quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.0% → 85.3%

Time: 3.3s

Alternatives: 9

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{+126}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-62}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e+126)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.05e-62)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+126) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-62) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d+126)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.05d-62) then
        tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e+126) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-62) {
		tmp = (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e+126:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.05e-62:
		tmp = (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e+126)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.05e-62)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e+126)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.05e-62)
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e+126], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-62], N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.9999999999999997e126

   1. Initial program 51.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 97.1

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 97.1%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -3.9999999999999997e126 < b_2 < 1.05e-62

   1. Initial program 79.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   if 1.05e-62 < b_2

   1. Initial program 17.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 86.7

         \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]

   4. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{-1}{2} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      6. lower-*.f64 86.8

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   6. Applied rewrites 86.8%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-63}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6e-90)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 3e-63)
     (/ (+ (- b_2) (sqrt (* (- a) c))) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-90) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3e-63) {
		tmp = (-b_2 + sqrt((-a * c))) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-6d-90)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 3d-63) then
        tmp = (-b_2 + sqrt((-a * c))) / a
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-90) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3e-63) {
		tmp = (-b_2 + Math.sqrt((-a * c))) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6e-90:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 3e-63:
		tmp = (-b_2 + math.sqrt((-a * c))) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6e-90)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 3e-63)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6e-90)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 3e-63)
		tmp = (-b_2 + sqrt((-a * c))) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6e-90], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3e-63], N[(N[((-b$95$2) + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.00000000000000041e-90

   1. Initial program 72.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 84.5

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 84.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -6.00000000000000041e-90 < b_2 < 2.99999999999999979e-63

   1. Initial program 71.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\left(-1 \cdot a\right) \cdot \color{blue}{c}}}{a} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot c}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{c}}}{a} \]

      4. lower-neg.f64 67.0

         \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}{a} \]

   4. Applied rewrites 67.0%

      \[\leadsto \frac{\left(-b\_2\right) + \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if 2.99999999999999979e-63 < b_2

   1. Initial program 17.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 86.7

         \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]

   4. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{-1}{2} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      6. lower-*.f64 86.7

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   6. Applied rewrites 86.7%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6 \cdot 10^{-90}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-63}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6e-90)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 3e-63) (/ (sqrt (* (- a) c)) a) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-90) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3e-63) {
		tmp = sqrt((-a * c)) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-6d-90)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 3d-63) then
        tmp = sqrt((-a * c)) / a
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-90) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 3e-63) {
		tmp = Math.sqrt((-a * c)) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6e-90:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 3e-63:
		tmp = math.sqrt((-a * c)) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6e-90)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 3e-63)
		tmp = Float64(sqrt(Float64(Float64(-a) * c)) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6e-90)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 3e-63)
		tmp = sqrt((-a * c)) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6e-90], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3e-63], N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.00000000000000041e-90

   1. Initial program 72.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 84.5

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 84.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -6.00000000000000041e-90 < b_2 < 2.99999999999999979e-63

   1. Initial program 71.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 65.8

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

   4. Applied rewrites 65.8%

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

   if 2.99999999999999979e-63 < b_2

   1. Initial program 17.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 86.7

         \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]

   4. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{-1}{2} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      6. lower-*.f64 86.7

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   6. Applied rewrites 86.7%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\sqrt{-c}}{\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.8e-108)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 2.2e-115) (/ (sqrt (- c)) (sqrt a)) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.8e-108) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2.2e-115) {
		tmp = sqrt(-c) / sqrt(a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-7.8d-108)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 2.2d-115) then
        tmp = sqrt(-c) / sqrt(a)
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.8e-108) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2.2e-115) {
		tmp = Math.sqrt(-c) / Math.sqrt(a);
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.8e-108:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 2.2e-115:
		tmp = math.sqrt(-c) / math.sqrt(a)
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.8e-108)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 2.2e-115)
		tmp = Float64(sqrt(Float64(-c)) / sqrt(a));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.8e-108)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 2.2e-115)
		tmp = sqrt(-c) / sqrt(a);
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.8e-108], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.2e-115], N[(N[Sqrt[(-c)], $MachinePrecision] / N[Sqrt[a], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.79999999999999989e-108

   1. Initial program 72.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 83.0

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 83.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -7.79999999999999989e-108 < b_2 < 2.1999999999999999e-115

   1. Initial program 72.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 9.3

         \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]

   4. Applied rewrites 9.3%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{-1}{2} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      6. lower-*.f64 9.3

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   6. Applied rewrites 9.3%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   7. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. mul-1-neg N/A

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

      4. lower-sqrt.f64 N/A

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

      5. distribute-neg-frac 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 32.9

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

   9. Applied rewrites 32.9%

      \[\leadsto \color{blue}{\sqrt{\frac{-c}{a}}} \]
   10. 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

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-sqrt.f64 43.4

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

   11. Applied rewrites 43.4%

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

   if 2.1999999999999999e-115 < b_2

   1. Initial program 20.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 82.3

         \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]

