The quadratic formula (r2)

Percentage Accurate: 52.2% → 85.0%

Time: 4.4s

Alternatives: 11

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% 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: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-130}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{+51}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.6e-130)
   (- (/ c b))
   (if (<= b 1.42e+51)
     (/ (- (- b) (sqrt (fma (* -4.0 a) c (* b b)))) (+ a a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-130) {
		tmp = -(c / b);
	} else if (b <= 1.42e+51) {
		tmp = (-b - sqrt(fma((-4.0 * a), c, (b * b)))) / (a + a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.6e-130)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 1.42e+51)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.6e-130], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 1.42e+51], N[(N[((-b) - N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.60000000000000058e-130

   1. Initial program 20.2%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   if -8.60000000000000058e-130 < b < 1.41999999999999998e51

   1. Initial program 82.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 82.1

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 82.1

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

   3. Applied rewrites 82.1%

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

   if 1.41999999999999998e51 < b

   1. Initial program 61.5%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

Alternative 2: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-23}:\\ \;\;\;\;-0.5 \cdot \frac{\sqrt{\left(a \cdot -4\right) \cdot c} + b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e-123)
   (- (/ c b))
   (if (<= b 4e-23)
     (* -0.5 (/ (+ (sqrt (* (* a -4.0) c)) b) a))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-123) {
		tmp = -(c / b);
	} else if (b <= 4e-23) {
		tmp = -0.5 * ((sqrt(((a * -4.0) * c)) + b) / a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.5d-123)) then
        tmp = -(c / b)
    else if (b <= 4d-23) then
        tmp = (-0.5d0) * ((sqrt(((a * (-4.0d0)) * c)) + b) / a)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-123) {
		tmp = -(c / b);
	} else if (b <= 4e-23) {
		tmp = -0.5 * ((Math.sqrt(((a * -4.0) * c)) + b) / a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e-123:
		tmp = -(c / b)
	elif b <= 4e-23:
		tmp = -0.5 * ((math.sqrt(((a * -4.0) * c)) + b) / a)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-123)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 4e-23)
		tmp = Float64(-0.5 * Float64(Float64(sqrt(Float64(Float64(a * -4.0) * c)) + b) / a));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e-123)
		tmp = -(c / b);
	elseif (b <= 4e-23)
		tmp = -0.5 * ((sqrt(((a * -4.0) * c)) + b) / a);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-123], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 4e-23], N[(-0.5 * N[(N[(N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.50000000000000011e-123

   1. Initial program 19.7%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 82.5

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

   4. Applied rewrites 82.5%

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

   if -7.50000000000000011e-123 < b < 3.99999999999999984e-23

   1. Initial program 79.8%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(\sqrt{\frac{c}{a}} \cdot \sqrt{-4}\right) + \frac{-1}{2} \cdot \frac{b}{a}} \]
   3. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\sqrt{\frac{c}{a}} \cdot \sqrt{-4} + \frac{b}{a}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(\sqrt{\frac{c}{a}} \cdot \sqrt{-4} + \frac{b}{a}\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \left(\sqrt{\frac{c}{a}} \cdot \sqrt{-4} + \color{blue}{\frac{b}{a}}\right) \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{-1}{2} \cdot \left(\sqrt{\frac{c}{a} \cdot -4} + \frac{\color{blue}{b}}{a}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \left(\sqrt{\frac{c}{a} \cdot -4} + \frac{\color{blue}{b}}{a}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \left(\sqrt{\frac{c}{a} \cdot -4} + \frac{b}{a}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \left(\sqrt{\frac{c}{a} \cdot -4} + \frac{b}{a}\right) \]

      8. lower-/.f64 32.6

         \[\leadsto -0.5 \cdot \left(\sqrt{\frac{c}{a} \cdot -4} + \frac{b}{\color{blue}{a}}\right) \]

   4. Applied rewrites 32.6%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\sqrt{\frac{c}{a} \cdot -4} + \frac{b}{a}\right)} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{-1}{2} \cdot \frac{b + \sqrt{a \cdot c} \cdot \sqrt{-4}}{\color{blue}{a}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{b + \sqrt{a \cdot c} \cdot \sqrt{-4}}{a} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{a \cdot c} \cdot \sqrt{-4} + b}{a} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{a \cdot c} \cdot \sqrt{-4} + b}{a} \]

      4. sqrt-unprod N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left(a \cdot c\right) \cdot -4} + b}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{-4 \cdot \left(a \cdot c\right)} + b}{a} \]

      6. associate-*r* N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left(-4 \cdot a\right) \cdot c} + b}{a} \]

      9. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{\sqrt{\left(a \cdot -4\right) \cdot c} + b}{a} \]

      10. lower-*.f64 67.2

          \[\leadsto -0.5 \cdot \frac{\sqrt{\left(a \cdot -4\right) \cdot c} + b}{a} \]

