quadm (p42, negative)

Percentage Accurate: 52.6% → 85.2%

Time: 4.0s

Alternatives: 10

Speedup: 2.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

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

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

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

Java

public static double code(double a, double b, double c) {
	return (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-107}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+108}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-107)
   (- (/ c b))
   (if (<= b 3e+108)
     (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-107) {
		tmp = -(c / b);
	} else if (b <= 3e+108) {
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.7d-107)) then
        tmp = -(c / b)
    else if (b <= 3d+108) then
        tmp = (-b - sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-107) {
		tmp = -(c / b);
	} else if (b <= 3e+108) {
		tmp = (-b - Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-107:
		tmp = -(c / b)
	elif b <= 3e+108:
		tmp = (-b - math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-107)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 3e+108)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.7e-107)
		tmp = -(c / b);
	elseif (b <= 3e+108)
		tmp = (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-107], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 3e+108], N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7000000000000003e-107

   1. Initial program 19.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c}{b} \]

      3. lower-/.f64 82.6

         \[\leadsto -\frac{c}{b} \]

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -3.7000000000000003e-107 < b < 2.99999999999999984e108

   1. Initial program 82.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   if 2.99999999999999984e108 < b

   1. Initial program 53.8%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1} \cdot \frac{b}{a} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{c}{b} + \frac{-1 \cdot b}{\color{blue}{a}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \frac{\mathsf{neg}\left(b\right)}{a} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{c}{b} + \frac{-b}{a} \]

      7. lower-/.f64 95.7

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\frac{c}{b} + \frac{-b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-107}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-107)
   (- (/ c b))
   (if (<= b 4.6e-48)
     (/ (- (- b) (sqrt (* -4.0 (* c a)))) (+ a a))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-107) {
		tmp = -(c / b);
	} else if (b <= 4.6e-48) {
		tmp = (-b - sqrt((-4.0 * (c * a)))) / (a + a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.7d-107)) then
        tmp = -(c / b)
    else if (b <= 4.6d-48) then
        tmp = (-b - sqrt(((-4.0d0) * (c * a)))) / (a + a)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-107) {
		tmp = -(c / b);
	} else if (b <= 4.6e-48) {
		tmp = (-b - Math.sqrt((-4.0 * (c * a)))) / (a + a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-107:
		tmp = -(c / b)
	elif b <= 4.6e-48:
		tmp = (-b - math.sqrt((-4.0 * (c * a)))) / (a + a)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-107)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 4.6e-48)
		tmp = Float64(Float64(Float64(-b) - sqrt(Float64(-4.0 * Float64(c * a)))) / Float64(a + a));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.7e-107)
		tmp = -(c / b);
	elseif (b <= 4.6e-48)
		tmp = (-b - sqrt((-4.0 * (c * a)))) / (a + a);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-107], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 4.6e-48], N[(N[((-b) - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7000000000000003e-107

   1. Initial program 19.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c}{b} \]

      3. lower-/.f64 82.6

         \[\leadsto -\frac{c}{b} \]

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -3.7000000000000003e-107 < b < 4.6000000000000001e-48

   1. Initial program 78.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot \color{blue}{a}\right)}}{2 \cdot a} \]

      3. lower-*.f64 68.3

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot \color{blue}{a}\right)}}{2 \cdot a} \]

   4. Applied rewrites 68.3%

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}{\color{blue}{2 \cdot a}} \]

      2. count-2-rev N/A

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}{\color{blue}{a + a}} \]

      3. lower-+.f64 68.3

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}{\color{blue}{a + a}} \]

   6. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right)}}{a + a}} \]

   if 4.6000000000000001e-48 < b

   1. Initial program 68.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1} \cdot \frac{b}{a} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{c}{b} + \frac{-1 \cdot b}{\color{blue}{a}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \frac{\mathsf{neg}\left(b\right)}{a} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{c}{b} + \frac{-b}{a} \]

