Quadratic roots, narrow range

Percentage Accurate: 55.0% → 99.3%

Time: 4.6s

Alternatives: 10

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: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot c\right) \cdot -4\\ \frac{t\_0}{\left(\sqrt{\mathsf{fma}\left(b, b, t\_0\right)} + b\right) \cdot \left(a + a\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a c) -4.0)))
   (/ t_0 (* (+ (sqrt (fma b b t_0)) b) (+ a a)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) * -4.0;
	return t_0 / ((sqrt(fma(b, b, t_0)) + b) * (a + a));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) * -4.0)
	return Float64(t_0 / Float64(Float64(sqrt(fma(b, b, t_0)) + b) * Float64(a + a)))
end

Wolfram

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

Derivation

1. Initial program 55.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

3. Applied rewrites 56.5%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

8. Applied rewrites 99.3%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 99.3

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

10. Applied rewrites 99.3%

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

Alternative 2: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 4\right) \cdot \frac{c}{b \cdot b}}, \left|b\right|, -b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b} + \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.002)
   (/ (fma (sqrt (- 1.0 (* (* a 4.0) (/ c (* b b))))) (fabs b) (- b)) (+ a a))
   (* -1.0 (+ (/ (* (* c c) a) (* (* 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)) <= -0.002) {
		tmp = fma(sqrt((1.0 - ((a * 4.0) * (c / (b * b))))), fabs(b), -b) / (a + a);
	} else {
		tmp = -1.0 * ((((c * c) * a) / ((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)) <= -0.002)
		tmp = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(a * 4.0) * Float64(c / Float64(b * b))))), abs(b), Float64(-b)) / Float64(a + a));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * b)) + 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], -0.002], N[(N[(N[Sqrt[N[(1.0 - N[(N[(a * 4.0), $MachinePrecision] * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[b], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 55.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-to-mult N/A

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

      6. sqrt-prod N/A

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

      7. lift-*.f64 N/A

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

      8. rem-sqrt-square-rev N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 55.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 55.8

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

   5. Applied rewrites 55.8%

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

   if -2e-3 < (/.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 55.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.9

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

   7. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 81.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.9

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.9

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 81.9

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

   9. Applied rewrites 81.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 81.9

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-/l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 81.9

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

   11. Applied rewrites 81.9%

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

Alternative 3: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.002:\\ \;\;\;\;\left(\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -4\right)}}{-b} + 1\right) \cdot \frac{-b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b} + \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.002)
   (* (+ (/ (sqrt (fma b b (* (* c a) -4.0))) (- b)) 1.0) (/ (- b) (+ a a)))
   (* -1.0 (+ (/ (* (* c c) a) (* (* 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)) <= -0.002) {
		tmp = ((sqrt(fma(b, b, ((c * a) * -4.0))) / -b) + 1.0) * (-b / (a + a));
	} else {
		tmp = -1.0 * ((((c * c) * a) / ((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)) <= -0.002)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * a) * -4.0))) / Float64(-b)) + 1.0) * Float64(Float64(-b) / Float64(a + a)));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * b)) + 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], -0.002], N[(N[(N[(N[Sqrt[N[(b * b + N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-b)), $MachinePrecision] + 1.0), $MachinePrecision] * N[((-b) / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 55.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 54.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 54.9

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

   5. Applied rewrites 54.9%

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

   if -2e-3 < (/.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 55.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.9

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

   7. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 81.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.9

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.9

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 81.9

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

   9. Applied rewrites 81.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 81.9

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-/l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 81.9

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

   11. Applied rewrites 81.9%

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

Alternative 4: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.002:\\ \;\;\;\;\frac{1}{\frac{a + a}{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b} + \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.002)
   (/ 1.0 (/ (+ a a) (- (sqrt (fma b b (* (* a c) -4.0))) b)))
   (* -1.0 (+ (/ (* (* c c) a) (* (* 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)) <= -0.002) {
		tmp = 1.0 / ((a + a) / (sqrt(fma(b, b, ((a * c) * -4.0))) - b));
	} else {
		tmp = -1.0 * ((((c * c) * a) / ((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)) <= -0.002)
		tmp = Float64(1.0 / Float64(Float64(a + a) / Float64(sqrt(fma(b, b, Float64(Float64(a * c) * -4.0))) - b)));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * b)) + 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], -0.002], N[(1.0 / N[(N[(a + a), $MachinePrecision] / N[(N[Sqrt[N[(b * b + N[(N[(a * c), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 55.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 55.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 55.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. sub-flip-reverse N/A

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

      12. lower--.f64 55.0

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

   3. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 55.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 55.1

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

   5. Applied rewrites 55.1%

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

   if -2e-3 < (/.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 55.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.9

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

   7. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 81.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.9

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.9

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 81.9

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

   9. Applied rewrites 81.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 81.9

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-/l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 81.9

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

   11. Applied rewrites 81.9%

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

Alternative 5: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.002:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b} + \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.002)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (* -1.0 (+ (/ (* (* c c) a) (* (* 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)) <= -0.002) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = -1.0 * ((((c * c) * a) / ((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)) <= -0.002)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * b)) + 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], -0.002], 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[(-1.0 * N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $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)) < -2e-3

   1. Initial program 55.0%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 55.1

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

   3. Applied rewrites 55.1%

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

   if -2e-3 < (/.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 55.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.9

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

   7. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 81.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.9

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.9

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 81.9

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

   9. Applied rewrites 81.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 81.9

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

       4. lift-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. associate-/l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 81.9

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

   11. Applied rewrites 81.9%

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

Alternative 6: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.002:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.002)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 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)) <= -0.002) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * 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)) <= -0.002)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * 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], -0.002], 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[(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)) < -2e-3

   1. Initial program 55.0%

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

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 55.1

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

   3. Applied rewrites 55.1%

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

   if -2e-3 < (/.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 55.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.9

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

   7. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 81.9

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 81.9

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

   9. Applied rewrites 81.9%

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

Alternative 7: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.002:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.002)
   (* (- (sqrt (fma -4.0 (* c a) (* b b))) b) (/ 0.5 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)) <= -0.002) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) - b) * (0.5 / 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)) <= -0.002)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b) * Float64(0.5 / 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], -0.002], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / 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)) < -2e-3

   1. Initial program 55.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

   3. Applied rewrites 55.0%

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

   if -2e-3 < (/.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 55.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.9

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

   7. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 81.9

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 81.9

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

   9. Applied rewrites 81.9%

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

Alternative 8: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.002:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.002)
   (/ (- (sqrt (fma -4.0 (* c a) (* b 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)) <= -0.002) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * 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)) <= -0.002)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * 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], -0.002], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $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)) < -2e-3

   1. Initial program 55.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 55.0

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 55.0

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

      17. lift-*.f64 N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 55.0

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

   3. Applied rewrites 55.0%

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

   if -2e-3 < (/.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 55.0%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.9

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

   7. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 81.9

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-*.f64 81.9

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

   9. Applied rewrites 81.9%

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

Alternative 9: 81.9% 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 55.0%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 91.0%

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

5. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 81.9

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

7. Applied rewrites 81.9%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-neg.f64 81.9

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. associate-/l* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 81.9

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

9. Applied rewrites 81.9%

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

Alternative 10: 64.8% 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 55.0%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.8

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

4. Applied rewrites 64.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 64.8

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

6. Applied rewrites 64.8%

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

Reproduce

herbie shell --seed 2025151 
(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)))