   7. Applied rewrites 67.2%

      \[\leadsto -0.5 \cdot \frac{\sqrt{\left(a \cdot -4\right) \cdot c} + b}{\color{blue}{a}} \]

   if 3.99999999999999984e-23 < b

   1. Initial program 67.4%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

Alternative 3: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{\left(a \cdot -4\right) \cdot c}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e-123)
   (- (/ c b))
   (if (<= b 3.2e-23)
     (/ (- (sqrt (* (* a -4.0) c))) (+ a a))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-123) {
		tmp = -(c / b);
	} else if (b <= 3.2e-23) {
		tmp = -sqrt(((a * -4.0) * c)) / (a + a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-7.5d-123)) then
        tmp = -(c / b)
    else if (b <= 3.2d-23) then
        tmp = -sqrt(((a * (-4.0d0)) * c)) / (a + a)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-123) {
		tmp = -(c / b);
	} else if (b <= 3.2e-23) {
		tmp = -Math.sqrt(((a * -4.0) * c)) / (a + a);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.5e-123:
		tmp = -(c / b)
	elif b <= 3.2e-23:
		tmp = -math.sqrt(((a * -4.0) * c)) / (a + a)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-123)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 3.2e-23)
		tmp = Float64(Float64(-sqrt(Float64(Float64(a * -4.0) * c))) / Float64(a + a));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.5e-123)
		tmp = -(c / b);
	elseif (b <= 3.2e-23)
		tmp = -sqrt(((a * -4.0) * c)) / (a + a);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-123], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 3.2e-23], N[((-N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]) / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.50000000000000011e-123

   1. Initial program 19.7%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 82.5

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

   4. Applied rewrites 82.5%

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

   if -7.50000000000000011e-123 < b < 3.19999999999999976e-23

   1. Initial program 79.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 79.8

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

      14. lift-*.f64 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 79.8

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

   3. Applied rewrites 79.8%

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

   4. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{-4}\right)}}{a + a} \]
   5. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto \frac{-\sqrt{a \cdot c} \cdot \sqrt{-4}}{a + a} \]

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 65.2

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

   6. Applied rewrites 65.2%

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

   if 3.19999999999999976e-23 < b

   1. Initial program 67.4%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

Alternative 4: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-130}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-23}:\\ \;\;\;\;-\left(-\sqrt{\frac{\frac{-1}{c}}{a}}\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.6e-130)
   (- (/ c b))
   (if (<= b 3.2e-23)
     (- (* (- (sqrt (/ (/ -1.0 c) a))) c))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-130) {
		tmp = -(c / b);
	} else if (b <= 3.2e-23) {
		tmp = -(-sqrt(((-1.0 / c) / a)) * c);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.6d-130)) then
        tmp = -(c / b)
    else if (b <= 3.2d-23) then
        tmp = -(-sqrt((((-1.0d0) / c) / a)) * c)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-130) {
		tmp = -(c / b);
	} else if (b <= 3.2e-23) {
		tmp = -(-Math.sqrt(((-1.0 / c) / a)) * c);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.6e-130:
		tmp = -(c / b)
	elif b <= 3.2e-23:
		tmp = -(-math.sqrt(((-1.0 / c) / a)) * c)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.6e-130)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 3.2e-23)
		tmp = Float64(-Float64(Float64(-sqrt(Float64(Float64(-1.0 / c) / a))) * c));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.6e-130)
		tmp = -(c / b);
	elseif (b <= 3.2e-23)
		tmp = -(-sqrt(((-1.0 / c) / a)) * c);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.6e-130], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 3.2e-23], (-N[((-N[Sqrt[N[(N[(-1.0 / c), $MachinePrecision] / a), $MachinePrecision]], $MachinePrecision]) * c), $MachinePrecision]), N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.60000000000000058e-130

   1. Initial program 20.2%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   if -8.60000000000000058e-130 < b < 3.19999999999999976e-23

