The quadratic formula (r1)

Percentage Accurate: 52.4% → 85.2%

Time: 4.0s

Alternatives: 8

Speedup: 2.7×

Specification

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.4% accurate, 1.0× speedup

Math

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

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(Float64(4.0 * 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[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{+110}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.16e+110)
   (/ (- b) a)
   (if (<= b 2.65e-48)
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e+110) {
		tmp = -b / a;
	} else if (b <= 2.65e-48) {
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	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 <= (-1.16d+110)) then
        tmp = -b / a
    else if (b <= 2.65d-48) then
        tmp = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e+110) {
		tmp = -b / a;
	} else if (b <= 2.65e-48) {
		tmp = (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.16e+110:
		tmp = -b / a
	elif b <= 2.65e-48:
		tmp = (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.16e+110)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.65e-48)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.16e+110)
		tmp = -b / a;
	elseif (b <= 2.65e-48)
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.16e+110], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.65e-48], N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.16e110

   1. Initial program 53.5%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 96.4

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

   4. Applied rewrites 96.4%

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

   if -1.16e110 < b < 2.65e-48

   1. Initial program 78.5%

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

   if 2.65e-48 < b

   1. Initial program 16.4%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 2: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -10.5:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -10.5)
   (/ (- b) a)
   (if (<= b 2.65e-48)
     (/ (+ (sqrt (* (* c a) -4.0)) (- b)) (+ a a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -10.5) {
		tmp = -b / a;
	} else if (b <= 2.65e-48) {
		tmp = (sqrt(((c * a) * -4.0)) + -b) / (a + a);
	} else {
		tmp = -c / b;
	}
	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 = -b / a
    else if (b <= 2.65d-48) then
        tmp = (sqrt(((c * a) * (-4.0d0))) + -b) / (a + a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -10.5) {
		tmp = -b / a;
	} else if (b <= 2.65e-48) {
		tmp = (Math.sqrt(((c * a) * -4.0)) + -b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -10.5:
		tmp = -b / a
	elif b <= 2.65e-48:
		tmp = (math.sqrt(((c * a) * -4.0)) + -b) / (a + a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -10.5)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.65e-48)
		tmp = Float64(Float64(sqrt(Float64(Float64(c * a) * -4.0)) + Float64(-b)) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -10.5)
		tmp = -b / a;
	elseif (b <= 2.65e-48)
		tmp = (sqrt(((c * a) * -4.0)) + -b) / (a + a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -10.5], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.65e-48], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -10.5

   1. Initial program 65.6%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 90.7

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

   4. Applied rewrites 90.7%

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

   if -10.5 < b < 2.65e-48

   1. Initial program 74.8%

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

   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 62.6

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

   4. Applied rewrites 62.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 62.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 62.6

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

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(c \cdot a\right) \cdot -4} + \left(-b\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(2 \cdot a\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(c \cdot a\right) \cdot -4} + \left(-b\right)}{\mathsf{Rewrite=>}\left(count-2-rev, \left(a + a\right)\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(c \cdot a\right) \cdot -4} + \left(-b\right)}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(a + a\right)\right)} \]

   6. Applied rewrites 62.6%

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

   if 2.65e-48 < b

   1. Initial program 16.4%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 3: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot a\right) \cdot -4}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-111)
   (/ (- b) a)
   (if (<= b 2.65e-48) (/ (sqrt (* (* c a) -4.0)) (+ a a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-111) {
		tmp = -b / a;
	} else if (b <= 2.65e-48) {
		tmp = sqrt(((c * a) * -4.0)) / (a + a);
	} else {
		tmp = -c / b;
	}
	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 <= (-1.6d-111)) then
        tmp = -b / a
    else if (b <= 2.65d-48) then
        tmp = sqrt(((c * a) * (-4.0d0))) / (a + a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-111) {
		tmp = -b / a;
	} else if (b <= 2.65e-48) {
		tmp = Math.sqrt(((c * a) * -4.0)) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-111:
		tmp = -b / a
	elif b <= 2.65e-48:
		tmp = math.sqrt(((c * a) * -4.0)) / (a + a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-111)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.65e-48)
		tmp = Float64(sqrt(Float64(Float64(c * a) * -4.0)) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e-111)
		tmp = -b / a;
	elseif (b <= 2.65e-48)
		tmp = sqrt(((c * a) * -4.0)) / (a + a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-111], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.65e-48], N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5999999999999999e-111

