ABCF->ab-angle b

Percentage Accurate: 19.2% → 40.6%

Time: 9.1s

Alternatives: 10

Speedup: 9.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.2% 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: 40.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0}\\ t_2 := -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_0))
        (t_2 (* -1.0 (pow (* (/ F C) -1.0) 0.5))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -2e-170)
       t_1
       (if (<= t_1 0.0)
         t_2
         (if (<= t_1 INFINITY)
           (sqrt
            (*
             (/
              (* F (- (+ A C) (sqrt (fma B B (* (- A C) (- A C))))))
              (fma B B (* -4.0 (* A C))))
             2.0))
           t_2))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
	double t_2 = -1.0 * pow(((F / C) * -1.0), 0.5);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -2e-170) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt((((F * ((A + C) - sqrt(fma(B, B, ((A - C) * (A - C)))))) / fma(B, B, (-4.0 * (A * C)))) * 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = 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)
	t_2 = Float64(-1.0 * (Float64(Float64(F / C) * -1.0) ^ 0.5))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -2e-170)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B, B, Float64(Float64(A - C) * Float64(A - C)))))) / fma(B, B, Float64(-4.0 * Float64(A * C)))) * 2.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-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]}, Block[{t$95$2 = N[(-1.0 * N[Power[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -2e-170], t$95$1, If[LessEqual[t$95$1, 0.0], t$95$2, If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -1.99999999999999997e-170 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0 or +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 27.4

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

   6. Applied rewrites 27.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 27.5

         \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5} \]

   8. Applied rewrites 27.5%

      \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{\color{blue}{0.5}} \]

   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))) < -1.99999999999999997e-170

   1. Initial program 19.2%

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

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

   1. Initial program 19.2%

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

   2. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   7. Applied rewrites 5.0%

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

Alternative 2: 40.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5}\\ t_2 := {B}^{2} - t\_0\\ t_3 := \frac{-\sqrt{\left(2 \cdot \left(t\_2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_2}\\ t_4 := B \cdot B - t\_0\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t\_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (* -1.0 (pow (* (/ F C) -1.0) 0.5)))
        (t_2 (- (pow B 2.0) t_0))
        (t_3
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_2 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_2))
        (t_4 (- (* B B) t_0)))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 -2e-170)
       (/
        (-
         (sqrt
          (*
           (* 2.0 (* t_4 F))
           (- (+ A C) (sqrt (fma (- A C) (- A C) (* B B)))))))
        t_4)
       (if (<= t_3 0.0)
         t_1
         (if (<= t_3 INFINITY)
           (sqrt
            (*
             (/
              (* F (- (+ A C) (sqrt (fma B B (* (- A C) (- A C))))))
              (fma B B (* -4.0 (* A C))))
             2.0))
           t_1))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = -1.0 * pow(((F / C) * -1.0), 0.5);
	double t_2 = pow(B, 2.0) - t_0;
	double t_3 = -sqrt(((2.0 * (t_2 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_2;
	double t_4 = (B * B) - t_0;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= -2e-170) {
		tmp = -sqrt(((2.0 * (t_4 * F)) * ((A + C) - sqrt(fma((A - C), (A - C), (B * B)))))) / t_4;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((((F * ((A + C) - sqrt(fma(B, B, ((A - C) * (A - C)))))) / fma(B, B, (-4.0 * (A * C)))) * 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(-1.0 * (Float64(Float64(F / C) * -1.0) ^ 0.5))
	t_2 = Float64((B ^ 2.0) - t_0)
	t_3 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_2)
	t_4 = Float64(Float64(B * B) - t_0)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_3 <= -2e-170)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))) / t_4);
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= Inf)
		tmp = sqrt(Float64(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B, B, Float64(Float64(A - C) * Float64(A - C)))))) / fma(B, B, Float64(-4.0 * Float64(A * C)))) * 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[Power[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * 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$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(B * B), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$1, If[LessEqual[t$95$3, -2e-170], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, Infinity], N[Sqrt[N[(N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0 or -1.99999999999999997e-170 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0 or +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 27.4

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

   6. Applied rewrites 27.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 27.5

         \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5} \]

   8. Applied rewrites 27.5%

      \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{\color{blue}{0.5}} \]

   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))) < -1.99999999999999997e-170

   1. Initial program 19.2%

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

   3. Applied rewrites 19.2%

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

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

   1. Initial program 19.2%

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

   2. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   7. Applied rewrites 5.0%