   1. Initial program 80.1%

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

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-neg.f64 65.0

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 65.0

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

   7. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 65.8

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

   9. Applied rewrites 65.8%

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

   if 3.19999999999999976e-23 < b

   1. Initial program 67.4%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

Alternative 5: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-130}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-23}:\\ \;\;\;\;-\left(-\sqrt{\frac{-1}{c \cdot a}}\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.6e-130)
   (- (/ c b))
   (if (<= b 3.2e-23)
     (- (* (- (sqrt (/ -1.0 (* c a)))) c))
     (+ (/ c b) (/ (- b) a)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-130) {
		tmp = -(c / b);
	} else if (b <= 3.2e-23) {
		tmp = -(-sqrt((-1.0 / (c * a))) * c);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.6d-130)) then
        tmp = -(c / b)
    else if (b <= 3.2d-23) then
        tmp = -(-sqrt(((-1.0d0) / (c * a))) * c)
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.6e-130) {
		tmp = -(c / b);
	} else if (b <= 3.2e-23) {
		tmp = -(-Math.sqrt((-1.0 / (c * a))) * c);
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.6e-130:
		tmp = -(c / b)
	elif b <= 3.2e-23:
		tmp = -(-math.sqrt((-1.0 / (c * a))) * c)
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.6e-130)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 3.2e-23)
		tmp = Float64(-Float64(Float64(-sqrt(Float64(-1.0 / Float64(c * a)))) * c));
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.6e-130)
		tmp = -(c / b);
	elseif (b <= 3.2e-23)
		tmp = -(-sqrt((-1.0 / (c * a))) * c);
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.6e-130], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 3.2e-23], (-N[((-N[Sqrt[N[(-1.0 / N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * c), $MachinePrecision]), N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.60000000000000058e-130

   1. Initial program 20.2%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   if -8.60000000000000058e-130 < b < 3.19999999999999976e-23

   1. Initial program 80.1%

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

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 65.5%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-neg.f64 65.0

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 65.0

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

   7. Applied rewrites 65.0%

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

   if 3.19999999999999976e-23 < b

   1. Initial program 67.4%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

Alternative 6: 71.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\frac{c}{a}}\\ \mathbf{if}\;b \leq -1.42 \cdot 10^{-149}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{-217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.9 \cdot 10^{-230}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} + \frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (/ c a)))))
   (if (<= b -1.42e-149)
     (- (/ c b))
     (if (<= b -3.2e-217)
       t_0
       (if (<= b 7.9e-230) (- t_0) (+ (/ c b) (/ (- b) a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(-(c / a));
	double tmp;
	if (b <= -1.42e-149) {
		tmp = -(c / b);
	} else if (b <= -3.2e-217) {
		tmp = t_0;
	} else if (b <= 7.9e-230) {
		tmp = -t_0;
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-(c / a))
    if (b <= (-1.42d-149)) then
        tmp = -(c / b)
    else if (b <= (-3.2d-217)) then
        tmp = t_0
    else if (b <= 7.9d-230) then
        tmp = -t_0
    else
        tmp = (c / b) + (-b / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(-(c / a));
	double tmp;
	if (b <= -1.42e-149) {
		tmp = -(c / b);
	} else if (b <= -3.2e-217) {
		tmp = t_0;
	} else if (b <= 7.9e-230) {
		tmp = -t_0;
	} else {
		tmp = (c / b) + (-b / a);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(-(c / a))
	tmp = 0
	if b <= -1.42e-149:
		tmp = -(c / b)
	elif b <= -3.2e-217:
		tmp = t_0
	elif b <= 7.9e-230:
		tmp = -t_0
	else:
		tmp = (c / b) + (-b / a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-Float64(c / a)))
	tmp = 0.0
	if (b <= -1.42e-149)
		tmp = Float64(-Float64(c / b));
	elseif (b <= -3.2e-217)
		tmp = t_0;
	elseif (b <= 7.9e-230)
		tmp = Float64(-t_0);
	else
		tmp = Float64(Float64(c / b) + Float64(Float64(-b) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(-(c / a));
	tmp = 0.0;
	if (b <= -1.42e-149)
		tmp = -(c / b);
	elseif (b <= -3.2e-217)
		tmp = t_0;
	elseif (b <= 7.9e-230)
		tmp = -t_0;
	else
		tmp = (c / b) + (-b / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[b, -1.42e-149], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, -3.2e-217], t$95$0, If[LessEqual[b, 7.9e-230], (-t$95$0), N[(N[(c / b), $MachinePrecision] + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.42e-149

   1. Initial program 21.3%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   if -1.42e-149 < b < -3.2000000000000001e-217

   1. Initial program 72.3%

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

   2. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 32.2

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

   4. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.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 32.2

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

   6. Applied rewrites 32.2%

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

   if -3.2000000000000001e-217 < b < 7.89999999999999994e-230

   1. Initial program 76.6%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 4.7

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

   4. Applied rewrites 4.7%

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

   5. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 40.4

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

   10. Applied rewrites 40.4%

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

   if 7.89999999999999994e-230 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 73.2

