Quadratic roots, medium range

Percentage Accurate: 31.1% → 95.7%

Time: 7.4s

Alternatives: 15

Speedup: 4.6×

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: 31.1% 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: 95.7% accurate, 0.2× speedup

Math

\[\frac{-c}{b} - \frac{\mathsf{fma}\left(\frac{5}{{b}^{6} \cdot a}, {\left(c \cdot a\right)}^{4}, \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, \left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot 2\right)\right)}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (-
  (/ (- c) b)
  (/
   (fma
    (/ 5.0 (* (pow b 6.0) a))
    (pow (* c a) 4.0)
    (fma
     (* c c)
     (/ a (* b b))
     (* (* (* (* (pow b -4.0) c) (* (* a a) c)) c) 2.0)))
   b)))

C

double code(double a, double b, double c) {
	return (-c / b) - (fma((5.0 / (pow(b, 6.0) * a)), pow((c * a), 4.0), fma((c * c), (a / (b * b)), ((((pow(b, -4.0) * c) * ((a * a) * c)) * c) * 2.0))) / b);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) / b) - Float64(fma(Float64(5.0 / Float64((b ^ 6.0) * a)), (Float64(c * a) ^ 4.0), fma(Float64(c * c), Float64(a / Float64(b * b)), Float64(Float64(Float64(Float64((b ^ -4.0) * c) * Float64(Float64(a * a) * c)) * c) * 2.0))) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[((-c) / b), $MachinePrecision] - N[(N[(N[(5.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{b}} \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(\frac{1}{4}, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(\frac{1}{4}, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\left(\frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) + \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   4. associate--l+ N/A

      \[\leadsto \frac{\left(-c\right) - \left(\frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) + \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)\right)}{b} \]

   5. associate--r+ N/A

      \[\leadsto \frac{\left(\left(-c\right) - \frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right)\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\left(\left(-c\right) - \frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right)\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

8. Applied rewrites 95.7%

   \[\leadsto \frac{\left(\left(-c\right) - \left({\left(c \cdot a\right)}^{4} \cdot 0.25\right) \cdot \frac{20}{{b}^{6} \cdot a}\right) - \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left(\left(-c\right) - \left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a}\right) - \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{\color{blue}{b}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\left(\left(-c\right) - \left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a}\right) - \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{\left(\left(-c\right) - \left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a}\right) - \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b} \]

   4. associate--l- N/A

      \[\leadsto \frac{\left(-c\right) - \left(\left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a} + \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)\right)}{b} \]

   5. div-sub N/A

      \[\leadsto \frac{-c}{b} - \color{blue}{\frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a} + \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b}} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{-c}{b} - \color{blue}{\frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a} + \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b}} \]

10. Applied rewrites 95.7%

    \[\leadsto \frac{-c}{b} - \color{blue}{\frac{\mathsf{fma}\left(\frac{5}{{b}^{6} \cdot a}, {\left(c \cdot a\right)}^{4}, \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, \left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot 2\right)\right)}{b}} \]
11. Add Preprocessing

Alternative 2: 95.7% accurate, 0.2× speedup

Math

\[\frac{\left(-\mathsf{fma}\left(\frac{5}{{b}^{6} \cdot a}, {\left(c \cdot a\right)}^{4}, c\right)\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, \left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot 2\right)}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (- (fma (/ 5.0 (* (pow b 6.0) a)) (pow (* c a) 4.0) c))
   (fma
    (* c c)
    (/ a (* b b))
    (* (* (* (* (pow b -4.0) c) (* (* a a) c)) c) 2.0)))
  b))