   4. Applied rewrites 82.3%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{-1}{2} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      6. lower-*.f64 82.4

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   6. Applied rewrites 82.4%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 71.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.8e-109)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.85e-115) (sqrt (- (/ c a))) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.8e-109) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-115) {
		tmp = sqrt(-(c / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5.8d-109)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.85d-115) then
        tmp = sqrt(-(c / a))
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.8e-109) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.85e-115) {
		tmp = Math.sqrt(-(c / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.8e-109:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.85e-115:
		tmp = math.sqrt(-(c / a))
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.8e-109)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.85e-115)
		tmp = sqrt(Float64(-Float64(c / a)));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.8e-109)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.85e-115)
		tmp = sqrt(-(c / a));
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.8e-109], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-115], N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.8e-109

   1. Initial program 72.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 82.9

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 82.9%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   if -5.8e-109 < b_2 < 1.85e-115

   1. Initial program 72.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 9.6

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 9.6%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Taylor expanded in a around inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-/.f64 33.0

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

   7. Applied rewrites 33.0%

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

   if 1.85e-115 < b_2

   1. Initial program 20.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 82.4

         \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]

   4. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{-1}{2} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      6. lower-*.f64 82.4

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   6. Applied rewrites 82.4%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 53.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e+69)
   (/ (- b_2) a)
   (if (<= b_2 1.85e-115) (sqrt (- (/ c a))) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e+69) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.85e-115) {
		tmp = sqrt(-(c / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.2d+69)) then
        tmp = -b_2 / a
    else if (b_2 <= 1.85d-115) then
        tmp = sqrt(-(c / a))
    else
        tmp = ((-0.5d0) * c) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e+69) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.85e-115) {
		tmp = Math.sqrt(-(c / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e+69:
		tmp = -b_2 / a
	elif b_2 <= 1.85e-115:
		tmp = math.sqrt(-(c / a))
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e+69)
		tmp = Float64(Float64(-b_2) / a);
	elseif (b_2 <= 1.85e-115)
		tmp = sqrt(Float64(-Float64(c / a)));
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e+69)
		tmp = -b_2 / a;
	elseif (b_2 <= 1.85e-115)
		tmp = sqrt(-(c / a));
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e+69], N[((-b$95$2) / a), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-115], N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.2000000000000001e69

   1. Initial program 61.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in c around inf

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right) \cdot \color{blue}{c}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right) \cdot \color{blue}{c}}{a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c}} \cdot \sqrt{-1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c}} \cdot \sqrt{-1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-1 \cdot b\_2}{c}\right) \cdot c}{a} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{\mathsf{neg}\left(b\_2\right)}{c}\right) \cdot c}{a} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}{a} \]

      12. lower-/.f64 29.1

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}{a} \]

   4. Applied rewrites 29.1%

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}}{a} \]

   5. Taylor expanded in b_2 around inf

      \[\leadsto \frac{-1 \cdot \color{blue}{b\_2}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      2. lift-neg.f64 45.5

         \[\leadsto \frac{-b\_2}{a} \]

   7. Applied rewrites 45.5%

      \[\leadsto \frac{-b\_2}{a} \]

   if -1.2000000000000001e69 < b_2 < 1.85e-115

   1. Initial program 79.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 29.4

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 29.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Taylor expanded in a around inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-/.f64 26.6

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

   7. Applied rewrites 26.6%

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

   if 1.85e-115 < b_2

   1. Initial program 20.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 82.4

         \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]

   4. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{-1}{2} \]

      3. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      6. lower-*.f64 82.4

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   6. Applied rewrites 82.4%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 53.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.85 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e+69)
   (/ (- b_2) a)
   (if (<= b_2 1.85e-115) (sqrt (- (/ c a))) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e+69) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.85e-115) {
		tmp = sqrt(-(c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.2d+69)) then
        tmp = -b_2 / a
    else if (b_2 <= 1.85d-115) then
        tmp = sqrt(-(c / a))
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e+69) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.85e-115) {
		tmp = Math.sqrt(-(c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e+69:
		tmp = -b_2 / a
	elif b_2 <= 1.85e-115:
		tmp = math.sqrt(-(c / a))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e+69)
		tmp = Float64(Float64(-b_2) / a);
	elseif (b_2 <= 1.85e-115)
		tmp = sqrt(Float64(-Float64(c / a)));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e+69)
		tmp = -b_2 / a;
	elseif (b_2 <= 1.85e-115)
		tmp = sqrt(-(c / a));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e+69], N[((-b$95$2) / a), $MachinePrecision], If[LessEqual[b$95$2, 1.85e-115], N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.2000000000000001e69

   1. Initial program 61.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in c around inf

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right) \cdot \color{blue}{c}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right) \cdot \color{blue}{c}}{a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c}} \cdot \sqrt{-1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c}} \cdot \sqrt{-1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-1 \cdot b\_2}{c}\right) \cdot c}{a} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{\mathsf{neg}\left(b\_2\right)}{c}\right) \cdot c}{a} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}{a} \]

