right) \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (* 0.25 (* (/ (sqrt 2.0) C) (sqrt (* F (fma -4.0 C (* 4.0 C)))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return 0.25 * ((sqrt(2.0) / C) * sqrt((F * fma(-4.0, C, (4.0 * C)))));
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(0.25 * Float64(Float64(sqrt(2.0) / C) * sqrt(Float64(F * fma(-4.0, C, Float64(4.0 * C))))))
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(0.25 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / C), $MachinePrecision] * N[Sqrt[N[(F * N[(-4.0 * C + N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. distribute-lft-in N/A

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

   6. lower-fma.f64 N/A

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

3. Applied rewrites 25.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 21.0%

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

6. Taylor expanded in A around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 3.8

      \[\leadsto 0.25 \cdot \left(\frac{\sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C, 4 \cdot C\right)}\right) \]

8. Applied rewrites 3.8%

   \[\leadsto \color{blue}{0.25 \cdot \left(\frac{\sqrt{2}}{C} \cdot \sqrt{F \cdot \mathsf{fma}\left(-4, C, 4 \cdot C\right)}\right)} \]
9. Add Preprocessing

Alternative 13: 13.7% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: A, B, 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, 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
    code = (-1.0d0) * (sqrt((f / b)) * sqrt(2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

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

   4. associate--l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 18.6

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

   7. lift-sqrt.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. unpow2 N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-hypot.f64 23.7

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

3. Applied rewrites 23.7%

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

5. Applied rewrites 12.1%

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

7. Applied rewrites 8.3%

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

8. Taylor expanded in B around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 13.7

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

10. Applied rewrites 13.7%

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

Reproduce

herbie shell --seed 2025107 
(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))))