Quadratic roots, full range

Percentage Accurate: 52.0% → 84.8%

Time: 7.6s

Alternatives: 9

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\right) + \mathsf{fma}\left(a \cdot \frac{c}{b}, 2, -b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}{2 \cdot a} + \frac{b}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+76)
   (/ (+ (- b) (fma (* a (/ c b)) 2.0 (- b))) (* 2.0 a))
   (if (<= b 1.9e-88)
     (+ (/ (sqrt (fma a (* -4.0 c) (* b b))) (* 2.0 a)) (/ b (* -2.0 a)))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+76) {
		tmp = (-b + fma((a * (c / b)), 2.0, -b)) / (2.0 * a);
	} else if (b <= 1.9e-88) {
		tmp = (sqrt(fma(a, (-4.0 * c), (b * b))) / (2.0 * a)) + (b / (-2.0 * a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+76)
		tmp = Float64(Float64(Float64(-b) + fma(Float64(a * Float64(c / b)), 2.0, Float64(-b))) / Float64(2.0 * a));
	elseif (b <= 1.9e-88)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * b))) / Float64(2.0 * a)) + Float64(b / Float64(-2.0 * a)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+76], N[(N[((-b) + N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 2.0 + (-b)), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-88], N[(N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] + N[(b / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999996e76

   1. Initial program 54.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.2%

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

   if -2.49999999999999996e76 < b < 1.90000000000000006e-88

   1. Initial program 82.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 82.5

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

   4. Applied rewrites 82.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-neg.f64 N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-in N/A

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

      19. metadata-eval N/A

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

      20. lift-*.f64 82.5

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

   6. Applied rewrites 82.5%

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

   if 1.90000000000000006e-88 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 2: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\right) + \mathsf{fma}\left(a \cdot \frac{c}{b}, 2, -b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.5e+76)
   (/ (+ (- b) (fma (* a (/ c b)) 2.0 (- b))) (* 2.0 a))
   (if (<= b 1.9e-88)
     (/ (- (sqrt (fma a (* -4.0 c) (* b b))) b) (+ a a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.5e+76) {
		tmp = (-b + fma((a * (c / b)), 2.0, -b)) / (2.0 * a);
	} else if (b <= 1.9e-88) {
		tmp = (sqrt(fma(a, (-4.0 * c), (b * b))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.5e+76)
		tmp = Float64(Float64(Float64(-b) + fma(Float64(a * Float64(c / b)), 2.0, Float64(-b))) / Float64(2.0 * a));
	elseif (b <= 1.9e-88)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.5e+76], N[(N[((-b) + N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * 2.0 + (-b)), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.9e-88], N[(N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.49999999999999996e76

   1. Initial program 54.7%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 97.2%

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

   if -2.49999999999999996e76 < b < 1.90000000000000006e-88

   1. Initial program 82.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 82.5

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

   4. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 82.5

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

   6. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

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

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-down N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. pow2 N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lift-neg.f64 N/A

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

      18. lower--.f64 51.4

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

   8. Applied rewrites 82.5%

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

   if 1.90000000000000006e-88 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 3: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+29}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+29)
   (- (fma (/ (- c) (* b b)) b (/ b a)))
   (if (<= b 1.9e-88)
     (/ (- (sqrt (fma a (* -4.0 c) (* b b))) b) (+ a a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+29) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 1.9e-88) {
		tmp = (sqrt(fma(a, (-4.0 * c), (b * b))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+29)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 1.9e-88)
		tmp = Float64(Float64(sqrt(fma(a, Float64(-4.0 * c), Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+29], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 1.9e-88], N[(N[(N[Sqrt[N[(a * N[(-4.0 * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5e29

   1. Initial program 59.9%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -1.5e29 < b < 1.90000000000000006e-88

   1. Initial program 81.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 81.4

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

   4. Applied rewrites 81.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 81.4

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

   6. Applied rewrites 81.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

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

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-down N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. pow2 N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lift-neg.f64 N/A

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

      18. lower--.f64 57.3

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

   8. Applied rewrites 81.4%

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

   if 1.90000000000000006e-88 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{-116}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{-88}:\\ \;\;\;\;\frac{\sqrt{-4 \cdot \left(c \cdot a\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.18e-116)
   (- (fma (/ (- c) (* b b)) b (/ b a)))
   (if (<= b 1.9e-88) (/ (- (sqrt (* -4.0 (* c a))) b) (+ a a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.18e-116) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 1.9e-88) {
		tmp = (sqrt((-4.0 * (c * a))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.18e-116)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 1.9e-88)
		tmp = Float64(Float64(sqrt(Float64(-4.0 * Float64(c * a))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.18e-116], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 1.9e-88], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1800000000000001e-116

   1. Initial program 69.4%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

   if -1.1800000000000001e-116 < b < 1.90000000000000006e-88

   1. Initial program 75.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 75.6

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

   4. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.6

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

   6. Applied rewrites 75.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. unpow1 N/A

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

      4. sqr-pow N/A

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

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

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-prod-down N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. sqr-neg-rev N/A

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

      13. pow2 N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lift-neg.f64 N/A

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

      18. lower--.f64 74.8

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 75.6

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

   11. Applied rewrites 75.6%

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

   if 1.90000000000000006e-88 < b

   1. Initial program 19.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 84.5

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

   5. Applied rewrites 84.5%

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

Alternative 5: 66.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-309}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5e-309) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5e-309) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5d-309) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5e-309) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5e-309:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5e-309)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5e-309)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5e-309], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9999999999999995e-309

   1. Initial program 73.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 69.5

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

   5. Applied rewrites 69.5%

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

   if 4.9999999999999995e-309 < b

   1. Initial program 29.7%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 69.9

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

   5. Applied rewrites 69.9%

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

Alternative 6: 42.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -1.05e-304) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-304) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.05d-304)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-304) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.05e-304:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.05e-304)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.05e-304)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.05e-304], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.05000000000000004e-304

   1. Initial program 73.7%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if -1.05000000000000004e-304 < b

   1. Initial program 29.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 19.1

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

   5. Applied rewrites 19.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

      11. frac-add N/A

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \color{blue}{0}}{a} \]

      5. metadata-eval N/A

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

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-/.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-inverses 17.4

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

   10. Applied rewrites 17.4%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 10.7% 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 53.5%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 33.0

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

5. Applied rewrites 33.0%

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

7. Applied rewrites 9.1%

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

Alternative 8: 2.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in b around -inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 39.3

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

5. Applied rewrites 39.3%

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

7. Applied rewrites 2.6%

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

9. Applied rewrites 2.6%

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

Alternative 9: 10.9% accurate, 50.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 9.9

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

5. Applied rewrites 9.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. frac-2neg N/A

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

   9. lift-neg.f64 N/A

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

   10. remove-double-neg N/A

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

   11. frac-add N/A

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

7. Applied rewrites 4.1%

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

8. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-inverses N/A

      \[\leadsto \frac{\frac{-1}{4} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

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

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

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

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

   9. lower-/.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. +-inverses 9.4

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

10. Applied rewrites 9.4%

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

11. Final simplification 9.4%

    \[\leadsto 0 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))