ABCF->ab-angle b

Percentage Accurate: 19.4% → 46.6%

Time: 25.1s

Alternatives: 8

Speedup: N/A×

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: 19.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 46.6% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 2.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B\_m \cdot B\_m}{C}\right) + A\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= B_m 2.9e-17)
     (/
      (sqrt (* (* 2.0 (* t_0 F)) (+ (+ A (* -0.5 (/ (* B_m B_m) C))) A)))
      (- t_0))
     (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A (hypot A B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 2.9e-17) {
		tmp = sqrt(((2.0 * (t_0 * F)) * ((A + (-0.5 * ((B_m * B_m) / C))) + A))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 2.9e-17) {
		tmp = Math.sqrt(((2.0 * (t_0 * F)) * ((A + (-0.5 * ((B_m * B_m) / C))) + A))) / -t_0;
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (A - Math.hypot(A, B_m))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	tmp = 0
	if B_m <= 2.9e-17:
		tmp = math.sqrt(((2.0 * (t_0 * F)) * ((A + (-0.5 * ((B_m * B_m) / C))) + A))) / -t_0
	else:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (A - math.hypot(A, B_m))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 2.9e-17)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + Float64(-0.5 * Float64(Float64(B_m * B_m) / C))) + A))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - hypot(A, B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	tmp = 0.0;
	if (B_m <= 2.9e-17)
		tmp = sqrt(((2.0 * (t_0 * F)) * ((A + (-0.5 * ((B_m * B_m) / C))) + A))) / -t_0;
	else
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.9e-17], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.9000000000000003e-17

   1. Initial program 16.0%

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

   3. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 17.8

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

   5. Applied rewrites 17.8%

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

   if 2.9000000000000003e-17 < B

   1. Initial program 20.1%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 47.9

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

   5. Applied rewrites 47.9%

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

4. Final simplification 25.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + -0.5 \cdot \frac{B \cdot B}{C}\right) + A\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.1% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 10^{-32}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right), 2 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) + A\right)\right)\right)\right)}}{-\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1e-32)
   (/
    (sqrt
     (fma
      -8.0
      (* A (* C (* F (+ A A))))
      (* 2.0 (* (* B_m B_m) (* F (+ (+ A (* 2.0 A)) A))))))
    (- (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A (hypot A B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e-32) {
		tmp = sqrt(fma(-8.0, (A * (C * (F * (A + A)))), (2.0 * ((B_m * B_m) * (F * ((A + (2.0 * A)) + A)))))) / -(pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1e-32)
		tmp = Float64(sqrt(fma(-8.0, Float64(A * Float64(C * Float64(F * Float64(A + A)))), Float64(2.0 * Float64(Float64(B_m * B_m) * Float64(F * Float64(Float64(A + Float64(2.0 * A)) + A)))))) / Float64(-Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - hypot(A, B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1e-32], N[(N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(F * N[(N[(A + N[(2.0 * A), $MachinePrecision]), $MachinePrecision] + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.00000000000000006e-32

   1. Initial program 16.0%

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

   3. Taylor expanded in A around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 9.9

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

   5. Applied rewrites 9.9%

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

   6. Taylor expanded in C around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 14.3%

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

   9. Taylor expanded in B around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift--.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. lower-*.f64 N/A

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

   11. Applied rewrites 15.9%

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

   if 1.00000000000000006e-32 < B

   1. Initial program 19.8%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 46.0

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

   5. Applied rewrites 46.0%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10^{-32}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-8, A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right), 2 \cdot \left(\left(B \cdot B\right) \cdot \left(F \cdot \left(\left(A + 2 \cdot A\right) + A\right)\right)\right)\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.4% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-182}:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot \frac{F}{C}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-182)
   (sqrt (* (* -0.5 (/ F C)) 2.0))
   (* (/ (sqrt 2.0) (- B_m)) (sqrt (* F (- A (hypot A B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-182) {
		tmp = sqrt(((-0.5 * (F / C)) * 2.0));
	} else {
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-182) {
		tmp = Math.sqrt(((-0.5 * (F / C)) * 2.0));
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * (A - Math.hypot(A, B_m))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-182:
		tmp = math.sqrt(((-0.5 * (F / C)) * 2.0))
	else:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * (A - math.hypot(A, B_m))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-182)
		tmp = sqrt(Float64(Float64(-0.5 * Float64(F / C)) * 2.0));
	else
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(F * Float64(A - hypot(A, B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-182)
		tmp = sqrt(((-0.5 * (F / C)) * 2.0));
	else
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * (A - hypot(A, B_m))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-182], N[Sqrt[N[(N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-182

   1. Initial program 11.4%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 15.8%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 20.6

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

   8. Applied rewrites 20.6%

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

   if 1e-182 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 19.5%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 21.1

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

   5. Applied rewrites 21.1%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-182}:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot \frac{F}{C}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = -Math.sqrt((((F * ((A + C) - Math.hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((-0.5 * (F / C)) * 2.0));
	} else {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt((F * -B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0
	tmp = 0
	if t_1 <= 0.0:
		tmp = -math.sqrt((((F * ((A + C) - math.hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0))
	elif t_1 <= math.inf:
		tmp = math.sqrt(((-0.5 * (F / C)) * 2.0))
	else:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt((F * -B_m))
	return tmp

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0;
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = -sqrt((((F * ((A + C) - hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	elseif (t_1 <= Inf)
		tmp = sqrt(((-0.5 * (F / C)) * 2.0));
	else
		tmp = (sqrt(2.0) / -B_m) * sqrt((F * -B_m));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 31.8%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 44.8%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 35.7%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 17.6