C

double code(double a, double b, double c) {
	return (-fma((5.0 / (pow(b, 6.0) * a)), pow((c * a), 4.0), c) - fma((c * c), (a / (b * b)), ((((pow(b, -4.0) * c) * ((a * a) * c)) * c) * 2.0))) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-fma(Float64(5.0 / Float64((b ^ 6.0) * a)), (Float64(c * a) ^ 4.0), c)) - fma(Float64(c * c), Float64(a / Float64(b * b)), Float64(Float64(Float64(Float64((b ^ -4.0) * c) * Float64(Float64(a * a) * c)) * c) * 2.0))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[((-N[(N[(5.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] + c), $MachinePrecision]) - N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{b}} \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(\frac{1}{4}, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(\frac{1}{4}, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\left(\frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) + \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   4. associate--l+ N/A

      \[\leadsto \frac{\left(-c\right) - \left(\frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) + \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)\right)}{b} \]

   5. associate--r+ N/A

      \[\leadsto \frac{\left(\left(-c\right) - \frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right)\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\left(\left(-c\right) - \frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right)\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

8. Applied rewrites 95.7%

   \[\leadsto \frac{\left(\left(-c\right) - \left({\left(c \cdot a\right)}^{4} \cdot 0.25\right) \cdot \frac{20}{{b}^{6} \cdot a}\right) - \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b} \]
9. Step-by-step derivation

10. Applied rewrites 95.7%

    \[\leadsto \frac{\left(-\mathsf{fma}\left(\frac{5}{{b}^{6} \cdot a}, {\left(c \cdot a\right)}^{4}, c\right)\right) - \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, \left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot 2\right)}{\color{blue}{b}} \]
11. Add Preprocessing

Alternative 3: 95.7% accurate, 0.2× speedup

Math

\[\frac{\left(-c\right) - \mathsf{fma}\left(\frac{5}{{b}^{6} \cdot a}, {\left(c \cdot a\right)}^{4}, \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, \left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot 2\right)\right)}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (- c)
   (fma
    (/ 5.0 (* (pow b 6.0) a))
    (pow (* c a) 4.0)
    (fma
     (* c c)
     (/ a (* b b))
     (* (* (* (* (pow b -4.0) c) (* (* a a) c)) c) 2.0))))
  b))

C

double code(double a, double b, double c) {
	return (-c - fma((5.0 / (pow(b, 6.0) * a)), pow((c * a), 4.0), fma((c * c), (a / (b * b)), ((((pow(b, -4.0) * c) * ((a * a) * c)) * c) * 2.0)))) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) - fma(Float64(5.0 / Float64((b ^ 6.0) * a)), (Float64(c * a) ^ 4.0), fma(Float64(c * c), Float64(a / Float64(b * b)), Float64(Float64(Float64(Float64((b ^ -4.0) * c) * Float64(Float64(a * a) * c)) * c) * 2.0)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[((-c) - N[(N[(5.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[Power[b, -4.0], $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{b}} \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(\frac{1}{4}, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(\frac{1}{4}, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\left(\frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) + \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   4. associate--l+ N/A

      \[\leadsto \frac{\left(-c\right) - \left(\frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) + \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)\right)}{b} \]

   5. associate--r+ N/A

      \[\leadsto \frac{\left(\left(-c\right) - \frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right)\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\left(\left(-c\right) - \frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right)\right) - \left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b} - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

8. Applied rewrites 95.7%

   \[\leadsto \frac{\left(\left(-c\right) - \left({\left(c \cdot a\right)}^{4} \cdot 0.25\right) \cdot \frac{20}{{b}^{6} \cdot a}\right) - \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b} \]
9. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\left(\left(-c\right) - \left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a}\right) - \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\left(\left(-c\right) - \left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a}\right) - \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)}{b} \]

   3. associate--l- N/A

      \[\leadsto \frac{\left(-c\right) - \left(\left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a} + \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)\right)}{b} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\left(-c\right) - \left(\left({\left(c \cdot a\right)}^{4} \cdot \frac{1}{4}\right) \cdot \frac{20}{{b}^{6} \cdot a} + \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)\right)}{b} \]