      7. lower-/.f64 87.5

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\frac{c}{b} + \frac{-b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10.5:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-40}:\\ \;\;\;\;-\left(-\sqrt{\frac{\frac{-1}{a}}{c}}\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -10.5)
   (- (/ c b))
   (if (<= b 9.2e-40)
     (- (* (- (sqrt (/ (/ -1.0 a) c))) c))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -10.5) {
		tmp = -(c / b);
	} else if (b <= 9.2e-40) {
		tmp = -(-sqrt(((-1.0 / a) / c)) * c);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-10.5d0)) then
        tmp = -(c / b)
    else if (b <= 9.2d-40) then
        tmp = -(-sqrt((((-1.0d0) / a) / c)) * c)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -10.5) {
		tmp = -(c / b);
	} else if (b <= 9.2e-40) {
		tmp = -(-Math.sqrt(((-1.0 / a) / c)) * c);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -10.5:
		tmp = -(c / b)
	elif b <= 9.2e-40:
		tmp = -(-math.sqrt(((-1.0 / a) / c)) * c)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -10.5)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 9.2e-40)
		tmp = Float64(-Float64(Float64(-sqrt(Float64(Float64(-1.0 / a) / c))) * c));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -10.5)
		tmp = -(c / b);
	elseif (b <= 9.2e-40)
		tmp = -(-sqrt(((-1.0 / a) / c)) * c);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -10.5], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 9.2e-40], (-N[((-N[Sqrt[N[(N[(-1.0 / a), $MachinePrecision] / c), $MachinePrecision]], $MachinePrecision]) * c), $MachinePrecision]), N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -10.5

   1. Initial program 13.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c}{b} \]

      3. lower-/.f64 90.7

         \[\leadsto -\frac{c}{b} \]

   4. Applied rewrites 90.7%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -10.5 < b < 9.2e-40

   1. Initial program 71.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(c \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) + \frac{1}{2} \cdot \frac{b}{a \cdot c}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -c \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) + \frac{1}{2} \cdot \frac{b}{a \cdot c}\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) + \frac{1}{2} \cdot \frac{b}{a \cdot c}\right) \cdot c \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) + \frac{1}{2} \cdot \frac{b}{a \cdot c}\right) \cdot c \]

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{b}{c \cdot a}, 0.5, -\sqrt{\frac{1}{c \cdot a} \cdot -1}\right) \cdot c} \]

   5. Taylor expanded in a around inf

      \[\leadsto -\left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right)\right) \cdot c \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right)\right) \cdot c \]

      2. sqrt-unprod N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{a \cdot c} \cdot -1}\right)\right) \cdot c \]

      3. *-commutative N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      4. lift-*.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      5. lift-/.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      6. lift-*.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      7. lift-sqrt.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      8. lift-neg.f64 60.1

         \[\leadsto -\left(-\sqrt{\frac{1}{c \cdot a} \cdot -1}\right) \cdot c \]

      9. lift-*.f64 N/A

         \[\leadsto -\left(-\sqrt{\frac{1}{c \cdot a} \cdot -1}\right) \cdot c \]

      10. lift-/.f64 N/A

          \[\leadsto -\left(-\sqrt{\frac{1}{c \cdot a} \cdot -1}\right) \cdot c \]

      11. associate-*l/ N/A

          \[\leadsto -\left(-\sqrt{\frac{1 \cdot -1}{c \cdot a}}\right) \cdot c \]

      12. metadata-eval N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

      13. lift-*.f64 N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

      14. *-commutative N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{a \cdot c}}\right) \cdot c \]

      15. lower-/.f64 N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{a \cdot c}}\right) \cdot c \]

      16. *-commutative N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

      17. lift-*.f64 60.1

          \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

   7. Applied rewrites 60.1%

      \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

      2. *-commutative N/A

         \[\leadsto -\left(-\sqrt{\frac{-1}{a \cdot c}}\right) \cdot c \]

      3. lower-/.f64 N/A

         \[\leadsto -\left(-\sqrt{\frac{-1}{a \cdot c}}\right) \cdot c \]

