ABCF->ab-angle a

Percentage Accurate: 18.9% → 61.7%

Time: 14.7s

Alternatives: 17

Speedup: 16.9×

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: 18.9% 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: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := -t\_0\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right)}}{t\_1}\\ t_3 := \mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{\left(\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot \left(\sqrt{t\_3} \cdot \sqrt{F}\right)\right) \cdot \sqrt{2}}{t\_1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C} \cdot \sqrt{t\_3 \cdot F}\right) \cdot \sqrt{2}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (- t_0))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* t_0 F))
            (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))))
          t_1))
        (t_3 (fma (* -4.0 C) A (* B_m B_m))))
   (if (<= t_2 -1e-217)
     (/
      (*
       (* (sqrt (+ (+ (hypot (- A C) B_m) A) C)) (* (sqrt t_3) (sqrt F)))
       (sqrt 2.0))
      t_1)
     (if (<= t_2 INFINITY)
       (/
        (*
         (* (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) (sqrt (* t_3 F)))
         (sqrt 2.0))
        t_1)
       (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F))))))

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 = -t_0;
	double t_2 = sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))))) / t_1;
	double t_3 = fma((-4.0 * C), A, (B_m * B_m));
	double tmp;
	if (t_2 <= -1e-217) {
		tmp = ((sqrt(((hypot((A - C), B_m) + A) + C)) * (sqrt(t_3) * sqrt(F))) * sqrt(2.0)) / t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = ((sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) * sqrt((t_3 * F))) * sqrt(2.0)) / t_1;
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	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(-t_0)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))))) / t_1)
	t_3 = fma(Float64(-4.0 * C), A, Float64(B_m * B_m))
	tmp = 0.0
	if (t_2 <= -1e-217)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * Float64(sqrt(t_3) * sqrt(F))) * sqrt(2.0)) / t_1);
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) * sqrt(Float64(t_3 * F))) * sqrt(2.0)) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = 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$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-217], N[(N[(N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$3], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$3 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $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))) < -1.00000000000000008e-217

   1. Initial program 51.8%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 67.1%

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

   9. Applied rewrites 79.3%

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

   if -1.00000000000000008e-217 < (/.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 15.4%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 24.2%

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

   7. Applied rewrites 33.4%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 34.1%

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

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

   1. Initial program 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

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 20.0

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

   5. Applied rewrites 20.0%

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

   7. Applied rewrites 29.0%

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

   9. Applied rewrites 29.1%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq -1 \cdot 10^{-217}:\\ \;\;\;\;\frac{\left(\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \left(\sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right)} \cdot \sqrt{F}\right)\right) \cdot \sqrt{2}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \leq \infty:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B\_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := -t\_0\\ \mathbf{if}\;B\_m \leq 1.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (- t_0)))
   (if (<= B_m 1.1e-137)
     (/ (sqrt (* (* 2.0 (* t_0 F)) (* 2.0 C))) t_1)
     (if (<= B_m 2.15e+54)
       (/
        (*
         (sqrt (+ (+ (hypot B_m (- A C)) A) C))
         (sqrt (* (* 2.0 F) (fma (* -4.0 A) C (* B_m B_m)))))
        t_1)
       (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F))))))

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 = -t_0;
	double tmp;
	if (B_m <= 1.1e-137) {
		tmp = sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_1;
	} else if (B_m <= 2.15e+54) {
		tmp = (sqrt(((hypot(B_m, (A - C)) + A) + C)) * sqrt(((2.0 * F) * fma((-4.0 * A), C, (B_m * B_m))))) / t_1;
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	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(-t_0)
	tmp = 0.0
	if (B_m <= 1.1e-137)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(2.0 * C))) / t_1);
	elseif (B_m <= 2.15e+54)
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C)) * sqrt(Float64(Float64(2.0 * F) * fma(Float64(-4.0 * A), C, Float64(B_m * B_m))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[B$95$m, 1.1e-137], N[(N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 2.15e+54], N[(N[(N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.1000000000000001e-137