10. Applied rewrites 95.7%

    \[\leadsto \frac{\left(-c\right) - \mathsf{fma}\left(\frac{5}{{b}^{6} \cdot a}, {\left(c \cdot a\right)}^{4}, \mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, \left(\left(\left({b}^{-4} \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right)\right) \cdot c\right) \cdot 2\right)\right)}{b} \]
11. Add Preprocessing

Alternative 4: 94.2% accurate, 0.3× speedup

Math

\[\mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -1.0
  (/ c b)
  (*
   a
   (fma
    -2.0
    (/ (* a (pow c 3.0)) (pow b 5.0))
    (* -1.0 (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	return fma(-1.0, (c / b), (a * fma(-2.0, ((a * pow(c, 3.0)) / pow(b, 5.0)), (-1.0 * (pow(c, 2.0) / pow(b, 3.0))))));
}

Julia

function code(a, b, c)
	return fma(-1.0, Float64(c / b), Float64(a * fma(-2.0, Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0)), Float64(-1.0 * Float64((c ^ 2.0) / (b ^ 3.0))))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 * N[(c / b), $MachinePrecision] + N[(a * N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{b}} \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

5. Taylor expanded in a around 0

   \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{\color{blue}{b}}, a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   2. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{5}} + -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]

7. Applied rewrites 94.2%

   \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{c}{b}}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right) \]
8. Add Preprocessing

Alternative 5: 94.2% accurate, 0.3× speedup

Math

\[\frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (*
    a
    (- (* -2.0 (/ (* a (pow c 3.0)) (pow b 4.0))) (/ (pow c 2.0) (pow b 2.0))))
   c)
  b))

C

double code(double a, double b, double c) {
	return ((a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 4.0))) - (pow(c, 2.0) / pow(b, 2.0)))) - 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 = ((a * (((-2.0d0) * ((a * (c ** 3.0d0)) / (b ** 4.0d0))) - ((c ** 2.0d0) / (b ** 2.0d0)))) - c) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((a * ((-2.0 * ((a * Math.pow(c, 3.0)) / Math.pow(b, 4.0))) - (Math.pow(c, 2.0) / Math.pow(b, 2.0)))) - c) / b;
}

Python

def code(a, b, c):
	return ((a * ((-2.0 * ((a * math.pow(c, 3.0)) / math.pow(b, 4.0))) - (math.pow(c, 2.0) / math.pow(b, 2.0)))) - c) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0))) - Float64((c ^ 2.0) / (b ^ 2.0)))) - c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((a * ((-2.0 * ((a * (c ^ 3.0)) / (b ^ 4.0))) - ((c ^ 2.0) / (b ^ 2.0)))) - c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{b}} \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

7. Taylor expanded in a around 0

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]
8. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]

9. Applied rewrites 94.2%

   \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]
10. Add Preprocessing

Alternative 6: 94.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (-
    (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 4.0))) (/ a (pow b 2.0))))
    1.0))
  b))

C

double code(double a, double b, double c) {
	return (c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 4.0))) - (a / pow(b, 2.0)))) - 1.0)) / 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 * ((c * (((-2.0d0) * (((a ** 2.0d0) * c) / (b ** 4.0d0))) - (a / (b ** 2.0d0)))) - 1.0d0)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((c * ((-2.0 * ((Math.pow(a, 2.0) * c) / Math.pow(b, 4.0))) - (a / Math.pow(b, 2.0)))) - 1.0)) / b;
}

Python

def code(a, b, c):
	return (c * ((c * ((-2.0 * ((math.pow(a, 2.0) * c) / math.pow(b, 4.0))) - (a / math.pow(b, 2.0)))) - 1.0)) / b

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0))) - Float64(a / (b ^ 2.0)))) - 1.0)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((c * ((-2.0 * (((a ^ 2.0) * c) / (b ^ 4.0))) - (a / (b ^ 2.0)))) - 1.0)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 31.1%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{b}} \]

