quadm (p42, negative)

Percentage Accurate: 51.6% → 86.2%

Time: 4.5s

Alternatives: 12

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq -2.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\mathsf{fma}\left(-1, b, t\_0\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+100}:\\ \;\;\;\;-\left(\frac{b}{a + a} + \frac{t\_0}{2 \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b -2.6e+52)
     (/ (- c) b)
     (if (<= b -2.8e-96)
       (/ (/ (* (* a c) 4.0) (fma -1.0 b t_0)) (* 2.0 a))
       (if (<= b 5e+100)
         (- (+ (/ b (+ a a)) (/ t_0 (* 2.0 a))))
         (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp;
	if (b <= -2.6e+52) {
		tmp = -c / b;
	} else if (b <= -2.8e-96) {
		tmp = (((a * c) * 4.0) / fma(-1.0, b, t_0)) / (2.0 * a);
	} else if (b <= 5e+100) {
		tmp = -((b / (a + a)) + (t_0 / (2.0 * a)));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (b <= -2.6e+52)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= -2.8e-96)
		tmp = Float64(Float64(Float64(Float64(a * c) * 4.0) / fma(-1.0, b, t_0)) / Float64(2.0 * a));
	elseif (b <= 5e+100)
		tmp = Float64(-Float64(Float64(b / Float64(a + a)) + Float64(t_0 / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.6e+52], N[((-c) / b), $MachinePrecision], If[LessEqual[b, -2.8e-96], N[(N[(N[(N[(a * c), $MachinePrecision] * 4.0), $MachinePrecision] / N[(-1.0 * b + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+100], (-N[(N[(b / N[(a + a), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[((-b) / a), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.6e52

   1. Initial program 11.2%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 91.8

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

   5. Applied rewrites 91.8%

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

   if -2.6e52 < b < -2.80000000000000015e-96

   1. Initial program 38.2%

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

      1. lift-neg.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -2.80000000000000015e-96 < b < 4.9999999999999999e100

   1. Initial program 78.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.9

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

   6. Applied rewrites 78.9%

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

   if 4.9999999999999999e100 < b

   1. Initial program 48.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. 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. lower-/.f64 N/A

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

      4. lift-neg.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq -2.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+100}:\\ \;\;\;\;-\left(\frac{b}{a + a} + \frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{2 \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+100}:\\ \;\;\;\;-\left(\frac{b}{a + a} + \frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{2 \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-64)
   (/ (- c) b)
   (if (<= b 5e+100)
     (- (+ (/ b (+ a a)) (/ (sqrt (fma (* -4.0 a) c (* b b))) (* 2.0 a))))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-64) {
		tmp = -c / b;
	} else if (b <= 5e+100) {
		tmp = -((b / (a + a)) + (sqrt(fma((-4.0 * a), c, (b * b))) / (2.0 * a)));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-64)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5e+100)
		tmp = Float64(-Float64(Float64(b / Float64(a + a)) + Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) / Float64(2.0 * a))));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-64], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5e+100], (-N[(N[(b / N[(a + a), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.2999999999999999e-64

   1. Initial program 15.6%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 86.6

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

   5. Applied rewrites 86.6%

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

   if -3.2999999999999999e-64 < b < 4.9999999999999999e100

   1. Initial program 77.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 77.2

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

   6. Applied rewrites 77.2%

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

   if 4.9999999999999999e100 < b

   1. Initial program 48.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. 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. lower-/.f64 N/A

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

      4. lift-neg.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+100}:\\ \;\;\;\;-\left(\frac{b}{a + a} + \frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{2 \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-64)
   (/ (- c) b)
   (if (<= b 5e+100)
     (/ (- (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (+ a a))
     (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-64) {
		tmp = -c / b;
	} else if (b <= 5e+100) {
		tmp = (-b - sqrt(fma((c * a), -4.0, (b * b)))) / (a + a);
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-64)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 5e+100)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-64], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 5e+100], N[(N[((-b) - N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + 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 < -3.2999999999999999e-64

   1. Initial program 15.6%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 86.6

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

   5. Applied rewrites 86.6%

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

   if -3.2999999999999999e-64 < b < 4.9999999999999999e100

   1. Initial program 77.2%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      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. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 77.2

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

   6. Applied rewrites 77.2%

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

   if 4.9999999999999999e100 < b

   1. Initial program 48.2%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. 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. lower-/.f64 N/A

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

      4. lift-neg.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-64)
   (/ (- c) b)
   (if (<= b 1.55e-59)
     (/ (+ b (sqrt (* (* a c) -4.0))) (- (+ a a)))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-64) {
		tmp = -c / b;
	} else if (b <= 1.55e-59) {
		tmp = (b + sqrt(((a * c) * -4.0))) / -(a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.3d-64)) then
        tmp = -c / b
    else if (b <= 1.55d-59) then
        tmp = (b + sqrt(((a * c) * (-4.0d0)))) / -(a + a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-64) {
		tmp = -c / b;
	} else if (b <= 1.55e-59) {
		tmp = (b + Math.sqrt(((a * c) * -4.0))) / -(a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.3e-64:
		tmp = -c / b
	elif b <= 1.55e-59:
		tmp = (b + math.sqrt(((a * c) * -4.0))) / -(a + a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-64)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.55e-59)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(a * c) * -4.0))) / Float64(-Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.3e-64)
		tmp = -c / b;
	elseif (b <= 1.55e-59)
		tmp = (b + sqrt(((a * c) * -4.0))) / -(a + a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-64], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.55e-59], N[(N[(b + N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(a + a), $MachinePrecision])), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.2999999999999999e-64