      4. associate-/r* N/A

         \[\leadsto -\left(-\sqrt{\frac{\frac{-1}{a}}{c}}\right) \cdot c \]

      5. lower-/.f64 N/A

         \[\leadsto -\left(-\sqrt{\frac{\frac{-1}{a}}{c}}\right) \cdot c \]

      6. lower-/.f64 60.9

         \[\leadsto -\left(-\sqrt{\frac{\frac{-1}{a}}{c}}\right) \cdot c \]

   9. Applied rewrites 60.9%

      \[\leadsto -\left(-\sqrt{\frac{\frac{-1}{a}}{c}}\right) \cdot c \]

   if 9.2e-40 < b

   1. Initial program 68.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1} \cdot \frac{b}{a} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{c}{b} + \frac{-1 \cdot b}{\color{blue}{a}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \frac{\mathsf{neg}\left(b\right)}{a} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{c}{b} + \frac{-b}{a} \]

      7. lower-/.f64 88.1

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\frac{c}{b} + \frac{-b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-107}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-40}:\\ \;\;\;\;-\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-107)
   (- (/ c b))
   (if (<= b 9.2e-40)
     (- (* (- (sqrt (/ -1.0 (* c a)))) c))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-107) {
		tmp = -(c / b);
	} else if (b <= 9.2e-40) {
		tmp = -(-sqrt((-1.0 / (c * a))) * c);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.7d-107)) then
        tmp = -(c / b)
    else if (b <= 9.2d-40) then
        tmp = -(-sqrt(((-1.0d0) / (c * a))) * c)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-107) {
		tmp = -(c / b);
	} else if (b <= 9.2e-40) {
		tmp = -(-Math.sqrt((-1.0 / (c * a))) * c);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-107:
		tmp = -(c / b)
	elif b <= 9.2e-40:
		tmp = -(-math.sqrt((-1.0 / (c * a))) * c)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-107)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 9.2e-40)
		tmp = Float64(-Float64(Float64(-sqrt(Float64(-1.0 / Float64(c * a)))) * c));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.7e-107)
		tmp = -(c / b);
	elseif (b <= 9.2e-40)
		tmp = -(-sqrt((-1.0 / (c * a))) * c);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-107], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 9.2e-40], (-N[((-N[Sqrt[N[(-1.0 / N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * c), $MachinePrecision]), N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7000000000000003e-107

   1. Initial program 19.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c}{b} \]

      3. lower-/.f64 82.6

         \[\leadsto -\frac{c}{b} \]

   4. Applied rewrites 82.6%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -3.7000000000000003e-107 < b < 9.2e-40

   1. Initial program 78.4%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(c \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) + \frac{1}{2} \cdot \frac{b}{a \cdot c}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -c \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) + \frac{1}{2} \cdot \frac{b}{a \cdot c}\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) + \frac{1}{2} \cdot \frac{b}{a \cdot c}\right) \cdot c \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right) + \frac{1}{2} \cdot \frac{b}{a \cdot c}\right) \cdot c \]

   4. Applied rewrites 65.8%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{b}{c \cdot a}, 0.5, -\sqrt{\frac{1}{c \cdot a} \cdot -1}\right) \cdot c} \]

   5. Taylor expanded in a around inf

      \[\leadsto -\left(-1 \cdot \left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right)\right) \cdot c \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{a \cdot c}} \cdot \sqrt{-1}\right)\right) \cdot c \]

      2. sqrt-unprod N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{a \cdot c} \cdot -1}\right)\right) \cdot c \]

      3. *-commutative N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      4. lift-*.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      5. lift-/.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      6. lift-*.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      7. lift-sqrt.f64 N/A

         \[\leadsto -\left(\mathsf{neg}\left(\sqrt{\frac{1}{c \cdot a} \cdot -1}\right)\right) \cdot c \]

      8. lift-neg.f64 65.5

         \[\leadsto -\left(-\sqrt{\frac{1}{c \cdot a} \cdot -1}\right) \cdot c \]

