Quadratic roots, narrow range

Percentage Accurate: 55.8% → 90.4%

Time: 6.1s

Alternatives: 9

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: 55.8% 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: 90.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

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 90.4%

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

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 90.4%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 90.4%

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

11. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. pow2 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. pow2 N/A

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

    8. lift-*.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 90.4

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

13. Applied rewrites 90.4%

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

Alternative 2: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.26:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(\frac{-2 \cdot \left(c \cdot a\right)}{b \cdot b} - 1\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.26)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (fma
    a
    (/ (* (- (/ (* -2.0 (* c a)) (* b b)) 1.0) (* c c)) (* (* 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)) <= -0.26) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma(a, (((((-2.0 * (c * a)) / (b * b)) - 1.0) * (c * c)) / ((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)) <= -0.26)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = fma(a, Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(c * a)) / Float64(b * b)) - 1.0) * Float64(c * c)) / 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], -0.26], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(N[(N[(-2.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(c * c), $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)) < -0.26000000000000001

   1. Initial program 55.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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-*l* 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 55.9

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

   3. Applied rewrites 55.9%

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

   if -0.26000000000000001 < (/.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 55.8%

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

   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 90.4%

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

   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 90.4%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 87.3

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

   10. Applied rewrites 87.3%

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

Alternative 3: 89.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.26:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{b \cdot b} - a}{\left(b \cdot b\right) \cdot b} \cdot c - \frac{1}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.26)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (*
    (-
     (* (/ (- (/ (* (* (* a a) c) -2.0) (* b b)) a) (* (* b b) b)) c)
     (/ 1.0 b))
    c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.26) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = ((((((((a * a) * c) * -2.0) / (b * b)) - a) / ((b * b) * b)) * c) - (1.0 / b)) * c;
	}
	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)) <= -0.26)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * -2.0) / Float64(b * b)) - a) / Float64(Float64(b * b) * b)) * c) - Float64(1.0 / b)) * c);
	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], -0.26], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $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)) < -0.26000000000000001

   1. Initial program 55.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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-*l* 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 55.9

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

   3. Applied rewrites 55.9%

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

   if -0.26000000000000001 < (/.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 55.8%

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

   2. Taylor expanded in c around 0

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

   4. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(c, \mathsf{fma}\left(\frac{\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot c}{a \cdot b}, -0.25, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), -\frac{a}{\left(b \cdot b\right) \cdot b}\right) \cdot c - \frac{1}{b}\right) \cdot c} \]

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow3 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 87.1

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

   7. Applied rewrites 87.1%

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

Alternative 4: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.000139:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.000139)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (fma a (/ (- (* c c)) (* (* 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)) <= -0.000139) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma(a, (-(c * c) / ((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)) <= -0.000139)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = fma(a, Float64(Float64(-Float64(c * c)) / 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], -0.000139], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(a * N[((-N[(c * c), $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)) < -1.38999999999999991e-4

   1. Initial program 55.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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-*l* 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 55.9

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

   3. Applied rewrites 55.9%

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

   if -1.38999999999999991e-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 55.8%

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

   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 90.4%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow3 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 81.1

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

   7. Applied rewrites 81.1%

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

Alternative 5: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.000139:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.000139)
   (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (+ a a))
   (fma a (/ (- (* c c)) (* (* 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)) <= -0.000139) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) / (a + a);
	} else {
		tmp = fma(a, (-(c * c) / ((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)) <= -0.000139)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = fma(a, Float64(Float64(-Float64(c * c)) / 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], -0.000139], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(a * N[((-N[(c * c), $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)) < -1.38999999999999991e-4

   1. Initial program 55.8%

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

   3. Applied rewrites 55.8%

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

   if -1.38999999999999991e-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 55.8%

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

   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 90.4%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow3 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 81.1

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

   7. Applied rewrites 81.1%

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

Alternative 6: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.000139:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.000139)
   (/ (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (+ a a))
   (- (/ (fma (* c c) (/ a (* 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)) <= -0.000139) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) / (a + a);
	} else {
		tmp = -(fma((c * c), (a / (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)) <= -0.000139)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = Float64(-Float64(fma(Float64(c * c), Float64(a / 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], -0.000139], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[(c * c), $MachinePrecision] * N[(a / 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)) < -1.38999999999999991e-4

   1. Initial program 55.8%

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

   3. Applied rewrites 55.8%

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

   if -1.38999999999999991e-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 55.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-1 \cdot \frac{c}{b}} \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-1 \cdot \frac{c}{b}} \]

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 81.1

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

   4. Applied rewrites 81.1%

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

   6. Applied rewrites 81.1%

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

Alternative 7: 81.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-1 \cdot \frac{c}{b}} \]

   2. lower-+.f64 N/A

      \[\leadsto -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-1 \cdot \frac{c}{b}} \]

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow3 N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r/ N/A

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

   16. mul-1-neg N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 81.1

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

4. Applied rewrites 81.1%

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

6. Applied rewrites 81.1%

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

Alternative 8: 80.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.8%

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

2. Taylor expanded in c around 0

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

4. Applied rewrites 90.3%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(c, \mathsf{fma}\left(\frac{\left(\frac{\left(a \cdot a\right) \cdot \left(a \cdot a\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot c}{a \cdot b}, -0.25, \frac{-2 \cdot \left(a \cdot a\right)}{{b}^{5}}\right), -\frac{a}{\left(b \cdot b\right) \cdot b}\right) \cdot c - \frac{1}{b}\right) \cdot c} \]

5. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 80.9

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

7. Applied rewrites 80.9%

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

Alternative 9: 64.0% 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 55.8%

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

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 64.0

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

4. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2025126 
(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)))