   1. Initial program 21.9%

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

   3. Taylor expanded in A around -inf

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

      1. lower-*.f64 19.3

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

   5. Applied rewrites 19.3%

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

   if 1.1000000000000001e-137 < B < 2.14999999999999988e54

   1. Initial program 40.5%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. unpow-prod-down N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 60.7%

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

   if 2.14999999999999988e54 < B

   1. Initial program 7.8%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.9%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.1 \cdot 10^{-137}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(B, A - C\right) + A\right) + C} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4 \cdot A, C, B \cdot B\right)}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot F}\\ \mathbf{if}\;B\_m \leq 8.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C} \cdot t\_0\right) \cdot \sqrt{2}}{-\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(\sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C} \cdot t\_0\right) \cdot \sqrt{2}}{-\left(B\_m \cdot B\_m\right) \cdot \mathsf{fma}\left(\frac{-4 \cdot A}{B\_m}, \frac{C}{B\_m}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) F))))
   (if (<= B_m 8.2e-66)
     (/
      (* (* (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C)) t_0) (sqrt 2.0))
      (- (- (pow B_m 2.0) (* (* 4.0 A) C))))
     (if (<= B_m 2.15e+54)
       (/
        (* (* (sqrt (+ (+ (hypot (- A C) B_m) A) C)) t_0) (sqrt 2.0))
        (- (* (* B_m B_m) (fma (/ (* -4.0 A) B_m) (/ C B_m) 1.0))))
       (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt((fma((-4.0 * C), A, (B_m * B_m)) * F));
	double tmp;
	if (B_m <= 8.2e-66) {
		tmp = ((sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) * t_0) * sqrt(2.0)) / -(pow(B_m, 2.0) - ((4.0 * A) * C));
	} else if (B_m <= 2.15e+54) {
		tmp = ((sqrt(((hypot((A - C), B_m) + A) + C)) * t_0) * sqrt(2.0)) / -((B_m * B_m) * fma(((-4.0 * A) / B_m), (C / B_m), 1.0));
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	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 = sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * F))
	tmp = 0.0
	if (B_m <= 8.2e-66)
		tmp = Float64(Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) * t_0) * sqrt(2.0)) / Float64(-Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))));
	elseif (B_m <= 2.15e+54)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C)) * t_0) * sqrt(2.0)) / Float64(-Float64(Float64(B_m * B_m) * fma(Float64(Float64(-4.0 * A) / B_m), Float64(C / B_m), 1.0))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B$95$m, 8.2e-66], N[(N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.15e+54], N[(N[(N[(N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / (-N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(N[(N[(-4.0 * A), $MachinePrecision] / B$95$m), $MachinePrecision] * N[(C / B$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.19999999999999996e-66

   1. Initial program 23.3%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 31.8%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 19.4%

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

   if 8.19999999999999996e-66 < B < 2.14999999999999988e54

   1. Initial program 39.4%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 51.2%

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

   7. Applied rewrites 61.2%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 61.1

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

   10. Applied rewrites 61.1%

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

   if 2.14999999999999988e54 < B

   1. Initial program 7.8%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 46.2

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 66.7%

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

   9. Applied rewrites 66.9%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{\left(\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2}}{-\left(B \cdot B\right) \cdot \mathsf{fma}\left(\frac{-4 \cdot A}{B}, \frac{C}{B}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, C\right) + C} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B\_m \cdot B\_m\right) \cdot F}\right) \cdot \sqrt{2}}{-\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.4e-54)
   (/
    (*
     (*
      (sqrt (+ (fma -0.5 (/ (* B_m B_m) A) C) C))
      (sqrt (* (fma (* -4.0 C) A (* B_m B_m)) F)))
     (sqrt 2.0))
    (- (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.4e-54) {
		tmp = ((sqrt((fma(-0.5, ((B_m * B_m) / A), C) + C)) * sqrt((fma((-4.0 * C), A, (B_m * B_m)) * F))) * sqrt(2.0)) / -(pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.4e-54)
		tmp = Float64(Float64(Float64(sqrt(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), C) + C)) * sqrt(Float64(fma(Float64(-4.0 * C), A, Float64(B_m * B_m)) * F))) * sqrt(2.0)) / Float64(-Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.4e-54], N[(N[(N[(N[Sqrt[N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[(-4.0 * C), $MachinePrecision] * A + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $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[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.3999999999999999e-54