4. Applied rewrites 95.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

7. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

9. Applied rewrites 94.1%

   \[\leadsto \frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]
10. Add Preprocessing

Alternative 7: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1000:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, c, -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1000.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/ (fma -1.0 c (* -1.0 (/ (* a (pow c 2.0)) (pow b 2.0)))) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1000.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = fma(-1.0, c, (-1.0 * ((a * pow(c, 2.0)) / pow(b, 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1000.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(fma(-1.0, c, Float64(-1.0 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, 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], -1000.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * c + N[(-1.0 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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)) < -1e3

   1. Initial program 31.1%

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

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

      3. add-flip N/A

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

      4. flip-- N/A

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

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

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

   3. Applied rewrites 32.0%

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

   if -1e3 < (/.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 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 91.0%

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

   4. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, c, -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1000:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1000.0)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/ (* c (- (* -1.0 (/ (* a c) (pow b 2.0))) 1.0)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1000.0) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = (c * ((-1.0 * ((a * c) / pow(b, 2.0))) - 1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1000.0)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(-1.0 * Float64(Float64(a * c) / (b ^ 2.0))) - 1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, 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], -1000.0], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 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)) < -1e3

   1. Initial program 31.1%

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

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

      3. add-flip N/A

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

      4. flip-- N/A

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

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

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

   3. Applied rewrites 32.0%

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

   if -1e3 < (/.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 31.1%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{b}} \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 91.0%

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

   7. Applied rewrites 91.0%

      \[\leadsto \frac{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{-2 \cdot a}, a, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot 0.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1000.0)
   (/ (fma (/ b (* -2.0 a)) a (* (sqrt (fma (* c -4.0) a (* b b))) 0.5)) a)
   (/ (* c (- (* -1.0 (/ (* a c) (pow b 2.0))) 1.0)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1000.0) {
		tmp = fma((b / (-2.0 * a)), a, (sqrt(fma((c * -4.0), a, (b * b))) * 0.5)) / a;
	} else {
		tmp = (c * ((-1.0 * ((a * c) / pow(b, 2.0))) - 1.0)) / 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)) <= -1000.0)
		tmp = Float64(fma(Float64(b / Float64(-2.0 * a)), a, Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) * 0.5)) / a);
	else
		tmp = Float64(Float64(c * Float64(Float64(-1.0 * Float64(Float64(a * c) / (b ^ 2.0))) - 1.0)) / 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], -1000.0], N[(N[(N[(b / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision] * a + N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(N[(-1.0 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 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)) < -1e3

   1. Initial program 31.1%

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

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

      5. associate-/r* N/A

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

      6. add-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 32.6%

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

   if -1e3 < (/.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 31.1%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-1 \cdot c + \left(-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{4} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{\color{blue}{b}} \]

   4. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 91.0%

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

   7. Applied rewrites 91.0%

      \[\leadsto \frac{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{-2 \cdot a}, a, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot 0.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}, c \cdot a\right) \cdot -2}{b}}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1000.0)
   (/ (fma (/ b (* -2.0 a)) a (* (sqrt (fma (* c -4.0) a (* b b))) 0.5)) a)
   (/ (/ (* (fma a (* (* c c) (/ a (* b b))) (* c a)) -2.0) b) (+ a a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1000.0) {
		tmp = fma((b / (-2.0 * a)), a, (sqrt(fma((c * -4.0), a, (b * b))) * 0.5)) / a;
	} else {
		tmp = ((fma(a, ((c * c) * (a / (b * b))), (c * a)) * -2.0) / b) / (a + a);
	}
	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)) <= -1000.0)
		tmp = Float64(fma(Float64(b / Float64(-2.0 * a)), a, Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) * 0.5)) / a);
	else
		tmp = Float64(Float64(Float64(fma(a, Float64(Float64(c * c) * Float64(a / Float64(b * b))), Float64(c * a)) * -2.0) / b) / Float64(a + a));
	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], -1000.0], N[(N[(N[(b / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision] * a + N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(a * N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] / b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $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)) < -1e3

