Quadratic roots, medium range

Percentage Accurate: 32.0% → 99.4%

Time: 4.1s

Alternatives: 8

Speedup: 4.6×

Specification

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

Math

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

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: 32.0% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(a \cdot c\right)}{\left(\left(1 - \frac{1}{\frac{-b}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}\right) \cdot \left(-b\right)\right) \cdot \left(a + a\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 32.9%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 99.4

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 99.4

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

8. Applied rewrites 99.4%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lift-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 99.4

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

   14. lower-fma.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-fma.f64 99.4

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

10. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(a \cdot c\right)}{\left(\left(1 - \frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{-b}\right) \cdot \left(-b\right)\right) \cdot \left(a + a\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 32.9%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 99.4

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 99.4

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

8. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

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

   4. associate-/l/ N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 32.9%

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

4. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 99.4

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

   10. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \left(a + a\right)\right)\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \mathsf{Rewrite=>}\left(lift-+.f64, \left(a + a\right)\right)} \]

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) \cdot \mathsf{Rewrite<=}\left(lift-*.f64, \left(2 \cdot a\right)\right)} \]

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)\right)\right)} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(2 \cdot a\right)\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)} \]

   17. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\mathsf{Rewrite=>}\left(count-2-rev, \left(a + a\right)\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\mathsf{Rewrite<=}\left(lift-+.f64, \left(a + a\right)\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right)} \]

   19. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\left(a + a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, c \cdot a, \mathsf{Rewrite=>}\left(lift-*.f64, \left(b \cdot b\right)\right)\right)}\right)} \]

   20. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\left(a + a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{Rewrite=>}\left(lift-fma.f64, \left(-4 \cdot \left(c \cdot a\right) + b \cdot b\right)\right)}\right)} \]

   21. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\left(a + a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{Rewrite=>}\left(*-commutative, \left(\left(c \cdot a\right) \cdot -4\right)\right) + b \cdot b}\right)} \]

   22. lift-*.f64 N/A

       \[\leadsto \frac{\left(a \cdot c\right) \cdot 4}{\left(a + a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)\right)\right)}\right)} \]

8. Applied rewrites 99.4%

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

Alternative 4: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0002:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{1 - \left(a \cdot 4\right) \cdot \frac{c}{b \cdot b}}, \left|b\right|, -b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0002)
   (/
    (fma (sqrt (- 1.0 (* (* a 4.0) (/ c (* b b))))) (fabs b) (- b))
    (* 2.0 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.0002) {
		tmp = fma(sqrt((1.0 - ((a * 4.0) * (c / (b * b))))), fabs(b), -b) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.0002)
		tmp = Float64(fma(sqrt(Float64(1.0 - Float64(Float64(a * 4.0) * Float64(c / Float64(b * b))))), abs(b), Float64(-b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0002], 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[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2.0000000000000001e-4

   1. Initial program 32.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 32.5%

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

   if -2.0000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.0%

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

   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 80.9

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

   4. Applied rewrites 80.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 5: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0002:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0002)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.0002) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.0002)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0002], 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[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2.0000000000000001e-4

   1. Initial program 32.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 32.0

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

   3. Applied rewrites 32.0%

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

   if -2.0000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.0%

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

   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 80.9

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

   4. Applied rewrites 80.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 6: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0002:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0002)
   (* (- (sqrt (fma -4.0 (* c a) (* b b))) b) (/ 0.5 a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.0002) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) - b) * (0.5 / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.0002)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0002], 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[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2.0000000000000001e-4

   1. Initial program 32.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 32.0%

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

   if -2.0000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.0%

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

   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 80.9

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

   4. Applied rewrites 80.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 7: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0002:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0002)
   (/ (- (sqrt (fma -4.0 (* c a) (* b b))) b) (+ a a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.0002) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.0002)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0002], 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[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -2.0000000000000001e-4

   1. Initial program 32.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 32.0

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

      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 32.0

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

      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 32.0

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

   3. Applied rewrites 32.0%

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

   if -2.0000000000000001e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 32.0%

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

   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 80.9

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

   4. Applied rewrites 80.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 80.9

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

   6. Applied rewrites 80.9%

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

Alternative 8: 80.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

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 80.9

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

4. Applied rewrites 80.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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 80.9

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

6. Applied rewrites 80.9%

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

Reproduce

herbie shell --seed 2025140 
(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)))