Quadratic roots, narrow range

Percentage Accurate: 54.8% → 91.3%

Time: 5.3s

Alternatives: 8

Speedup: 4.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(c \cdot \frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -2 \cdot a\right)}{b \cdot b} - 1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

4. Applied rewrites 91.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

5. Taylor expanded in b around inf

   \[\leadsto \mathsf{fma}\left(a, \frac{\left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{4}} + -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right) - {c}^{2}}{\color{blue}{{b}^{3}}}, \frac{-c}{b}\right) \]

7. Applied rewrites 91.3%

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

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \frac{{c}^{2} \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right) \]

   2. pow2 N/A

      \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right) \]

   4. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} + -2 \cdot \frac{a}{{b}^{2}}\right) - 1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right) \]

10. Applied rewrites 91.3%

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

11. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. pow2 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. pow2 N/A

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

    8. lift-*.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 91.3

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

13. Applied rewrites 91.3%

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

Alternative 2: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.018:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right)}{b} + \frac{\frac{\left(c \cdot c\right) \cdot \left(-a\right)}{b \cdot b}}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.018)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (+ a a))
   (+
    (/ (fma (/ (* (* a a) (* (* c c) c)) (* (* b b) (* b b))) -2.0 (- c)) b)
    (/ (/ (* (* c c) (- a)) (* b b)) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.018) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (a + a);
	} else {
		tmp = (fma((((a * a) * ((c * c) * c)) / ((b * b) * (b * b))), -2.0, -c) / b) + ((((c * c) * -a) / (b * b)) / 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)) <= -0.018)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(a + a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), -2.0, Float64(-c)) / b) + Float64(Float64(Float64(Float64(c * c) * Float64(-a)) / Float64(b * b)) / 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], -0.018], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + (-c)), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] / b), $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)) < -0.0179999999999999986

   1. Initial program 78.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.4

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

   3. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -0.0179999999999999986 < (/.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 46.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 93.2%

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

   6. Applied rewrites 93.2%

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

Alternative 3: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.018:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(-2 \cdot \frac{a \cdot c}{b \cdot b} - 1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.018)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (+ a a))
   (fma
    a
    (/ (* (* c c) (- (* -2.0 (/ (* a c) (* b b))) 1.0)) (* (* b b) b))
    (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.018) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (a + a);
	} else {
		tmp = fma(a, (((c * c) * ((-2.0 * ((a * c) / (b * b))) - 1.0)) / ((b * b) * b)), (-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)) <= -0.018)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(a + a));
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * Float64(Float64(-2.0 * Float64(Float64(a * c) / Float64(b * b))) - 1.0)) / Float64(Float64(b * b) * b)), 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], -0.018], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $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)) < -0.0179999999999999986

   1. Initial program 78.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.4

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

   3. Applied rewrites 78.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -0.0179999999999999986 < (/.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 46.3%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

   5. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(a, \frac{\left(-5 \cdot \frac{{a}^{2} \cdot {c}^{4}}{{b}^{4}} + -2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}}\right) - {c}^{2}}{\color{blue}{{b}^{3}}}, \frac{-c}{b}\right) \]

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 93.2

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

   10. Applied rewrites 93.2%

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

Alternative 4: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00025)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (+ a a))
   (fma (/ (- (* c c)) (* (* b b) b)) 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)) <= -0.00025) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (a + a);
	} else {
		tmp = fma((-(c * c) / ((b * b) * b)), a, (-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)) <= -0.00025)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(a + a));
	else
		tmp = fma(Float64(Float64(-Float64(c * c)) / Float64(Float64(b * b) * b)), a, 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], -0.00025], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(c * c), $MachinePrecision]) / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $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)) < -2.5000000000000001e-4

   1. Initial program 75.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.5

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

   3. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -2.5000000000000001e-4 < (/.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 40.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{{c}^{2}}{{b}^{3}}, a, \frac{-c}{b}\right) \]
   6. Step-by-step derivation

      1. pow2 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow3 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

Alternative 5: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00025)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (+ a a))
   (- (/ (fma a (/ (* c c) (* b b)) c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00025) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (a + a);
	} else {
		tmp = -(fma(a, ((c * c) / (b * b)), 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)) <= -0.00025)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(a + a));
	else
		tmp = Float64(-Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), 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], -0.00025], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], (-N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $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)) < -2.5000000000000001e-4

   1. Initial program 75.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.5

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

   3. Applied rewrites 75.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -2.5000000000000001e-4 < (/.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 40.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

Alternative 6: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00025:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00025)
   (/ (+ (- b) (sqrt (fma -4.0 (* a c) (* b b)))) (+ a a))
   (- (/ (fma a (/ (* c c) (* b b)) c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00025) {
		tmp = (-b + sqrt(fma(-4.0, (a * c), (b * b)))) / (a + a);
	} else {
		tmp = -(fma(a, ((c * c) / (b * b)), 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)) <= -0.00025)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(-4.0, Float64(a * c), Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(-Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), 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], -0.00025], N[(N[((-b) + N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], (-N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $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)) < -2.5000000000000001e-4

   1. Initial program 75.3%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 75.3

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

   4. Applied rewrites 75.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.3

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

   6. Applied rewrites 75.3%

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

   7. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 75.3

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

   9. Applied rewrites 75.3%

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

   if -2.5000000000000001e-4 < (/.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 40.9%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

Alternative 7: 82.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ -\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

4. Applied rewrites 88.3%

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

5. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. pow2 N/A

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

   8. associate-/l* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 82.1

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

7. Applied rewrites 82.1%

   \[\leadsto -\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b} \]
8. Add Preprocessing

Alternative 8: 64.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 64.9

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

4. Applied rewrites 64.9%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))