Quadratic roots, medium range

Percentage Accurate: 31.7% → 99.4%

Time: 3.0s

Alternatives: 5

Speedup: 3.6×

Specification

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

Math

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

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: 31.7% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot a\right) \cdot -4\\ \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, b \cdot b - b \cdot b\right)}{\sqrt{\frac{t\_0 \cdot t\_0 - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(\frac{\left(-4 \cdot a\right) \cdot c}{b \cdot b} - 1\right) \cdot \left(b \cdot b\right)}} - \left(-b\right)}}{2 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c a) -4.0)))
   (/
    (/
     (fma (* a c) -4.0 (- (* b b) (* b b)))
     (-
      (sqrt
       (/
        (- (* t_0 t_0) (* (* b b) (* b b)))
        (* (- (/ (* (* -4.0 a) c) (* b b)) 1.0) (* b b))))
      (- b)))
    (* 2.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (c * a) * -4.0;
	return (fma((a * c), -4.0, ((b * b) - (b * b))) / (sqrt((((t_0 * t_0) - ((b * b) * (b * b))) / (((((-4.0 * a) * c) / (b * b)) - 1.0) * (b * b)))) - -b)) / (2.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * a) * -4.0)
	return Float64(Float64(fma(Float64(a * c), -4.0, Float64(Float64(b * b) - Float64(b * b))) / Float64(sqrt(Float64(Float64(Float64(t_0 * t_0) - Float64(Float64(b * b) * Float64(b * b))) / Float64(Float64(Float64(Float64(Float64(-4.0 * a) * c) / Float64(b * b)) - 1.0) * Float64(b * b)))) - Float64(-b))) / Float64(2.0 * a))
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. lift-+.f64 N/A

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

   2. +-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 32.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. flip-+ N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower--.f64 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in b around inf

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

   1. pow2 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

10. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{a + a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (fma (* c a) -4.0 0.0) (- (sqrt (fma -4.0 (* c a) (* b b))) (- b)))
  (+ a a)))

C

double code(double a, double b, double c) {
	return (fma((c * a), -4.0, 0.0) / (sqrt(fma(-4.0, (c * a), (b * b))) - -b)) / (a + a);
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(c * a), -4.0, 0.0) / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - Float64(-b))) / Float64(a + a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(c * a), $MachinePrecision] * -4.0 + 0.0), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - (-b)), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.7%

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

   1. lift-+.f64 N/A

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

   2. +-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 32.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 99.4

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

   4. lift--.f64 N/A

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

   5. +-inverses 99.4

      \[\leadsto \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(c \cdot a, -4, 0\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{2 \cdot a} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{\color{blue}{2 \cdot a}} \]

   2. count-2-rev N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{\color{blue}{a + a}} \]

   3. lift-+.f64 99.4

      \[\leadsto \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{\color{blue}{a + a}} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(c \cdot a, -4, 0\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - \left(-b\right)}}{\color{blue}{a + a}} \]
10. Add Preprocessing

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. lift-+.f64 N/A

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

   2. +-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 32.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 99.4

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

Alternative 4: 90.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. lift-+.f64 N/A

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

   2. +-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 32.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 99.4

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 90.9

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

10. Applied rewrites 90.9%

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

Alternative 5: 81.1% accurate, 3.6× 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 31.7%

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

   1. lift-+.f64 N/A

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

   2. +-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 32.7%

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

   1. lift--.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 99.4

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 81.1

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

10. Applied rewrites 81.1%

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

Reproduce

herbie shell --seed 2025107 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))