      12. lower-/.f64 29.1

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}{a} \]

   4. Applied rewrites 29.1%

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}}{a} \]

   5. Taylor expanded in b_2 around inf

      \[\leadsto \frac{-1 \cdot \color{blue}{b\_2}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      2. lift-neg.f64 45.5

         \[\leadsto \frac{-b\_2}{a} \]

   7. Applied rewrites 45.5%

      \[\leadsto \frac{-b\_2}{a} \]

   if -1.2000000000000001e69 < b_2 < 1.85e-115

   1. Initial program 79.8%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 29.4

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 29.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Taylor expanded in a around inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-/.f64 26.6

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

   7. Applied rewrites 26.6%

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

   if 1.85e-115 < b_2

   1. Initial program 20.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{\frac{-1}{2}} \]

      3. lower-/.f64 82.4

         \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]

   4. Applied rewrites 82.4%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 26.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e+69) (/ (- b_2) a) (sqrt (- (/ c a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e+69) {
		tmp = -b_2 / a;
	} else {
		tmp = sqrt(-(c / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.2d+69)) then
        tmp = -b_2 / a
    else
        tmp = sqrt(-(c / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e+69) {
		tmp = -b_2 / a;
	} else {
		tmp = Math.sqrt(-(c / a));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e+69:
		tmp = -b_2 / a
	else:
		tmp = math.sqrt(-(c / a))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e+69)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = sqrt(Float64(-Float64(c / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e+69)
		tmp = -b_2 / a;
	else
		tmp = sqrt(-(c / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e+69], N[((-b$95$2) / a), $MachinePrecision], N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.2000000000000001e69

   1. Initial program 61.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in c around inf

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right) \cdot \color{blue}{c}}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right) \cdot \color{blue}{c}}{a} \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c}} \cdot \sqrt{-1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c}} \cdot \sqrt{-1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      5. sqrt-unprod N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-1 \cdot b\_2}{c}\right) \cdot c}{a} \]

      10. mul-1-neg N/A

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{\mathsf{neg}\left(b\_2\right)}{c}\right) \cdot c}{a} \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}{a} \]

      12. lower-/.f64 29.1

          \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}{a} \]

   4. Applied rewrites 29.1%

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}}{a} \]

   5. Taylor expanded in b_2 around inf

      \[\leadsto \frac{-1 \cdot \color{blue}{b\_2}}{a} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      2. lift-neg.f64 45.5

         \[\leadsto \frac{-b\_2}{a} \]

   7. Applied rewrites 45.5%

      \[\leadsto \frac{-b\_2}{a} \]

   if -1.2000000000000001e69 < b_2

   1. Initial program 49.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   3. Step-by-step derivation

      1. lower-*.f64 15.5

         \[\leadsto \frac{-2 \cdot \color{blue}{b\_2}}{a} \]

   4. Applied rewrites 15.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]

   5. Taylor expanded in a around inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-/.f64 20.8

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

   7. Applied rewrites 20.8%

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

Alternative 9: 15.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{-b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (- b_2) a))

C

double code(double a, double b_2, double c) {
	return -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = -b_2 / a
end function

Java

public static double code(double a, double b_2, double c) {
	return -b_2 / a;
}

Python

def code(a, b_2, c):
	return -b_2 / a

Julia

function code(a, b_2, c)
	return Float64(Float64(-b_2) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -b_2 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[((-b$95$2) / a), $MachinePrecision]

Derivation

1. Initial program 52.0%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

2. Taylor expanded in c around inf

   \[\leadsto \frac{\color{blue}{c \cdot \left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right) \cdot \color{blue}{c}}{a} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(-1 \cdot \frac{b\_2}{c} + \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right) \cdot \color{blue}{c}}{a} \]

   3. +-commutative N/A

      \[\leadsto \frac{\left(\sqrt{\frac{a}{c}} \cdot \sqrt{-1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{\left(\sqrt{\frac{a}{c}} \cdot \sqrt{-1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

   5. sqrt-unprod N/A

      \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + -1 \cdot \frac{b\_2}{c}\right) \cdot c}{a} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-1 \cdot b\_2}{c}\right) \cdot c}{a} \]

   10. mul-1-neg N/A

       \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{\mathsf{neg}\left(b\_2\right)}{c}\right) \cdot c}{a} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}{a} \]

   12. lower-/.f64 20.6

       \[\leadsto \frac{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}{a} \]

4. Applied rewrites 20.6%

   \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{a}{c} \cdot -1} + \frac{-b\_2}{c}\right) \cdot c}}{a} \]

5. Taylor expanded in b_2 around inf

   \[\leadsto \frac{-1 \cdot \color{blue}{b\_2}}{a} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

   2. lift-neg.f64 15.1

      \[\leadsto \frac{-b\_2}{a} \]

7. Applied rewrites 15.1%

   \[\leadsto \frac{-b\_2}{a} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	end
	tmp_3 = tmp_2;
end

Wolfram

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

Reproduce

herbie shell --seed 2025112 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform c (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))