   1. Initial program 70.7%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 82.1

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

   4. Applied rewrites 82.1%

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

   if -1.5999999999999999e-111 < b < 2.65e-48

   1. Initial program 70.3%

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

   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 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 67.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 67.2

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

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(c \cdot a\right) \cdot -4} + \left(-b\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(2 \cdot a\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(c \cdot a\right) \cdot -4} + \left(-b\right)}{\mathsf{Rewrite=>}\left(count-2-rev, \left(a + a\right)\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(c \cdot a\right) \cdot -4} + \left(-b\right)}{\mathsf{Rewrite=>}\left(lower-+.f64, \left(a + a\right)\right)} \]

   6. Applied rewrites 67.2%

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

   7. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot c} \cdot \sqrt{-4}}}{a + a} \]
   8. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 66.6

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

   9. Applied rewrites 66.6%

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

   if 2.65e-48 < b

   1. Initial program 16.4%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 4: 79.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;\left(-c\right) \cdot \sqrt{\frac{-1}{c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-111)
   (/ (- b) a)
   (if (<= b 1.8e-35) (* (- c) (sqrt (/ -1.0 (* c a)))) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-111) {
		tmp = -b / a;
	} else if (b <= 1.8e-35) {
		tmp = -c * sqrt((-1.0 / (c * a)));
	} else {
		tmp = -c / b;
	}
	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 <= (-1.6d-111)) then
        tmp = -b / a
    else if (b <= 1.8d-35) then
        tmp = -c * sqrt(((-1.0d0) / (c * a)))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-111) {
		tmp = -b / a;
	} else if (b <= 1.8e-35) {
		tmp = -c * Math.sqrt((-1.0 / (c * a)));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-111:
		tmp = -b / a
	elif b <= 1.8e-35:
		tmp = -c * math.sqrt((-1.0 / (c * a)))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-111)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.8e-35)
		tmp = Float64(Float64(-c) * sqrt(Float64(-1.0 / Float64(c * a))));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.6e-111)
		tmp = -b / a;
	elseif (b <= 1.8e-35)
		tmp = -c * sqrt((-1.0 / (c * a)));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-111], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.8e-35], N[((-c) * N[Sqrt[N[(-1.0 / N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5999999999999999e-111

   1. Initial program 70.7%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 82.1

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

   4. Applied rewrites 82.1%

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

   if -1.5999999999999999e-111 < b < 1.80000000000000009e-35

   1. Initial program 69.4%

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

   2. Taylor expanded in c around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. sqrt-unprod N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 65.0