      9. lift-*.f64 N/A

         \[\leadsto -\left(-\sqrt{\frac{1}{c \cdot a} \cdot -1}\right) \cdot c \]

      10. lift-/.f64 N/A

          \[\leadsto -\left(-\sqrt{\frac{1}{c \cdot a} \cdot -1}\right) \cdot c \]

      11. associate-*l/ N/A

          \[\leadsto -\left(-\sqrt{\frac{1 \cdot -1}{c \cdot a}}\right) \cdot c \]

      12. metadata-eval N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

      13. lift-*.f64 N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

      14. *-commutative N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{a \cdot c}}\right) \cdot c \]

      15. lower-/.f64 N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{a \cdot c}}\right) \cdot c \]

      16. *-commutative N/A

          \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

      17. lift-*.f64 65.5

          \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

   7. Applied rewrites 65.5%

      \[\leadsto -\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c \]

   if 9.2e-40 < b

   1. Initial program 68.3%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1} \cdot \frac{b}{a} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{c}{b} + \frac{-1 \cdot b}{\color{blue}{a}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \frac{\mathsf{neg}\left(b\right)}{a} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{c}{b} + \frac{-b}{a} \]

      7. lower-/.f64 88.1

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 88.1%

      \[\leadsto \color{blue}{\frac{c}{b} + \frac{-b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 71.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-122}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-122)
   (- (/ c b))
   (if (<= b 9.5e-137) (- (sqrt (- (/ c a)))) (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-122) {
		tmp = -(c / b);
	} else if (b <= 9.5e-137) {
		tmp = -sqrt(-(c / a));
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.5d-122)) then
        tmp = -(c / b)
    else if (b <= 9.5d-137) then
        tmp = -sqrt(-(c / a))
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-122) {
		tmp = -(c / b);
	} else if (b <= 9.5e-137) {
		tmp = -Math.sqrt(-(c / a));
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-122:
		tmp = -(c / b)
	elif b <= 9.5e-137:
		tmp = -math.sqrt(-(c / a))
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-122)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 9.5e-137)
		tmp = Float64(-sqrt(Float64(-Float64(c / a))));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e-122)
		tmp = -(c / b);
	elseif (b <= 9.5e-137)
		tmp = -sqrt(-(c / a));
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-122], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 9.5e-137], (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]), N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4999999999999998e-122

   1. Initial program 21.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c}{b} \]

      3. lower-/.f64 81.3

         \[\leadsto -\frac{c}{b} \]

   4. Applied rewrites 81.3%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -4.4999999999999998e-122 < b < 9.5000000000000007e-137

   1. Initial program 75.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} + \sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{b}{a}}, \sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b}{\color{blue}{a}}, \sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right) \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

      6. lower-/.f64 35.4

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

   4. Applied rewrites 35.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right)} \]

   5. Taylor expanded in c around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a}} \cdot \sqrt{-1} \]

      3. sqrt-prod N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      5. *-commutative N/A

         \[\leadsto -\sqrt{-1 \cdot \frac{c}{a}} \]

      6. mul-1-neg N/A

         \[\leadsto -\sqrt{\mathsf{neg}\left(\frac{c}{a}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto -\sqrt{-\frac{c}{a}} \]

      8. lift-/.f64 36.1

         \[\leadsto -\sqrt{-\frac{c}{a}} \]

   7. Applied rewrites 36.1%

      \[\leadsto -\sqrt{-\frac{c}{a}} \]

   if 9.5000000000000007e-137 < b

   1. Initial program 72.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{c}{b} + \color{blue}{-1} \cdot \frac{b}{a} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{c}{b} + \frac{-1 \cdot b}{\color{blue}{a}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{c}{b} + \frac{\mathsf{neg}\left(b\right)}{a} \]

      6. lift-neg.f64 N/A

         \[\leadsto \frac{c}{b} + \frac{-b}{a} \]