   1. Initial program 23.5%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 28.8%

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

   7. Applied rewrites 32.4%

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

   8. Taylor expanded in A around -inf

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

   10. Applied rewrites 19.0%

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

   if 4.3999999999999999e-54 < B

   1. Initial program 17.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 43.5

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

   5. Applied rewrites 43.5%

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

   7. Applied rewrites 58.1%

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

   9. Applied rewrites 58.3%

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

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, C\right) + C} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot C, A, B \cdot B\right) \cdot F}\right) \cdot \sqrt{2}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 7 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B\_m \cdot B\_m}{A}, 2 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \cdot \sqrt{2}}{-\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 7e-67)
   (/
    (*
     (sqrt
      (*
       (* (fma -0.5 (/ (* B_m B_m) A) (* 2.0 C)) F)
       (fma -4.0 (* C A) (* B_m B_m))))
     (sqrt 2.0))
    (- (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F))))

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 <= 7e-67) {
		tmp = (sqrt(((fma(-0.5, ((B_m * B_m) / A), (2.0 * C)) * F) * fma(-4.0, (C * A), (B_m * B_m)))) * sqrt(2.0)) / -(pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	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 <= 7e-67)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(-0.5, Float64(Float64(B_m * B_m) / A), Float64(2.0 * C)) * F) * fma(-4.0, Float64(C * A), Float64(B_m * B_m)))) * sqrt(2.0)) / Float64(-Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	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, 7e-67], N[(N[(N[Sqrt[N[(N[(N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $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[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.0000000000000001e-67

   1. Initial program 23.3%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 18.9%

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

   if 7.0000000000000001e-67 < B

   1. Initial program 17.7%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 42.1

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

   5. Applied rewrites 42.1%

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

   7. Applied rewrites 56.1%

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

   9. Applied rewrites 56.4%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(-0.5, \frac{B \cdot B}{A}, 2 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.26 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \cdot \sqrt{2}}{-\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.26e-55)
   (/
    (* (sqrt (* (* (* 2.0 C) F) (fma -4.0 (* C A) (* B_m B_m)))) (sqrt 2.0))
    (- (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.26e-55) {
		tmp = (sqrt((((2.0 * C) * F) * fma(-4.0, (C * A), (B_m * B_m)))) * sqrt(2.0)) / -(pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.26e-55)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(2.0 * C) * F) * fma(-4.0, Float64(C * A), Float64(B_m * B_m)))) * sqrt(2.0)) / Float64(-Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.26e-55], N[(N[(N[Sqrt[N[(N[(N[(2.0 * C), $MachinePrecision] * F), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $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[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.2599999999999999e-55

   1. Initial program 23.6%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 28.9%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 18.2%

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

   if 1.2599999999999999e-55 < B

   1. Initial program 16.9%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 43.0

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

   5. Applied rewrites 43.0%

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

   7. Applied rewrites 57.4%

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

   9. Applied rewrites 57.7%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.26 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot C\right) \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{2}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot A, C, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(B\_m, A - C\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 A) C (* B_m B_m))))
   (if (<= B_m 6.5e+27)
     (/ (sqrt (* (+ (+ (hypot B_m (- A C)) A) C) (* (* 2.0 F) t_0))) (- t_0))
     (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((-4.0 * A), C, (B_m * B_m));
	double tmp;
	if (B_m <= 6.5e+27) {
		tmp = sqrt((((hypot(B_m, (A - C)) + A) + C) * ((2.0 * F) * t_0))) / -t_0;
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(-4.0 * A), C, Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 6.5e+27)
		tmp = Float64(sqrt(Float64(Float64(Float64(hypot(B_m, Float64(A - C)) + A) + C) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(-4.0 * A), $MachinePrecision] * C + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6.5e+27], N[(N[Sqrt[N[(N[(N[(N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 6.5000000000000005e27