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

   4. Applied rewrites 73.2%

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

Alternative 7: 71.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\frac{c}{a}}\\ \mathbf{if}\;b \leq -1.42 \cdot 10^{-149}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{-217}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 7.9 \cdot 10^{-230}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (/ c a)))))
   (if (<= b -1.42e-149)
     (- (/ c b))
     (if (<= b -3.2e-217) t_0 (if (<= b 7.9e-230) (- t_0) (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(-(c / a));
	double tmp;
	if (b <= -1.42e-149) {
		tmp = -(c / b);
	} else if (b <= -3.2e-217) {
		tmp = t_0;
	} else if (b <= 7.9e-230) {
		tmp = -t_0;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(-(c / a))
    if (b <= (-1.42d-149)) then
        tmp = -(c / b)
    else if (b <= (-3.2d-217)) then
        tmp = t_0
    else if (b <= 7.9d-230) then
        tmp = -t_0
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(-(c / a));
	double tmp;
	if (b <= -1.42e-149) {
		tmp = -(c / b);
	} else if (b <= -3.2e-217) {
		tmp = t_0;
	} else if (b <= 7.9e-230) {
		tmp = -t_0;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(-(c / a))
	tmp = 0
	if b <= -1.42e-149:
		tmp = -(c / b)
	elif b <= -3.2e-217:
		tmp = t_0
	elif b <= 7.9e-230:
		tmp = -t_0
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(-Float64(c / a)))
	tmp = 0.0
	if (b <= -1.42e-149)
		tmp = Float64(-Float64(c / b));
	elseif (b <= -3.2e-217)
		tmp = t_0;
	elseif (b <= 7.9e-230)
		tmp = Float64(-t_0);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(-(c / a));
	tmp = 0.0;
	if (b <= -1.42e-149)
		tmp = -(c / b);
	elseif (b <= -3.2e-217)
		tmp = t_0;
	elseif (b <= 7.9e-230)
		tmp = -t_0;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[b, -1.42e-149], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, -3.2e-217], t$95$0, If[LessEqual[b, 7.9e-230], (-t$95$0), N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.42e-149

   1. Initial program 21.3%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   if -1.42e-149 < b < -3.2000000000000001e-217

   1. Initial program 72.3%

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

   2. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 32.2

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

   4. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.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 32.2

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

   6. Applied rewrites 32.2%

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

   if -3.2000000000000001e-217 < b < 7.89999999999999994e-230

   1. Initial program 76.6%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 4.7

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

   4. Applied rewrites 4.7%

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

   5. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 37.2

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

   7. Applied rewrites 37.2%

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

   8. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-/.f64 40.4

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

   10. Applied rewrites 40.4%

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

   if 7.89999999999999994e-230 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 72.7

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

   4. Applied rewrites 72.7%

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

Alternative 8: 70.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.42 \cdot 10^{-149}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-234}:\\ \;\;\;\;\sqrt{-\frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.42e-149)
   (- (/ c b))
   (if (<= b 6.8e-234) (sqrt (- (/ c a))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.42e-149) {
		tmp = -(c / b);
	} else if (b <= 6.8e-234) {
		tmp = sqrt(-(c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.42d-149)) then
        tmp = -(c / b)
    else if (b <= 6.8d-234) then
        tmp = sqrt(-(c / a))
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.42e-149) {
		tmp = -(c / b);
	} else if (b <= 6.8e-234) {
		tmp = Math.sqrt(-(c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.42e-149:
		tmp = -(c / b)
	elif b <= 6.8e-234:
		tmp = math.sqrt(-(c / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.42e-149)
		tmp = Float64(-Float64(c / b));
	elseif (b <= 6.8e-234)
		tmp = sqrt(Float64(-Float64(c / a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.42e-149)
		tmp = -(c / b);
	elseif (b <= 6.8e-234)
		tmp = sqrt(-(c / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.42e-149], (-N[(c / b), $MachinePrecision]), If[LessEqual[b, 6.8e-234], N[Sqrt[(-N[(c / a), $MachinePrecision])], $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.42e-149

   1. Initial program 21.3%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 80.6

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

   4. Applied rewrites 80.6%

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

   if -1.42e-149 < b < 6.79999999999999971e-234

   1. Initial program 75.2%

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

   2. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 35.4

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

   4. Applied rewrites 35.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.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 35.4

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

   6. Applied rewrites 35.4%

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

   if 6.79999999999999971e-234 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 72.5

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

   4. Applied rewrites 72.5%

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

Alternative 9: 67.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = -(c / b)
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 31.1%

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

   2. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 67.8

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

   4. Applied rewrites 67.8%

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

   if -1.999999999999994e-310 < b

   1. Initial program 73.2%

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

   2. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 66.8

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

   4. Applied rewrites 66.8%

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

Alternative 10: 35.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 35.0

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

4. Applied rewrites 35.0%

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

Alternative 11: 10.3% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in c around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. mul-1-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lower-/.f64 34.8

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

4. Applied rewrites 34.8%

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

5. Taylor expanded in a around inf

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

   1. lift-/.f64 10.3

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

7. Applied rewrites 10.3%

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

Developer Target 1: 70.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - (4.0d0 * (a * c))))
    if (b < 0.0d0) then
        tmp = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-b - t_0) / (2.0d0 * a)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025114 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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