      7. lower-/.f64 81.4

         \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\frac{c}{b} + \frac{-b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-122}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 9.5 \cdot 10^{-137}:\\ \;\;\;\;-\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e-122)
   (- (/ c b))
   (if (<= b 9.5e-137) (- (sqrt (- (/ c a)))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-122) {
		tmp = -(c / b);
	} else if (b <= 9.5e-137) {
		tmp = -sqrt(-(c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4.5d-122)) then
        tmp = -(c / b)
    else if (b <= 9.5d-137) then
        tmp = -sqrt(-(c / a))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e-122) {
		tmp = -(c / b);
	} else if (b <= 9.5e-137) {
		tmp = -Math.sqrt(-(c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.5e-122:
		tmp = -(c / b)
	elif b <= 9.5e-137:
		tmp = -math.sqrt(-(c / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e-122)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 9.5e-137)
		tmp = Float64(-sqrt(Float64(-Float64(c / a))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.5e-122)
		tmp = -(c / b);
	elseif (b <= 9.5e-137)
		tmp = -sqrt(-(c / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e-122], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 9.5e-137], (-N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]), N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.4999999999999998e-122

   1. Initial program 21.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c}{b} \]

      3. lower-/.f64 81.3

         \[\leadsto -\frac{c}{b} \]

   4. Applied rewrites 81.3%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -4.4999999999999998e-122 < b < 9.5000000000000007e-137

   1. Initial program 75.1%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} + \sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\frac{b}{a}}, \sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b}{\color{blue}{a}}, \sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right) \]

      3. sqrt-unprod N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

      6. lower-/.f64 35.4

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

   4. Applied rewrites 35.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right)} \]

   5. Taylor expanded in c around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\sqrt{\frac{c}{a}} \cdot \sqrt{-1}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a}} \cdot \sqrt{-1} \]

      3. sqrt-prod N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      5. *-commutative N/A

         \[\leadsto -\sqrt{-1 \cdot \frac{c}{a}} \]

      6. mul-1-neg N/A

         \[\leadsto -\sqrt{\mathsf{neg}\left(\frac{c}{a}\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto -\sqrt{-\frac{c}{a}} \]

      8. lift-/.f64 36.1

         \[\leadsto -\sqrt{-\frac{c}{a}} \]

   7. Applied rewrites 36.1%

      \[\leadsto -\sqrt{-\frac{c}{a}} \]

   if 9.5000000000000007e-137 < b

   1. Initial program 72.0%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot b}{\color{blue}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{-b}{a} \]

      4. lower-/.f64 81.0

         \[\leadsto \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 81.0%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{-155}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-214}:\\ \;\;\;\;\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.45e-155)
   (- (/ c b))
   (if (<= b 1.45e-214) (sqrt (- (/ c a))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-155) {
		tmp = -(c / b);
	} else if (b <= 1.45e-214) {
		tmp = sqrt(-(c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.45d-155)) then
        tmp = -(c / b)
    else if (b <= 1.45d-214) then
        tmp = sqrt(-(c / a))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e-155) {
		tmp = -(c / b);
	} else if (b <= 1.45e-214) {
		tmp = Math.sqrt(-(c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.45e-155:
		tmp = -(c / b)
	elif b <= 1.45e-214:
		tmp = math.sqrt(-(c / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.45e-155)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.45e-214)
		tmp = sqrt(Float64(-Float64(c / a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.45e-155)
		tmp = -(c / b);
	elseif (b <= 1.45e-214)
		tmp = sqrt(-(c / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.45e-155], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.45e-214], N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.45000000000000005e-155

   1. Initial program 23.7%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c}{b} \]

      3. lower-/.f64 78.4

         \[\leadsto -\frac{c}{b} \]

   4. Applied rewrites 78.4%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -1.45000000000000005e-155 < b < 1.44999999999999993e-214

   1. Initial program 75.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{2 \cdot a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot \color{blue}{a}\right)}}{2 \cdot a} \]

      3. lower-*.f64 75.5

         \[\leadsto \frac{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot \color{blue}{a}\right)}}{2 \cdot a} \]

   4. Applied rewrites 75.5%

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

   5. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   6. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{-1 \cdot \frac{c}{a}} \]