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

Alternative 3: 38.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot 2}\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{-\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_1}\\ t_3 := -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+244}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;-1 \cdot t\_0\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0
         (sqrt
          (*
           (/
            (* F (- (+ A C) (sqrt (fma B B (* (- A C) (- A C))))))
            (fma B B (* -4.0 (* A C))))
           2.0)))
        (t_1 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_2
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_1 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_1))
        (t_3 (* -1.0 (pow (* (/ F C) -1.0) 0.5))))
   (if (<= t_2 -5e+244)
     t_3
     (if (<= t_2 -2e-170)
       (* -1.0 t_0)
       (if (<= t_2 0.0) t_3 (if (<= t_2 INFINITY) t_0 t_3))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = sqrt((((F * ((A + C) - sqrt(fma(B, B, ((A - C) * (A - C)))))) / fma(B, B, (-4.0 * (A * C)))) * 2.0));
	double t_1 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_2 = -sqrt(((2.0 * (t_1 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_1;
	double t_3 = -1.0 * pow(((F / C) * -1.0), 0.5);
	double tmp;
	if (t_2 <= -5e+244) {
		tmp = t_3;
	} else if (t_2 <= -2e-170) {
		tmp = -1.0 * t_0;
	} else if (t_2 <= 0.0) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = sqrt(Float64(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B, B, Float64(Float64(A - C) * Float64(A - C)))))) / fma(B, B, Float64(-4.0 * Float64(A * C)))) * 2.0))
	t_1 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_2 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_1)
	t_3 = Float64(-1.0 * (Float64(Float64(F / C) * -1.0) ^ 0.5))
	tmp = 0.0
	if (t_2 <= -5e+244)
		tmp = t_3;
	elseif (t_2 <= -2e-170)
		tmp = Float64(-1.0 * t_0);
	elseif (t_2 <= 0.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_0;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 * N[Power[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+244], t$95$3, If[LessEqual[t$95$2, -2e-170], N[(-1.0 * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$3, If[LessEqual[t$95$2, Infinity], t$95$0, t$95$3]]]]]]]]

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))) < -5.00000000000000022e244 or -1.99999999999999997e-170 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0 or +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 27.4

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

   6. Applied rewrites 27.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 27.5

         \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5} \]

   8. Applied rewrites 27.5%

      \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{\color{blue}{0.5}} \]

   if -5.00000000000000022e244 < (/.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))) < -1.99999999999999997e-170

   1. Initial program 19.2%

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

   2. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 17.9%

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

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

   1. Initial program 19.2%

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

   2. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   7. Applied rewrites 5.0%

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

Alternative 4: 30.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -3.2 \cdot 10^{-181}:\\ \;\;\;\;\sqrt{\frac{F \cdot \left(\left(A + C\right) - \sqrt{\mathsf{fma}\left(B, B, \left(A - C\right) \cdot \left(A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)} \cdot 2}\\ \mathbf{elif}\;C \leq 5.5 \cdot 10^{-91}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C -3.2e-181)
   (sqrt
    (*
     (/
      (* F (- (+ A C) (sqrt (fma B B (* (- A C) (- A C))))))
      (fma B B (* -4.0 (* A C))))
     2.0))
   (if (<= C 5.5e-91)
     (* -1.0 (* (/ (sqrt 2.0) B) (sqrt (* F (- A (sqrt (fma A A (* B B))))))))
     (* -1.0 (pow (* (/ F C) -1.0) 0.5)))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -3.2e-181) {
		tmp = sqrt((((F * ((A + C) - sqrt(fma(B, B, ((A - C) * (A - C)))))) / fma(B, B, (-4.0 * (A * C)))) * 2.0));
	} else if (C <= 5.5e-91) {
		tmp = -1.0 * ((sqrt(2.0) / B) * sqrt((F * (A - sqrt(fma(A, A, (B * B)))))));
	} else {
		tmp = -1.0 * pow(((F / C) * -1.0), 0.5);
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= -3.2e-181)
		tmp = sqrt(Float64(Float64(Float64(F * Float64(Float64(A + C) - sqrt(fma(B, B, Float64(Float64(A - C) * Float64(A - C)))))) / fma(B, B, Float64(-4.0 * Float64(A * C)))) * 2.0));
	elseif (C <= 5.5e-91)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B))))))));
	else
		tmp = Float64(-1.0 * (Float64(Float64(F / C) * -1.0) ^ 0.5));
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, -3.2e-181], N[Sqrt[N[(N[(N[(F * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(B * B + N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(B * B + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[C, 5.5e-91], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[Power[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -3.2000000000000002e-181

