Quadratic roots, medium range

Percentage Accurate: 30.7% → 99.6%

Time: 6.1s

Alternatives: 9

Speedup: 4.3×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

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: 30.7% accurate, 1.0× speedup

Math

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

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: 99.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

3. Applied rewrites 30.2%

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

4. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 99.4%

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

10. Applied rewrites 99.6%

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

Alternative 2: 99.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  (*
 (/ (/ (- c) a) (fma (sqrt (fma (* c a) -4.0 (* b b))) 1.0 b))
 (+ a a)))

C

double code(double a, double b, double c) {
	return ((-c / a) / fma(sqrt(fma((c * a), -4.0, (b * b))), 1.0, b)) * (a + a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(-c) / a) / fma(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))), 1.0, b)) * Float64(a + a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[((-c) / a), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0 + b), $MachinePrecision]), $MachinePrecision] * N[(a + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.7%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

3. Applied rewrites 30.2%

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

4. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 99.4%

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

6. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac N/A

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

   5. lift-neg.f64 N/A

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

   6. lower-/.f64 99.4%

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

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-c}{\color{blue}{a}}}{\mathsf{Rewrite=>}\left(lift--.f64, \left(\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a} - \frac{b}{-2 \cdot a}\right)\right)} \]

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-c}{\color{blue}{a}}}{\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{-2 \cdot a}\right)\right)} \]

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

       \[\leadsto \frac{\frac{-c}{\color{blue}{a}}}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{a + a}, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, \mathsf{neg}\left(\frac{b}{-2 \cdot a}\right)\right)\right)\right)} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{-c}{a}}{\mathsf{fma}\left(\frac{0.5}{a}, \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}, \frac{b}{a + a}\right)}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. add-to-fraction N/A

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

   5. associate-/r/ N/A

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

10. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. flip-+ N/A

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

   6. lower-unsound-/.f64 N/A

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

3. Applied rewrites 30.2%

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

4. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 99.4%

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

6. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac N/A

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

   5. lift-neg.f64 N/A

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

   6. lower-/.f64 99.4%

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

   7. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-c}{\color{blue}{a}}}{\mathsf{Rewrite=>}\left(lift--.f64, \left(\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a} - \frac{b}{-2 \cdot a}\right)\right)} \]

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-c}{\color{blue}{a}}}{\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a}\right)\right) + \left(\mathsf{neg}\left(\frac{b}{-2 \cdot a}\right)\right)} \]

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

       \[\leadsto \frac{\frac{-c}{\color{blue}{a}}}{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(\frac{1}{a + a}, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}, \mathsf{neg}\left(\frac{b}{-2 \cdot a}\right)\right)\right)\right)} \]

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{\frac{-c}{a}}{\mathsf{fma}\left(\frac{0.5}{a}, \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}, \frac{b}{a + a}\right)}} \]
9. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. add-flip N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. metadata-eval N/A

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

   6. associate-/r* N/A

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

   7. count-2-rev N/A

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

   8. lift-+.f64 N/A

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

   9. mult-flip-rev N/A

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

   10. lift-/.f64 N/A

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

   11. distribute-neg-frac N/A

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

   12. sub-div N/A

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

   13. lower--.f32 N/A

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

   14. lower-unsound--.f32 N/A

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

   15. lower-/.f64 N/A

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

10. Applied rewrites 99.4%

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

Alternative 4: 84.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(1 + \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{-b}\right) \cdot \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     -1.8e-5)
  (/
   (* (+ 1.0 (/ (sqrt (fma (* c -4.0) a (* b b))) (- b))) (- b))
   (* 2.0 a))
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.8e-5) {
		tmp = ((1.0 + (sqrt(fma((c * -4.0), a, (b * b))) / -b)) * -b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.8e-5)
		tmp = Float64(Float64(Float64(1.0 + Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) / Float64(-b))) * Float64(-b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -1.8e-5], N[(N[(N[(1.0 + N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-b)), $MachinePrecision]), $MachinePrecision] * (-b)), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.8e-5

   1. Initial program 30.7%

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

      1. lift-+.f64 N/A

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

      2. sum-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 30.6%

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

   if -1.8e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 30.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.8%

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.8%

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

   6. Applied rewrites 81.8%

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

Alternative 5: 84.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{a + a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     -1.8e-5)
  (/ 1.0 (/ (+ a a) (- (sqrt (fma b b (* (* -4.0 c) a))) b)))
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.8e-5) {
		tmp = 1.0 / ((a + a) / (sqrt(fma(b, b, ((-4.0 * c) * a))) - b));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.8e-5)
		tmp = Float64(1.0 / Float64(Float64(a + a) / Float64(sqrt(fma(b, b, Float64(Float64(-4.0 * c) * a))) - b)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -1.8e-5], N[(1.0 / N[(N[(a + a), $MachinePrecision] / N[(N[Sqrt[N[(b * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.8e-5

   1. Initial program 30.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 30.7%

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 30.7%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 30.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 30.8%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 30.8%

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

   5. Applied rewrites 30.8%

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

   if -1.8e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 30.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.8%

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.8%

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

   6. Applied rewrites 81.8%

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

Alternative 6: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     -1.8e-5)
  (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.8e-5) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.8e-5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -1.8e-5], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.8e-5

   1. Initial program 30.7%

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

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqr-neg-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 30.8%

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

   3. Applied rewrites 30.8%

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

   if -1.8e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 30.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.8%

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.8%

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

   6. Applied rewrites 81.8%

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

Alternative 7: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     -1.8e-5)
  (/ (- (sqrt (fma (* c a) -4.0 (* b b))) b) (+ a a))
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.8e-5) {
		tmp = (sqrt(fma((c * a), -4.0, (b * b))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.8e-5)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -1.8e-5], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.8e-5

   1. Initial program 30.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-2neg N/A

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

      5. mult-flip N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 32.0%

          \[\leadsto \mathsf{fma}\left(b, \frac{-0.5}{a}, \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\right) \]

   3. Applied rewrites 32.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a}\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 31.8%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{1}{\color{blue}{\frac{a + a}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}}}\right) \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 31.8%

         \[\leadsto \mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{1}{\frac{a + a}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)}}}\right) \]

   5. Applied rewrites 31.8%

      \[\leadsto \mathsf{fma}\left(b, \frac{-0.5}{a}, \color{blue}{\frac{1}{\frac{a + a}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}}}\right) \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. div-flip-rev N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. associate-/r* N/A

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

      13. count-2-rev N/A

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

      14. lift-+.f64 N/A

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

      15. *-commutative N/A

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

      16. mult-flip N/A

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

      17. sub-div N/A

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

   7. Applied rewrites 30.7%

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

   if -1.8e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 30.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.8%

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.8%

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

   6. Applied rewrites 81.8%

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

Alternative 8: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     -1.8e-5)
  (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (+ a a))
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1.8e-5) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1.8e-5)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -1.8e-5], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1.8e-5

   1. Initial program 30.7%

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

   3. Applied rewrites 30.7%

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

   if -1.8e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 30.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.8%

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

   4. Applied rewrites 81.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.8%

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

   6. Applied rewrites 81.8%

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

Alternative 9: 81.8% accurate, 4.3× speedup

Math

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

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 30.7%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 81.8%

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

4. Applied rewrites 81.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 81.8%

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

6. Applied rewrites 81.8%

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

Reproduce

herbie shell --seed 2025255 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))