   1. Initial program 15.6%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 86.6

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

   5. Applied rewrites 86.6%

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

   if -3.2999999999999999e-64 < b < 1.55e-59

   1. Initial program 71.6%

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      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. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 71.6

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

   4. Applied rewrites 71.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 71.6

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

   6. Applied rewrites 71.6%

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

   7. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 66.8

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

   9. Applied rewrites 66.8%

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

   if 1.55e-59 < b

   1. Initial program 63.4%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 89.8

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

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;\frac{b + \sqrt{\left(a \cdot c\right) \cdot -4}}{-\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-59}:\\ \;\;\;\;\frac{-\sqrt{-4 \cdot \left(c \cdot a\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-64)
   (/ (- c) b)
   (if (<= b 1.12e-59)
     (/ (- (sqrt (* -4.0 (* c a)))) (+ a a))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-64) {
		tmp = -c / b;
	} else if (b <= 1.12e-59) {
		tmp = -sqrt((-4.0 * (c * a))) / (a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.3d-64)) then
        tmp = -c / b
    else if (b <= 1.12d-59) then
        tmp = -sqrt(((-4.0d0) * (c * a))) / (a + a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-64) {
		tmp = -c / b;
	} else if (b <= 1.12e-59) {
		tmp = -Math.sqrt((-4.0 * (c * a))) / (a + a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.3e-64:
		tmp = -c / b
	elif b <= 1.12e-59:
		tmp = -math.sqrt((-4.0 * (c * a))) / (a + a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-64)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.12e-59)
		tmp = Float64(Float64(-sqrt(Float64(-4.0 * Float64(c * a)))) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.3e-64)
		tmp = -c / b;
	elseif (b <= 1.12e-59)
		tmp = -sqrt((-4.0 * (c * a))) / (a + a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-64], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.12e-59], N[((-N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.2999999999999999e-64

   1. Initial program 15.6%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 86.6

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

   5. Applied rewrites 86.6%

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

   if -3.2999999999999999e-64 < b < 1.1200000000000001e-59

   1. Initial program 71.6%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 65.8

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

   5. Applied rewrites 65.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 65.8

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

   7. Applied rewrites 65.8%

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

   if 1.1200000000000001e-59 < b

   1. Initial program 63.4%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 89.8

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

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.12 \cdot 10^{-59}:\\ \;\;\;\;\frac{-\sqrt{-4 \cdot \left(c \cdot a\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-145}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{-c}}{-\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.05e-145)
   (/ (- c) b)
   (if (<= b 8e-88) (/ (sqrt (- c)) (- (sqrt a))) (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-145) {
		tmp = -c / b;
	} else if (b <= 8e-88) {
		tmp = sqrt(-c) / -sqrt(a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.05d-145)) then
        tmp = -c / b
    else if (b <= 8d-88) then
        tmp = sqrt(-c) / -sqrt(a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-145) {
		tmp = -c / b;
	} else if (b <= 8e-88) {
		tmp = Math.sqrt(-c) / -Math.sqrt(a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.05e-145:
		tmp = -c / b
	elif b <= 8e-88:
		tmp = math.sqrt(-c) / -math.sqrt(a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.05e-145)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 8e-88)
		tmp = Float64(sqrt(Float64(-c)) / Float64(-sqrt(a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.05e-145)
		tmp = -c / b;
	elseif (b <= 8e-88)
		tmp = sqrt(-c) / -sqrt(a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.05e-145], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 8e-88], N[(N[Sqrt[(-c)], $MachinePrecision] / (-N[Sqrt[a], $MachinePrecision])), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.04999999999999996e-145

   1. Initial program 23.0%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 77.7

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

   5. Applied rewrites 77.7%

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

   if -1.04999999999999996e-145 < b < 7.99999999999999947e-88

   1. Initial program 76.2%

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

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   4. 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 24.8

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

   5. Applied rewrites 24.8%

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

   6. Taylor expanded in c around -inf

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

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 38.7

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

   8. Applied rewrites 38.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-sqrt.f64 57.6

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

   10. Applied rewrites 57.6%

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

   if 7.99999999999999947e-88 < b

   1. Initial program 64.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 86.4

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

   5. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 86.4

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

   7. Applied rewrites 86.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-145}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{-c}}{-\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e-146)
   (/ (- c) b)
   (if (<= b 1.6e-109) (- (sqrt (/ (- c) a))) (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-146) {
		tmp = -c / b;
	} else if (b <= 1.6e-109) {
		tmp = -sqrt((-c / a));
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d-146)) then
        tmp = -c / b
    else if (b <= 1.6d-109) then
        tmp = -sqrt((-c / a))
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-146) {
		tmp = -c / b;
	} else if (b <= 1.6e-109) {
		tmp = -Math.sqrt((-c / a));
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e-146:
		tmp = -c / b
	elif b <= 1.6e-109:
		tmp = -math.sqrt((-c / a))
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e-146)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.6e-109)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e-146)
		tmp = -c / b;
	elseif (b <= 1.6e-109)
		tmp = -sqrt((-c / a));
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e-146], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.6e-109], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0999999999999999e-146