   1. Initial program 25.1%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 31.2%

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

   if 6.5000000000000005e27 < B

   1. Initial program 11.2%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 44.0

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

   5. Applied rewrites 44.0%

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

   7. Applied rewrites 61.5%

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

   9. Applied rewrites 61.7%

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

Alternative 8: 50.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)} \cdot \sqrt{2}}{-\left({B\_m}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 5.8e-88)
   (/
    (* (sqrt (* -8.0 (* A (* (* C C) F)))) (sqrt 2.0))
    (- (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F))))

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 <= 5.8e-88) {
		tmp = (sqrt((-8.0 * (A * ((C * C) * F)))) * sqrt(2.0)) / -(pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.8e-88) {
		tmp = (Math.sqrt((-8.0 * (A * ((C * C) * F)))) * Math.sqrt(2.0)) / -(Math.pow(B_m, 2.0) - ((4.0 * A) * C));
	} else {
		tmp = (Math.sqrt((2.0 * (Math.hypot(C, B_m) + C))) / -B_m) * Math.sqrt(F);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.8e-88:
		tmp = (math.sqrt((-8.0 * (A * ((C * C) * F)))) * math.sqrt(2.0)) / -(math.pow(B_m, 2.0) - ((4.0 * A) * C))
	else:
		tmp = (math.sqrt((2.0 * (math.hypot(C, B_m) + C))) / -B_m) * math.sqrt(F)
	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 <= 5.8e-88)
		tmp = Float64(Float64(sqrt(Float64(-8.0 * Float64(A * Float64(Float64(C * C) * F)))) * sqrt(2.0)) / Float64(-Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	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 <= 5.8e-88)
		tmp = (sqrt((-8.0 * (A * ((C * C) * F)))) * sqrt(2.0)) / -((B_m ^ 2.0) - ((4.0 * A) * C));
	else
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.8e-88], N[(N[(N[Sqrt[N[(-8.0 * N[(A * N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $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[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.8000000000000003e-88

   1. Initial program 22.8%

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

   3. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in A around -inf

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

   8. Applied rewrites 13.0%

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

   if 5.8000000000000003e-88 < B

   1. Initial program 19.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 41.3

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

   5. Applied rewrites 41.3%

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

   7. Applied rewrites 55.6%

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

   9. Applied rewrites 55.9%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)} \cdot \sqrt{2}}{-\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.5 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\_m\right) + C\right)}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.5e+55)
   (*
    (sqrt
     (/ (* F (+ A (+ C (hypot B_m (- A C))))) (fma B_m B_m (* -4.0 (* A C)))))
    (- (sqrt 2.0)))
   (* (/ (sqrt (* 2.0 (+ (hypot C B_m) C))) (- B_m)) (sqrt F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.5e+55) {
		tmp = sqrt(((F * (A + (C + hypot(B_m, (A - C))))) / fma(B_m, B_m, (-4.0 * (A * C))))) * -sqrt(2.0);
	} else {
		tmp = (sqrt((2.0 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.5e+55)
		tmp = Float64(sqrt(Float64(Float64(F * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F));
	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, 3.5e+55], N[(N[Sqrt[N[(N[(F * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.5000000000000001e55