      4. mul-1-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\frac{c}{a}\right)} \]

      5. lower-neg.f64 N/A

         \[\leadsto \sqrt{-\frac{c}{a}} \]

      6. lift-/.f64 36.7

         \[\leadsto \sqrt{-\frac{c}{a}} \]

   7. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\sqrt{-\frac{c}{a}}} \]

   if 1.44999999999999993e-214 < b

   1. Initial program 72.5%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot b}{\color{blue}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{-b}{a} \]

      4. lower-/.f64 74.8

         \[\leadsto \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 74.8%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (- (/ c b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -(c / b);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = -(c / b)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -(c / b);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -(c / b)
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(-Float64(c / b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-310)
		tmp = -(c / b);
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], (-N[(c / b), $MachinePrecision]), N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 32.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{c}{b} \]

      3. lower-/.f64 66.5

         \[\leadsto -\frac{c}{b} \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{-\frac{c}{b}} \]

   if -3.999999999999988e-310 < b

   1. Initial program 72.9%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot b}{\color{blue}{a}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\right)}{a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{-b}{a} \]

      4. lower-/.f64 67.3

         \[\leadsto \frac{-b}{\color{blue}{a}} \]

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 34.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ -\frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (- (/ c b)))

C

double code(double a, double b, double c) {
	return -(c / b);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = -(c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -(c / b);
}

Python

def code(a, b, c):
	return -(c / b)

Julia

function code(a, b, c)
	return Float64(-Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -(c / b);
end

Wolfram

code[a_, b_, c_] := (-N[(c / b), $MachinePrecision])

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

2. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c}{b}\right) \]

   2. lower-neg.f64 N/A

      \[\leadsto -\frac{c}{b} \]

   3. lower-/.f64 34.6

      \[\leadsto -\frac{c}{b} \]

4. Applied rewrites 34.6%

   \[\leadsto \color{blue}{-\frac{c}{b}} \]
5. Add Preprocessing

Alternative 10: 10.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

public static double code(double a, double b, double c) {
	return c / b;
}

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / b;
end

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

2. Taylor expanded in c around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{c}{b} + \color{blue}{-1 \cdot \frac{b}{a}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{c}{b} + \color{blue}{-1} \cdot \frac{b}{a} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{c}{b} + \frac{-1 \cdot b}{\color{blue}{a}} \]

   5. mul-1-neg N/A

      \[\leadsto \frac{c}{b} + \frac{\mathsf{neg}\left(b\right)}{a} \]

   6. lift-neg.f64 N/A

      \[\leadsto \frac{c}{b} + \frac{-b}{a} \]

   7. lower-/.f64 34.7

      \[\leadsto \frac{c}{b} + \frac{-b}{\color{blue}{a}} \]

4. Applied rewrites 34.7%

   \[\leadsto \color{blue}{\frac{c}{b} + \frac{-b}{a}} \]

5. Taylor expanded in a around inf

   \[\leadsto \frac{c}{\color{blue}{b}} \]
6. Step-by-step derivation

   1. lift-/.f64 10.6

      \[\leadsto \frac{c}{b} \]

7. Applied rewrites 10.6%

   \[\leadsto \frac{c}{\color{blue}{b}} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{t\_2 - \frac{b}{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{2} + t\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ c (- t_2 (/ b 2.0))) (/ (+ (/ b 2.0) t_2) (- a)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = c / (t_2 - (b / 2.0));
	} else {
		tmp_1 = ((b / 2.0) + t_2) / -a;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = c / (t_2 - (b / 2.0))
	else:
		tmp_1 = ((b / 2.0) + t_2) / -a
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(c / Float64(t_2 - Float64(b / 2.0)));
	else
		tmp_1 = Float64(Float64(Float64(b / 2.0) + t_2) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = c / (t_2 - (b / 2.0));
	else
		tmp_2 = ((b / 2.0) + t_2) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(c / N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2025106 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform c (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ c (- sqtD (/ b 2))) (/ (+ (/ b 2) sqtD) (- a)))))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))