   1. Initial program 23.0%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 77.7

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

   5. Applied rewrites 77.7%

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

   if -2.0999999999999999e-146 < b < 1.6000000000000001e-109

   1. Initial program 73.4%

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

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   4. 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 26.8

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

   5. Applied rewrites 26.8%

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

   6. Taylor expanded in c around -inf

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

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 40.0

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

   8. Applied rewrites 40.0%

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

   if 1.6000000000000001e-109 < b

   1. Initial program 66.8%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 82.8

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

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 82.8

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

   7. Applied rewrites 82.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e-146)
   (/ (- c) b)
   (if (<= b 1.6e-109) (- (sqrt (/ (- c) a))) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e-146) {
		tmp = -c / b;
	} else if (b <= 1.6e-109) {
		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 <= (-2.1d-146)) then
        tmp = -c / b
    else if (b <= 1.6d-109) 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 <= -2.1e-146) {
		tmp = -c / b;
	} else if (b <= 1.6e-109) {
		tmp = -Math.sqrt((-c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e-146:
		tmp = -c / b
	elif b <= 1.6e-109:
		tmp = -math.sqrt((-c / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e-146)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 1.6e-109)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e-146)
		tmp = -c / b;
	elseif (b <= 1.6e-109)
		tmp = -sqrt((-c / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e-146], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 1.6e-109], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0999999999999999e-146

   1. Initial program 23.0%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 77.7

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

   5. Applied rewrites 77.7%

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

   if -2.0999999999999999e-146 < b < 1.6000000000000001e-109

   1. Initial program 73.4%

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

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   4. 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 26.8

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

   5. Applied rewrites 26.8%

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

   6. Taylor expanded in c around -inf

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

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 40.0

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

   8. Applied rewrites 40.0%

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

   if 1.6000000000000001e-109 < b

   1. Initial program 66.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. 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. lower-/.f64 N/A

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

      4. lift-neg.f64 82.7

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

   5. Applied rewrites 82.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-146}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-149}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-149)
   (/ (- c) b)
   (if (<= b 3.2e-160) (sqrt (/ (- c) a)) (/ (- b) a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-149) {
		tmp = -c / b;
	} else if (b <= 3.2e-160) {
		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 <= (-3.7d-149)) then
        tmp = -c / b
    else if (b <= 3.2d-160) 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 <= -3.7e-149) {
		tmp = -c / b;
	} else if (b <= 3.2e-160) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-149:
		tmp = -c / b
	elif b <= 3.2e-160:
		tmp = math.sqrt((-c / a))
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-149)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 3.2e-160)
		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 <= -3.7e-149)
		tmp = -c / b;
	elseif (b <= 3.2e-160)
		tmp = sqrt((-c / a));
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-149], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 3.2e-160], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.6999999999999999e-149

   1. Initial program 24.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 76.2

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

   5. Applied rewrites 76.2%

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

   if -3.6999999999999999e-149 < b < 3.20000000000000009e-160

   1. Initial program 74.0%

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

   3. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\sqrt{\frac{c}{a}} \cdot \sqrt{-1}} \]
   4. 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 31.0

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

   5. Applied rewrites 31.0%

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

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 31.0

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

   7. Applied rewrites 31.0%

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

   if 3.20000000000000009e-160 < b

   1. Initial program 66.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. 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. lower-/.f64 N/A

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

      4. lift-neg.f64 77.4

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

   5. Applied rewrites 77.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-149}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-160}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-272}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-272) (/ (- c) b) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-272) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-272)) then
        tmp = -c / b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-272) {
		tmp = -c / b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-272:
		tmp = -c / b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-272)
		tmp = Float64(Float64(-c) / b);
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e-272)
		tmp = -c / b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-272], N[((-c) / b), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.99999999999999972e-272

   1. Initial program 37.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. 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 61.7

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

   5. Applied rewrites 61.7%

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

   if -3.99999999999999972e-272 < b

   1. Initial program 65.3%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. 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. lower-/.f64 N/A

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

      4. lift-neg.f64 67.2

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

   5. Applied rewrites 67.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-272}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.4% accurate, 3.6× 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 51.5%

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

3. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. 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 31.8

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

5. Applied rewrites 31.8%

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

6. Final simplification 31.8%

   \[\leadsto \frac{-c}{b} \]
7. Add Preprocessing

Alternative 12: 10.4% accurate, 4.2× 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 51.5%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 35.0

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

5. Applied rewrites 35.0%

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

6. Taylor expanded in a around inf

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

   1. lift-/.f64 9.8

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

8. Applied rewrites 9.8%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025036 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64
  :herbie-expected 10

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

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