   1. Initial program 25.3%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 7.7

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

   5. Applied rewrites 7.7%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

   8. Applied rewrites 29.1%

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

   if 3.5000000000000001e55 < B

   1. Initial program 7.9%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 47.0

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

   5. Applied rewrites 47.0%

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

   7. Applied rewrites 67.9%

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

   9. Applied rewrites 68.1%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 2.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(B\_m, C\right) + C\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{-B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.6e-19)
   (* (/ (sqrt 2.0) (- B_m)) (sqrt (* (+ (hypot B_m C) C) F)))
   (* (/ (sqrt F) (- B_m)) (sqrt (* (+ (hypot C B_m) C) 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.6e-19) {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((hypot(B_m, C) + C) * F));
	} else {
		tmp = (sqrt(F) / -B_m) * sqrt(((hypot(C, B_m) + C) * 2.0));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.6e-19) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt(((Math.hypot(B_m, C) + C) * F));
	} else {
		tmp = (Math.sqrt(F) / -B_m) * Math.sqrt(((Math.hypot(C, B_m) + C) * 2.0));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 2.6e-19:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt(((math.hypot(B_m, C) + C) * F))
	else:
		tmp = (math.sqrt(F) / -B_m) * math.sqrt(((math.hypot(C, B_m) + C) * 2.0))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 2.6e-19)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(Float64(hypot(B_m, C) + C) * F)));
	else
		tmp = Float64(Float64(sqrt(F) / Float64(-B_m)) * sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 2.6e-19)
		tmp = (sqrt(2.0) / -B_m) * sqrt(((hypot(B_m, C) + C) * F));
	else
		tmp = (sqrt(F) / -B_m) * sqrt(((hypot(C, B_m) + C) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 2.6e-19], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.60000000000000013e-19

   1. Initial program 27.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 19.9

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

   5. Applied rewrites 19.9%

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

   if 2.60000000000000013e-19 < F

   1. Initial program 15.2%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 12.2

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

   5. Applied rewrites 12.2%

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

   7. Applied rewrites 22.1%

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

   9. Applied rewrites 22.2%

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

   11. Applied rewrites 22.0%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{-B} \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 37.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{\sqrt{2}}{-B\_m} \cdot \sqrt{\left(\mathsf{hypot}\left(B\_m, C\right) + C\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + B\_m\right)} \cdot \left(-\sqrt{F}\right)}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.8e+88)
   (* (/ (sqrt 2.0) (- B_m)) (sqrt (* (+ (hypot B_m C) C) F)))
   (/ (* (sqrt (* 2.0 (+ C 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 tmp;
	if (B_m <= 8.8e+88) {
		tmp = (sqrt(2.0) / -B_m) * sqrt(((hypot(B_m, C) + C) * F));
	} else {
		tmp = (sqrt((2.0 * (C + 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 tmp;
	if (B_m <= 8.8e+88) {
		tmp = (Math.sqrt(2.0) / -B_m) * Math.sqrt(((Math.hypot(B_m, C) + C) * F));
	} else {
		tmp = (Math.sqrt((2.0 * (C + 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):
	tmp = 0
	if B_m <= 8.8e+88:
		tmp = (math.sqrt(2.0) / -B_m) * math.sqrt(((math.hypot(B_m, C) + C) * F))
	else:
		tmp = (math.sqrt((2.0 * (C + 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)
	tmp = 0.0
	if (B_m <= 8.8e+88)
		tmp = Float64(Float64(sqrt(2.0) / Float64(-B_m)) * sqrt(Float64(Float64(hypot(B_m, C) + C) * F)));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(C + B_m))) * Float64(-sqrt(F))) / 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 <= 8.8e+88)
		tmp = (sqrt(2.0) / -B_m) * sqrt(((hypot(B_m, C) + C) * F));
	else
		tmp = (sqrt((2.0 * (C + 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_] := If[LessEqual[B$95$m, 8.8e+88], N[(N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 * N[(C + B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 8.80000000000000035e88