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

   4. Applied rewrites 65.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. sqrt-prod N/A

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

      5. lift-*.f64 N/A

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

      6. lift-sqrt.f64 65.2

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 65.2

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

   7. Applied rewrites 65.2%

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

   if 1.80000000000000009e-35 < b

   1. Initial program 15.7%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 88.1

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

   4. Applied rewrites 88.1%

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

Alternative 5: 72.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{-c}}{\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-114)
   (/ (- b) a)
   (if (<= b 7.8e-73) (/ (sqrt (- c)) (sqrt a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-114) {
		tmp = -b / a;
	} else if (b <= 7.8e-73) {
		tmp = sqrt(-c) / sqrt(a);
	} else {
		tmp = -c / b;
	}
	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 <= (-3.2d-114)) then
        tmp = -b / a
    else if (b <= 7.8d-73) then
        tmp = sqrt(-c) / sqrt(a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-114) {
		tmp = -b / a;
	} else if (b <= 7.8e-73) {
		tmp = Math.sqrt(-c) / Math.sqrt(a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.2e-114:
		tmp = -b / a
	elif b <= 7.8e-73:
		tmp = math.sqrt(-c) / math.sqrt(a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-114)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 7.8e-73)
		tmp = Float64(sqrt(Float64(-c)) / sqrt(a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.2e-114)
		tmp = -b / a;
	elseif (b <= 7.8e-73)
		tmp = sqrt(-c) / sqrt(a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-114], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 7.8e-73], N[(N[Sqrt[(-c)], $MachinePrecision] / N[Sqrt[a], $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.2000000000000002e-114

   1. Initial program 70.8%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 81.9

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

   4. Applied rewrites 81.9%

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

   if -3.2000000000000002e-114 < b < 7.79999999999999963e-73

   1. Initial program 72.5%

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

   2. Taylor expanded in a around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 33.7

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

   4. Applied rewrites 33.7%

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

   5. Taylor expanded in c around -inf

      \[\leadsto \sqrt{\frac{c}{a}} \cdot \color{blue}{\sqrt{-1}} \]
   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. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 33.4

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

   7. Applied rewrites 33.4%

      \[\leadsto \sqrt{\frac{-c}{a}} \]
   8. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 41.6

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

   9. Applied rewrites 41.6%

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

   if 7.79999999999999963e-73 < b

   1. Initial program 17.8%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 85.5

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

   4. Applied rewrites 85.5%

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

Alternative 6: 71.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-129}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-134}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.6e-129)
   (/ (- b) a)
   (if (<= b 1.9e-134) (sqrt (/ (- c) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-129) {
		tmp = -b / a;
	} else if (b <= 1.9e-134) {
		tmp = sqrt((-c / a));
	} else {
		tmp = -c / b;
	}
	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 <= (-8.6d-129)) then
        tmp = -b / a
    else if (b <= 1.9d-134) then
        tmp = sqrt((-c / a))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-129) {
		tmp = -b / a;
	} else if (b <= 1.9e-134) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.6e-129:
		tmp = -b / a
	elif b <= 1.9e-134:
		tmp = math.sqrt((-c / a))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.6e-129)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.9e-134)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.6e-129)
		tmp = -b / a;
	elseif (b <= 1.9e-134)
		tmp = sqrt((-c / a));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.6e-129], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.9e-134], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.59999999999999962e-129

   1. Initial program 71.3%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 80.6

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

   4. Applied rewrites 80.6%

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

   if -8.59999999999999962e-129 < b < 1.90000000000000001e-134

   1. Initial program 74.8%

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

   2. Taylor expanded in a around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 35.6

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

   4. Applied rewrites 35.6%

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

   5. Taylor expanded in c around -inf

      \[\leadsto \sqrt{\frac{c}{a}} \cdot \color{blue}{\sqrt{-1}} \]
   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. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 36.0

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

   7. Applied rewrites 36.0%

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

   if 1.90000000000000001e-134 < b

   1. Initial program 21.4%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 7: 66.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 72.9%

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

   2. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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 66.5

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

   4. Applied rewrites 66.5%

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

   if -3.999999999999988e-310 < b

   1. Initial program 31.7%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 67.3

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

   4. Applied rewrites 67.3%

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

Alternative 8: 34.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

2. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
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 34.8

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

4. Applied rewrites 34.8%

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

Developer Target 1: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t\_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* a (/ (- (- b) t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b < 0.0:
		tmp = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b < 0.0)
		tmp = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025106 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (! :herbie-platform c (let ((d (- (* b b) (* (* 4 a) c)))) (let ((r1 (/ (+ (- b) (sqrt d)) (* 2 a)))) (let ((r2 (/ (- (- b) (sqrt d)) (* 2 a)))) (if (< b 0) r1 (/ c (* a r2)))))))

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