   1. Initial program 19.2%

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

   2. Taylor expanded in C around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around -inf

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. sqrt-unprod N/A

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

   7. Applied rewrites 5.0%

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

   if -3.2000000000000002e-181 < C < 5.49999999999999965e-91

   1. Initial program 19.2%

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

   2. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 9.5

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

   4. Applied rewrites 9.5%

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

   if 5.49999999999999965e-91 < C

   1. Initial program 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 27.4

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

   6. Applied rewrites 27.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 27.5

         \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5} \]

   8. Applied rewrites 27.5%

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

Alternative 5: 28.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0}\\ t_2 := -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+244}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(A, A, B \cdot B\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_0))
        (t_2 (* -1.0 (pow (* (/ F C) -1.0) 0.5))))
   (if (<= t_1 -5e+244)
     t_2
     (if (<= t_1 -2e-170)
       (*
        -1.0
        (* (/ (sqrt 2.0) B) (sqrt (* F (- A (sqrt (fma A A (* B B))))))))
       t_2))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
	double t_2 = -1.0 * pow(((F / C) * -1.0), 0.5);
	double tmp;
	if (t_1 <= -5e+244) {
		tmp = t_2;
	} else if (t_1 <= -2e-170) {
		tmp = -1.0 * ((sqrt(2.0) / B) * sqrt((F * (A - sqrt(fma(A, A, (B * B)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = 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)
	t_2 = Float64(-1.0 * (Float64(Float64(F / C) * -1.0) ^ 0.5))
	tmp = 0.0
	if (t_1 <= -5e+244)
		tmp = t_2;
	elseif (t_1 <= -2e-170)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(2.0) / B) * sqrt(Float64(F * Float64(A - sqrt(fma(A, A, Float64(B * B))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-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]}, Block[{t$95$2 = N[(-1.0 * N[Power[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+244], t$95$2, If[LessEqual[t$95$1, -2e-170], N[(-1.0 * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * N[Sqrt[N[(F * N[(A - N[Sqrt[N[(A * A + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 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))) < -5.00000000000000022e244 or -1.99999999999999997e-170 < (/.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 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 27.4

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

   6. Applied rewrites 27.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 27.5

         \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5} \]

   8. Applied rewrites 27.5%

      \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{\color{blue}{0.5}} \]

   if -5.00000000000000022e244 < (/.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))) < -1.99999999999999997e-170

   1. Initial program 19.2%

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

   2. Taylor expanded in C around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 9.5

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

   4. Applied rewrites 9.5%

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

Alternative 6: 27.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0}\\ t_2 := -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+57}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B\right) \cdot F\right)\right) \cdot \left(-1 \cdot B\right)}}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_0))
        (t_2 (* -1.0 (pow (* (/ F C) -1.0) 0.5))))
   (if (<= t_1 -5e+57)
     t_2
     (if (<= t_1 -2e-170)
       (/ (- (sqrt (* (* 2.0 (* (* B B) F)) (* -1.0 B)))) (* B B))
       t_2))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
	double t_2 = -1.0 * pow(((F / C) * -1.0), 0.5);
	double tmp;
	if (t_1 <= -5e+57) {
		tmp = t_2;
	} else if (t_1 <= -2e-170) {
		tmp = -sqrt(((2.0 * ((B * B) * F)) * (-1.0 * B))) / (B * B);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    t_1 = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
    t_2 = (-1.0d0) * (((f / c) * (-1.0d0)) ** 0.5d0)
    if (t_1 <= (-5d+57)) then
        tmp = t_2
    else if (t_1 <= (-2d-170)) then
        tmp = -sqrt(((2.0d0 * ((b * b) * f)) * ((-1.0d0) * b))) / (b * b)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
	double t_2 = -1.0 * Math.pow(((F / C) * -1.0), 0.5);
	double tmp;
	if (t_1 <= -5e+57) {
		tmp = t_2;
	} else if (t_1 <= -2e-170) {
		tmp = -Math.sqrt(((2.0 * ((B * B) * F)) * (-1.0 * B))) / (B * B);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	t_1 = -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0
	t_2 = -1.0 * math.pow(((F / C) * -1.0), 0.5)
	tmp = 0
	if t_1 <= -5e+57:
		tmp = t_2
	elif t_1 <= -2e-170:
		tmp = -math.sqrt(((2.0 * ((B * B) * F)) * (-1.0 * B))) / (B * B)
	else:
		tmp = t_2
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = 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)
	t_2 = Float64(-1.0 * (Float64(Float64(F / C) * -1.0) ^ 0.5))
	tmp = 0.0
	if (t_1 <= -5e+57)
		tmp = t_2;
	elseif (t_1 <= -2e-170)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64(B * B) * F)) * Float64(-1.0 * B)))) / Float64(B * B));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
	t_2 = -1.0 * (((F / C) * -1.0) ^ 0.5);
	tmp = 0.0;
	if (t_1 <= -5e+57)
		tmp = t_2;
	elseif (t_1 <= -2e-170)
		tmp = -sqrt(((2.0 * ((B * B) * F)) * (-1.0 * B))) / (B * B);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-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]}, Block[{t$95$2 = N[(-1.0 * N[Power[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+57], t$95$2, If[LessEqual[t$95$1, -2e-170], N[((-N[Sqrt[N[(N[(2.0 * N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(-1.0 * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 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))) < -4.99999999999999972e57 or -1.99999999999999997e-170 < (/.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 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 27.4