   1. Initial program 24.8%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 7.7

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

   5. Applied rewrites 7.7%

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

   if 8.80000000000000035e88 < B

   1. Initial program 7.0%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 79.5%

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

   9. Applied rewrites 79.7%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 77.9%

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

4. Final simplification 20.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{\sqrt{2}}{-B} \cdot \sqrt{\left(\mathsf{hypot}\left(B, C\right) + C\right) \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + B\right)} \cdot \left(-\sqrt{F}\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.3% accurate, 3.3× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ (* (sqrt (* 2.0 (+ (hypot C B_m) C))) (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) {
	return (sqrt((2.0 * (hypot(C, B_m) + C))) * sqrt(F)) / -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 * (Math.hypot(C, B_m) + C))) * Math.sqrt(F)) / -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 * (math.hypot(C, B_m) + C))) * math.sqrt(F)) / -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(Float64(2.0 * Float64(hypot(C, B_m) + C))) * sqrt(F)) / Float64(-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 * (hypot(C, B_m) + C))) * sqrt(F)) / -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[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 21.5%

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

3. Taylor expanded in A around 0

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

   1. mul-1-neg N/A

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

   2. distribute-lft-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. unpow2 N/A

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

   13. unpow2 N/A

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

   14. lower-hypot.f64 16.3

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

5. Applied rewrites 16.3%

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

7. Applied rewrites 21.6%

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

9. Applied rewrites 21.7%

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

Alternative 13: 42.3% accurate, 3.3× speedup

Math

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

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 (+ (hypot C B_m) C))) (- B_m)) (sqrt F)))

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 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
}

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 * (Math.hypot(C, B_m) + C))) / -B_m) * Math.sqrt(F);
}

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 * (math.hypot(C, B_m) + C))) / -B_m) * math.sqrt(F)

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(Float64(2.0 * Float64(hypot(C, B_m) + C))) / Float64(-B_m)) * sqrt(F))
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 * (hypot(C, B_m) + C))) / -B_m) * sqrt(F);
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[N[(2.0 * N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 21.5%

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

3. Taylor expanded in A around 0

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

   1. mul-1-neg N/A

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

   2. distribute-lft-neg-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. unpow2 N/A

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

   13. unpow2 N/A

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

   14. lower-hypot.f64 16.3

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

5. Applied rewrites 16.3%

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

7. Applied rewrites 21.6%

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

9. Applied rewrites 21.7%

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

Alternative 14: 37.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + B\_m\right)} \cdot \left(-\sqrt{F}\right)}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{B\_m} \cdot \sqrt{C}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (<= C 4.3e+189)
   (/ (* (sqrt (* 2.0 (+ C B_m))) (- (sqrt F))) B_m)
   (* (* (/ -2.0 B_m) (sqrt C)) (sqrt F))))

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 (C <= 4.3e+189) {
		tmp = (sqrt((2.0 * (C + B_m))) * -sqrt(F)) / B_m;
	} else {
		tmp = ((-2.0 / B_m) * sqrt(C)) * sqrt(F);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 4.3d+189) then
        tmp = (sqrt((2.0d0 * (c + b_m))) * -sqrt(f)) / b_m
    else
        tmp = (((-2.0d0) / b_m) * sqrt(c)) * sqrt(f)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.3e+189) {
		tmp = (Math.sqrt((2.0 * (C + B_m))) * -Math.sqrt(F)) / B_m;
	} else {
		tmp = ((-2.0 / B_m) * Math.sqrt(C)) * Math.sqrt(F);
	}
	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 C <= 4.3e+189:
		tmp = (math.sqrt((2.0 * (C + B_m))) * -math.sqrt(F)) / B_m
	else:
		tmp = ((-2.0 / B_m) * math.sqrt(C)) * math.sqrt(F)
	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 (C <= 4.3e+189)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(C + B_m))) * Float64(-sqrt(F))) / B_m);
	else
		tmp = Float64(Float64(Float64(-2.0 / B_m) * sqrt(C)) * sqrt(F));
	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 (C <= 4.3e+189)
		tmp = (sqrt((2.0 * (C + B_m))) * -sqrt(F)) / B_m;
	else
		tmp = ((-2.0 / B_m) * sqrt(C)) * sqrt(F);
	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[C, 4.3e+189], N[(N[(N[Sqrt[N[(2.0 * N[(C + B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision] / B$95$m), $MachinePrecision], N[(N[(N[(-2.0 / B$95$m), $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 4.29999999999999998e189