   1. Initial program 31.1%

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

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

      5. associate-/r* N/A

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

      6. add-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 32.6%

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

   if -1e3 < (/.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 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 90.7%

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

   4. Applied rewrites 90.7%

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

   6. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(a, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}, c \cdot a\right) \cdot -2}{b}}{a + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{-2 \cdot a}, a, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot 0.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, c\right)\right) \cdot -2}{b}}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1000.0)
   (/ (fma (/ b (* -2.0 a)) a (* (sqrt (fma (* c -4.0) a (* b b))) 0.5)) a)
   (/ (/ (* (* a (fma (* (/ a (* b b)) c) c c)) -2.0) b) (+ a a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1000.0) {
		tmp = fma((b / (-2.0 * a)), a, (sqrt(fma((c * -4.0), a, (b * b))) * 0.5)) / a;
	} else {
		tmp = (((a * fma(((a / (b * b)) * c), c, c)) * -2.0) / b) / (a + a);
	}
	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)) <= -1000.0)
		tmp = Float64(fma(Float64(b / Float64(-2.0 * a)), a, Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) * 0.5)) / a);
	else
		tmp = Float64(Float64(Float64(Float64(a * fma(Float64(Float64(a / Float64(b * b)) * c), c, c)) * -2.0) / b) / Float64(a + a));
	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], -1000.0], N[(N[(N[(b / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision] * a + N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(a * N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] / b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $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)) < -1e3

   1. Initial program 31.1%

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

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

      5. associate-/r* N/A

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

      6. add-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 32.6%

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

   if -1e3 < (/.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 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 90.7%

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

   4. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.6%

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 90.6%

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

   8. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(a \cdot \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, c\right)\right) \cdot -2}{b}}{a + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1000:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, c\right)\right) \cdot -2}{b}}{a + a}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1000.0)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/ (/ (* (* a (fma (* (/ a (* b b)) c) c c)) -2.0) b) (+ a a))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1000.0) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (((a * fma(((a / (b * b)) * c), c, c)) * -2.0) / b) / (a + a);
	}
	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)) <= -1000.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(a * fma(Float64(Float64(a / Float64(b * b)) * c), c, c)) * -2.0) / b) / Float64(a + a));
	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], -1000.0], 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[(N[(N[(N[(a * N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + c), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] / b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $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)) < -1e3

   1. Initial program 31.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 31.1%

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

   3. Applied rewrites 31.1%

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

   if -1e3 < (/.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 31.1%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 90.7%

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

   4. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 90.6%

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 90.6%

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

   8. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{\frac{\left(a \cdot \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, c\right)\right) \cdot -2}{b}}{a + a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1e-9)
   (/ (+ (- 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)) <= -1e-9) {
		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)) <= -1e-9)
		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], -1e-9], 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)) < -1.00000000000000006e-9

   1. Initial program 31.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 31.1%

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

   3. Applied rewrites 31.1%

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

   if -1.00000000000000006e-9 < (/.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 31.1%

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

   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 81.5%

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

   4. Applied rewrites 81.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 81.5%

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

   6. Applied rewrites 81.5%

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

Alternative 14: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1e-9)
   (/ (- (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)) <= -1e-9) {
		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)) <= -1e-9)
		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], -1e-9], 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)) < -1.00000000000000006e-9

   1. Initial program 31.1%

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

   3. Applied rewrites 31.1%

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

   if -1.00000000000000006e-9 < (/.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 31.1%

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

   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 81.5%

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

   4. Applied rewrites 81.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 81.5%

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

   6. Applied rewrites 81.5%

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

Alternative 15: 81.5% accurate, 4.6× 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 31.1%

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

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 81.5%

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

4. Applied rewrites 81.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 81.5%

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

6. Applied rewrites 81.5%

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

Reproduce

herbie shell --seed 2025183 
(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)))