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

   6. Applied rewrites 27.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 27.5

         \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{0.5} \]

   8. Applied rewrites 27.5%

      \[\leadsto -1 \cdot {\left(\frac{F}{C} \cdot -1\right)}^{\color{blue}{0.5}} \]

   if -4.99999999999999972e57 < (/.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))) < -1.99999999999999997e-170

   1. Initial program 19.2%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 5.8

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

   4. Applied rewrites 5.8%

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

   5. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 6.1

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

   7. Applied rewrites 6.1%

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

   8. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 5.3

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

   10. Applied rewrites 5.3%

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

Alternative 7: 27.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0}\\ t_2 := -1 \cdot \sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+57}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(B \cdot B\right) \cdot F\right)\right) \cdot \left(-1 \cdot B\right)}}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_0))
        (t_2 (* -1.0 (sqrt (* (/ F C) -1.0)))))
   (if (<= t_1 -5e+57)
     t_2
     (if (<= t_1 -2e-170)
       (/ (- (sqrt (* (* 2.0 (* (* B B) F)) (* -1.0 B)))) (* B B))
       t_2))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
	double t_2 = -1.0 * sqrt(((F / C) * -1.0));
	double tmp;
	if (t_1 <= -5e+57) {
		tmp = t_2;
	} else if (t_1 <= -2e-170) {
		tmp = -sqrt(((2.0 * ((B * B) * F)) * (-1.0 * B))) / (B * B);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    t_1 = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
    t_2 = (-1.0d0) * sqrt(((f / c) * (-1.0d0)))
    if (t_1 <= (-5d+57)) then
        tmp = t_2
    else if (t_1 <= (-2d-170)) then
        tmp = -sqrt(((2.0d0 * ((b * b) * f)) * ((-1.0d0) * b))) / (b * b)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
	double t_2 = -1.0 * Math.sqrt(((F / C) * -1.0));
	double tmp;
	if (t_1 <= -5e+57) {
		tmp = t_2;
	} else if (t_1 <= -2e-170) {
		tmp = -Math.sqrt(((2.0 * ((B * B) * F)) * (-1.0 * B))) / (B * B);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	t_1 = -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0
	t_2 = -1.0 * math.sqrt(((F / C) * -1.0))
	tmp = 0
	if t_1 <= -5e+57:
		tmp = t_2
	elif t_1 <= -2e-170:
		tmp = -math.sqrt(((2.0 * ((B * B) * F)) * (-1.0 * B))) / (B * B)
	else:
		tmp = t_2
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = 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)
	t_2 = Float64(-1.0 * sqrt(Float64(Float64(F / C) * -1.0)))
	tmp = 0.0
	if (t_1 <= -5e+57)
		tmp = t_2;
	elseif (t_1 <= -2e-170)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64(B * B) * F)) * Float64(-1.0 * B)))) / Float64(B * B));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
	t_2 = -1.0 * sqrt(((F / C) * -1.0));
	tmp = 0.0;
	if (t_1 <= -5e+57)
		tmp = t_2;
	elseif (t_1 <= -2e-170)
		tmp = -sqrt(((2.0 * ((B * B) * F)) * (-1.0 * B))) / (B * B);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-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]}, Block[{t$95$2 = N[(-1.0 * N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+57], t$95$2, If[LessEqual[t$95$1, -2e-170], N[((-N[Sqrt[N[(N[(2.0 * N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(-1.0 * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 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))) < -4.99999999999999972e57 or -1.99999999999999997e-170 < (/.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 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 27.4

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

   6. Applied rewrites 27.4%

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

   if -4.99999999999999972e57 < (/.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))) < -1.99999999999999997e-170

