quadp (p42, positive)

Percentage Accurate: 52.7% → 85.8%

Time: 7.1s

Alternatives: 6

Speedup: 2.5×

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: 52.7% 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: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{b}{-2 \cdot a} - \frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+154)
   (/ (- b) a)
   (if (<= b 2.5e-81)
     (- (/ b (* -2.0 a)) (/ (sqrt (fma -4.0 (* c a) (* b b))) (* -2.0 a)))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+154) {
		tmp = -b / a;
	} else if (b <= 2.5e-81) {
		tmp = (b / (-2.0 * a)) - (sqrt(fma(-4.0, (c * a), (b * b))) / (-2.0 * a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.5e-81)
		tmp = Float64(Float64(b / Float64(-2.0 * a)) - Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) / Float64(-2.0 * a)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.5e-81], N[(N[(b / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000007e154

   1. Initial program 37.3%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -2.00000000000000007e154 < b < 2.4999999999999999e-81

   1. Initial program 83.8%

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

   4. Applied rewrites 83.8%

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

   if 2.4999999999999999e-81 < b

   1. Initial program 17.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

Alternative 2: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e+154)
   (/ (- b) a)
   (if (<= b 2.5e-81)
     (/ (- b (sqrt (fma -4.0 (* c a) (* b b)))) (* -2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e+154) {
		tmp = -b / a;
	} else if (b <= 2.5e-81) {
		tmp = (b - sqrt(fma(-4.0, (c * a), (b * b)))) / (-2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.5e-81)
		tmp = Float64(Float64(b - sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.5e-81], N[(N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.00000000000000007e154

   1. Initial program 37.3%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.0

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

   5. Applied rewrites 98.0%

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

   if -2.00000000000000007e154 < b < 2.4999999999999999e-81

   1. Initial program 83.8%

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

   4. Applied rewrites 83.8%

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

   if 2.4999999999999999e-81 < b

   1. Initial program 17.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

Alternative 3: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e-66)
   (/ (- b) a)
   (if (<= b 2.5e-81)
     (/ (- b (sqrt (* -4.0 (* a c)))) (* -2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.6e-66) {
		tmp = -b / a;
	} else if (b <= 2.5e-81) {
		tmp = (b - sqrt((-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 <= (-1.6d-66)) then
        tmp = -b / a
    else if (b <= 2.5d-81) then
        tmp = (b - sqrt(((-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 <= -1.6e-66) {
		tmp = -b / a;
	} else if (b <= 2.5e-81) {
		tmp = (b - Math.sqrt((-4.0 * (a * c)))) / (-2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.6e-66:
		tmp = -b / a
	elif b <= 2.5e-81:
		tmp = (b - math.sqrt((-4.0 * (a * c)))) / (-2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.6e-66)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.5e-81)
		tmp = Float64(Float64(b - sqrt(Float64(-4.0 * Float64(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 <= -1.6e-66)
		tmp = -b / a;
	elseif (b <= 2.5e-81)
		tmp = (b - sqrt((-4.0 * (a * c)))) / (-2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.6e-66], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.5e-81], N[(N[(b - N[Sqrt[N[(-4.0 * N[(a * 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 < -1.59999999999999991e-66

   1. Initial program 68.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if -1.59999999999999991e-66 < b < 2.4999999999999999e-81

   1. Initial program 73.2%

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 71.4

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

   7. Applied rewrites 71.4%

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

   if 2.4999999999999999e-81 < b

   1. Initial program 17.8%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 88.1

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

   5. Applied rewrites 88.1%

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

Alternative 4: 66.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-245}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.2e-245) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.2e-245) {
		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 <= 1.2d-245) 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 <= 1.2e-245) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.2e-245:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.2e-245)
		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 <= 1.2e-245)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.2e-245], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.2e-245

   1. Initial program 68.6%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 1.2e-245 < b

   1. Initial program 32.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 5: 43.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -2e-310) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	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 <= (-2d-310)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 67.4%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 36.2%

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

   4. Applied rewrites 19.0%

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

   5. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{\color{blue}{0} \cdot a}{a}, -1 \cdot \frac{c}{{b}^{2}}\right) \cdot b \]