   1. Initial program 23.8%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 18.1

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

   5. Applied rewrites 18.1%

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

   7. Applied rewrites 23.6%

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

   9. Applied rewrites 23.7%

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

   10. Taylor expanded in C around 0

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

   12. Applied rewrites 21.5%

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

   if 4.29999999999999998e189 < C

   1. Initial program 2.2%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 0.9

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

   5. Applied rewrites 0.9%

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

   7. Applied rewrites 4.7%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 0.9%

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

   12. Applied rewrites 4.6%

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

4. Final simplification 19.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(C + B\right)} \cdot \left(-\sqrt{F}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{B} \cdot \sqrt{C}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{B\_m} \cdot \sqrt{C}\right) \cdot \sqrt{F}\\ \end{array} \end{array} \]

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 (<= C 4.3e+189)
   (/ (sqrt (* F 2.0)) (- (sqrt B_m)))
   (* (* (/ -2.0 B_m) (sqrt C)) (sqrt F))))

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 (C <= 4.3e+189) {
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	} else {
		tmp = ((-2.0 / B_m) * sqrt(C)) * sqrt(F);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b_m, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 4.3d+189) then
        tmp = sqrt((f * 2.0d0)) / -sqrt(b_m)
    else
        tmp = (((-2.0d0) / b_m) * sqrt(c)) * sqrt(f)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 4.3e+189) {
		tmp = Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
	} else {
		tmp = ((-2.0 / B_m) * Math.sqrt(C)) * Math.sqrt(F);
	}
	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 C <= 4.3e+189:
		tmp = math.sqrt((F * 2.0)) / -math.sqrt(B_m)
	else:
		tmp = ((-2.0 / B_m) * math.sqrt(C)) * math.sqrt(F)
	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 (C <= 4.3e+189)
		tmp = Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)));
	else
		tmp = Float64(Float64(Float64(-2.0 / B_m) * sqrt(C)) * sqrt(F));
	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 (C <= 4.3e+189)
		tmp = sqrt((F * 2.0)) / -sqrt(B_m);
	else
		tmp = ((-2.0 / B_m) * sqrt(C)) * sqrt(F);
	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[C, 4.3e+189], N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision], N[(N[(N[(-2.0 / B$95$m), $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 4.29999999999999998e189

   1. Initial program 23.8%

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

   3. Taylor expanded in B around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 15.5

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

   5. Applied rewrites 15.5%

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

   7. Applied rewrites 15.6%

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

   9. Applied rewrites 22.0%

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

   if 4.29999999999999998e189 < C

   1. Initial program 2.2%

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

   3. Taylor expanded in A around 0

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 0.9

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

   5. Applied rewrites 0.9%

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

   7. Applied rewrites 4.7%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 0.9%

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

   12. Applied rewrites 4.6%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{\sqrt{F \cdot 2}}{-\sqrt{B}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-2}{B} \cdot \sqrt{C}\right) \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.0% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.5%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 14.1

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

5. Applied rewrites 14.1%

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

7. Applied rewrites 14.1%

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

9. Applied rewrites 19.8%

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

10. Final simplification 19.8%

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

Alternative 17: 26.5% accurate, 16.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.5%

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

3. Taylor expanded in B around inf

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-/.f64 14.1

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

5. Applied rewrites 14.1%

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

7. Applied rewrites 14.1%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))