   1. Initial program 19.2%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 5.8

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

   4. Applied rewrites 5.8%

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

   5. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 6.1

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

   7. Applied rewrites 6.1%

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

   8. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 5.3

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

   10. Applied rewrites 5.3%

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

Alternative 8: 27.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0}\\ t_2 := -1 \cdot \sqrt{\frac{F}{C} \cdot -1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+57}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-110}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(B \cdot B\right) \cdot B\right) \cdot \left(-2 \cdot F\right)}}{B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_1
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_0 F))
             (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
          t_0))
        (t_2 (* -1.0 (sqrt (* (/ F C) -1.0)))))
   (if (<= t_1 -5e+57)
     t_2
     (if (<= t_1 -5e-110)
       (/ (- (sqrt (* (* (* B B) B) (* -2.0 F)))) (* B B))
       t_2))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
	double t_2 = -1.0 * sqrt(((F / C) * -1.0));
	double tmp;
	if (t_1 <= -5e+57) {
		tmp = t_2;
	} else if (t_1 <= -5e-110) {
		tmp = -sqrt((((B * B) * B) * (-2.0 * F))) / (B * B);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(a, b, c, f)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    t_1 = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
    t_2 = (-1.0d0) * sqrt(((f / c) * (-1.0d0)))
    if (t_1 <= (-5d+57)) then
        tmp = t_2
    else if (t_1 <= (-5d-110)) then
        tmp = -sqrt((((b * b) * b) * ((-2.0d0) * f))) / (b * b)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	double t_1 = -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
	double t_2 = -1.0 * Math.sqrt(((F / C) * -1.0));
	double tmp;
	if (t_1 <= -5e+57) {
		tmp = t_2;
	} else if (t_1 <= -5e-110) {
		tmp = -Math.sqrt((((B * B) * B) * (-2.0 * F))) / (B * B);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	t_1 = -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0
	t_2 = -1.0 * math.sqrt(((F / C) * -1.0))
	tmp = 0
	if t_1 <= -5e+57:
		tmp = t_2
	elif t_1 <= -5e-110:
		tmp = -math.sqrt((((B * B) * B) * (-2.0 * F))) / (B * B)
	else:
		tmp = t_2
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_1 = 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)
	t_2 = Float64(-1.0 * sqrt(Float64(Float64(F / C) * -1.0)))
	tmp = 0.0
	if (t_1 <= -5e+57)
		tmp = t_2;
	elseif (t_1 <= -5e-110)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(B * B) * B) * Float64(-2.0 * F)))) / Float64(B * B));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	t_1 = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
	t_2 = -1.0 * sqrt(((F / C) * -1.0));
	tmp = 0.0;
	if (t_1 <= -5e+57)
		tmp = t_2;
	elseif (t_1 <= -5e-110)
		tmp = -sqrt((((B * B) * B) * (-2.0 * F))) / (B * B);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-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]}, Block[{t$95$2 = N[(-1.0 * N[Sqrt[N[(N[(F / C), $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+57], t$95$2, If[LessEqual[t$95$1, -5e-110], N[((-N[Sqrt[N[(N[(N[(B * B), $MachinePrecision] * B), $MachinePrecision] * N[(-2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(B * B), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 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))) < -4.99999999999999972e57 or -5e-110 < (/.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 19.2%

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

   2. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. sqrt-unprod N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-/.f64 27.4

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

   6. Applied rewrites 27.4%

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

   if -4.99999999999999972e57 < (/.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))) < -5e-110

   1. Initial program 19.2%

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

   2. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 5.0

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

   4. Applied rewrites 5.0%

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

   5. Taylor expanded in A around 0

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

      1. pow2 N/A

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

      2. lift-*.f64 4.7

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

   7. Applied rewrites 4.7%

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

   8. Taylor expanded in B around inf

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

      1. lower-*.f64 4.6

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

   10. Applied rewrites 4.6%

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

Alternative 9: 27.4% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.2%

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

2. Taylor expanded in A around -inf

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

   1. lower-*.f64 N/A

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

   2. sqrt-unprod N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-sqrt.f64 0.0

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

4. Applied rewrites 0.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-unprod N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-/.f64 27.4

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

6. Applied rewrites 27.4%

   \[\leadsto -1 \cdot \color{blue}{\sqrt{\frac{F}{C} \cdot -1}} \]
7. Add Preprocessing

Alternative 10: 2.0% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.2%

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

2. Taylor expanded in B around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. sqrt-pow2 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 2.0

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

4. Applied rewrites 2.0%

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

5. Taylor expanded in B around 0

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

   1. sqrt-unprod N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-/.f64 2.0

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

7. Applied rewrites 2.0%

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

Reproduce

herbie shell --seed 2025127 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))