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

   5. Applied rewrites 17.6%

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

   6. Taylor expanded in A around 0

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

      1. lower-*.f64 15.5

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

   8. Applied rewrites 15.5%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot \frac{F}{C}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{F \cdot \left(-B\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.4% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_1 = Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B_m, 2.0)))))) / -t_0;
	double tmp;
	if (t_1 <= 0.0) {
		tmp = -Math.sqrt((((F * ((A + C) - Math.hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = Math.sqrt(((-0.5 * (F / C)) * 2.0));
	} else {
		tmp = A * ((Math.sqrt(2.0) / (-A * B_m)) * Math.sqrt((F * (C - Math.hypot(B_m, C)))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	t_1 = math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B_m, 2.0)))))) / -t_0
	tmp = 0
	if t_1 <= 0.0:
		tmp = -math.sqrt((((F * ((A + C) - math.hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0))
	elif t_1 <= math.inf:
		tmp = math.sqrt(((-0.5 * (F / C)) * 2.0))
	else:
		tmp = A * ((math.sqrt(2.0) / (-A * B_m)) * math.sqrt((F * (C - math.hypot(B_m, C)))))
	return tmp

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	t_1 = sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B_m ^ 2.0)))))) / -t_0;
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = -sqrt((((F * ((A + C) - hypot(B_m, (A - C)))) / ((B_m * B_m) - (4.0 * (A * C)))) * 2.0));
	elseif (t_1 <= Inf)
		tmp = sqrt(((-0.5 * (F / C)) * 2.0));
	else
		tmp = A * ((sqrt(2.0) / (-A * B_m)) * sqrt((F * (C - hypot(B_m, C)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 31.8%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 44.8%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 35.7%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.2

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

   8. Applied rewrites 36.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 14.2%

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

   6. Taylor expanded in A around inf

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

   8. Applied rewrites 8.0%

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

   9. Taylor expanded in A around 0

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

       1. sqrt-prod N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

   11. Applied rewrites 11.2%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq 0:\\ \;\;\;\;-\sqrt{\frac{F \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}{B \cdot B - 4 \cdot \left(A \cdot C\right)} \cdot 2}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot \frac{F}{C}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \left(\frac{\sqrt{2}}{\left(-A\right) \cdot B} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.4% accurate, N/A× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot \frac{F}{C}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \left(\frac{\sqrt{2}}{\left(-A\right) \cdot B\_m} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-34)
   (sqrt (* (* -0.5 (/ F C)) 2.0))
   (* A (* (/ (sqrt 2.0) (* (- A) B_m)) (sqrt (* F (- C (hypot B_m C))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-34) {
		tmp = sqrt(((-0.5 * (F / C)) * 2.0));
	} else {
		tmp = A * ((sqrt(2.0) / (-A * B_m)) * sqrt((F * (C - hypot(B_m, C)))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-34) {
		tmp = Math.sqrt(((-0.5 * (F / C)) * 2.0));
	} else {
		tmp = A * ((Math.sqrt(2.0) / (-A * B_m)) * Math.sqrt((F * (C - Math.hypot(B_m, C)))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-34:
		tmp = math.sqrt(((-0.5 * (F / C)) * 2.0))
	else:
		tmp = A * ((math.sqrt(2.0) / (-A * B_m)) * math.sqrt((F * (C - math.hypot(B_m, C)))))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-34)
		tmp = sqrt(Float64(Float64(-0.5 * Float64(F / C)) * 2.0));
	else
		tmp = Float64(A * Float64(Float64(sqrt(2.0) / Float64(Float64(-A) * B_m)) * sqrt(Float64(F * Float64(C - hypot(B_m, C))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-34)
		tmp = sqrt(((-0.5 * (F / C)) * 2.0));
	else
		tmp = A * ((sqrt(2.0) / (-A * B_m)) * sqrt((F * (C - hypot(B_m, C)))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-34], N[Sqrt[N[(N[(-0.5 * N[(F / C), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(A * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[((-A) * B$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999986e-34

   1. Initial program 17.5%

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

   3. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   5. Applied rewrites 14.8%

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

   6. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 17.1

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

   8. Applied rewrites 17.1%

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

   if 1.99999999999999986e-34 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.7%

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

   3. Taylor expanded in A around 0

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

   5. Applied rewrites 21.4%

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

   6. Taylor expanded in A around inf

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

   8. Applied rewrites 15.8%

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

   9. Taylor expanded in A around 0

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

       1. sqrt-prod N/A

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

       2. pow2 N/A

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

       3. pow2 N/A

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

   11. Applied rewrites 18.6%

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

4. Final simplification 18.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot \frac{F}{C}\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;A \cdot \left(\frac{\sqrt{2}}{\left(-A\right) \cdot B} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 21.2% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in A around 0

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

5. Applied rewrites 12.8%

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

6. Taylor expanded in A around inf

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

8. Applied rewrites 9.5%

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

9. Taylor expanded in A around 0

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

    1. sqrt-prod N/A

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

    2. pow2 N/A

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

    3. pow2 N/A

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

11. Applied rewrites 11.8%

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

12. Final simplification 11.8%

    \[\leadsto A \cdot \left(\frac{\sqrt{2}}{\left(-A\right) \cdot B} \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}\right) \]
13. Add Preprocessing

Alternative 8: 2.0% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.1%

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

3. Taylor expanded in F around -inf

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. sqrt-unprod N/A

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

5. Applied rewrites 8.9%

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

6. Taylor expanded in C around 0

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. pow2 N/A

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

   4. pow2 N/A

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

   5. lift-hypot.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-*.f64 17.5

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

8. Applied rewrites 17.5%

   \[\leadsto \frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))