      12. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{\color{blue}{0 \cdot a}}{a}, -1 \cdot \frac{c}{{b}^{2}}\right) \cdot b \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \color{blue}{\frac{-1 \cdot c}{{b}^{2}}}\right) \cdot b \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{{b}^{2}}\right) \cdot b \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \color{blue}{\frac{\mathsf{neg}\left(c\right)}{{b}^{2}}}\right) \cdot b \]

      16. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \frac{\color{blue}{-c}}{{b}^{2}}\right) \cdot b \]

      17. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \frac{-c}{\color{blue}{b \cdot b}}\right) \cdot b \]

      18. lower-*.f64 44.0

          \[\leadsto \mathsf{fma}\left(\frac{-0.5}{a}, \frac{0 \cdot a}{a}, \frac{-c}{\color{blue}{b \cdot b}}\right) \cdot b \]

   7. Applied rewrites 44.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{a}, \frac{0 \cdot a}{a}, \frac{-c}{b \cdot b}\right) \cdot b} \]

   8. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. associate-*r/ N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} - \frac{b}{a} \cdot \frac{-1}{2}} \]

      7. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{b}{a} - \color{blue}{\frac{-1}{2} \cdot \frac{b}{a}} \]

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{b}{a}} \]

      10. distribute-rgt-out N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{b}{a} \cdot 0} \]

      13. lower-/.f64 16.0

          \[\leadsto \color{blue}{\frac{b}{a}} \cdot 0 \]

   10. Applied rewrites 16.0%

       \[\leadsto \color{blue}{\frac{b}{a} \cdot 0} \]

   11. Taylor expanded in a around 0

       \[\leadsto 0 \]
   12. Step-by-step derivation

   13. Applied rewrites 23.6%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 11.0% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 52.3%

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

4. Applied rewrites 34.2%

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

5. Taylor expanded in b around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r/ N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. distribute-rgt1-in N/A

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

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{\color{blue}{0} \cdot a}{a}, -1 \cdot \frac{c}{{b}^{2}}\right) \cdot b \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{\color{blue}{0 \cdot a}}{a}, -1 \cdot \frac{c}{{b}^{2}}\right) \cdot b \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \color{blue}{\frac{-1 \cdot c}{{b}^{2}}}\right) \cdot b \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{{b}^{2}}\right) \cdot b \]

   15. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \color{blue}{\frac{\mathsf{neg}\left(c\right)}{{b}^{2}}}\right) \cdot b \]

   16. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \frac{\color{blue}{-c}}{{b}^{2}}\right) \cdot b \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{2}}{a}, \frac{0 \cdot a}{a}, \frac{-c}{\color{blue}{b \cdot b}}\right) \cdot b \]

   18. lower-*.f64 22.4

       \[\leadsto \mathsf{fma}\left(\frac{-0.5}{a}, \frac{0 \cdot a}{a}, \frac{-c}{\color{blue}{b \cdot b}}\right) \cdot b \]

7. Applied rewrites 22.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.5}{a}, \frac{0 \cdot a}{a}, \frac{-c}{b \cdot b}\right) \cdot b} \]

8. Taylor expanded in a around 0

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

   1. div-add N/A

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

   2. associate-*r/ N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. mul-1-neg N/A

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

   6. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} - \frac{b}{a} \cdot \frac{-1}{2}} \]

   7. *-commutative N/A

      \[\leadsto \frac{-1}{2} \cdot \frac{b}{a} - \color{blue}{\frac{-1}{2} \cdot \frac{b}{a}} \]

   8. metadata-eval N/A

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

   9. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a} + \frac{1}{2} \cdot \frac{b}{a}} \]

   10. distribute-rgt-out N/A

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

   11. metadata-eval N/A

       \[\leadsto \frac{b}{a} \cdot \color{blue}{0} \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{b}{a} \cdot 0} \]

   13. lower-/.f64 9.0

       \[\leadsto \color{blue}{\frac{b}{a}} \cdot 0 \]

10. Applied rewrites 9.0%

    \[\leadsto \color{blue}{\frac{b}{a} \cdot 0} \]

11. Taylor expanded in a around 0

    \[\leadsto 0 \]
12. Step-by-step derivation

13. Applied rewrites 12.8%

    \[\leadsto 0 \]
14. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

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) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

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 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	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 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	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 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	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(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	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 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	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[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024359 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (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) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

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