Quadratic roots, medium range

Percentage Accurate: 31.7% → 95.3%

Time: 6.6s

Alternatives: 8

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: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\frac{a \cdot \mathsf{fma}\left(-5, a \cdot c, -2 \cdot \left(b \cdot b\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (*
   (-
    (* (/ (* a (fma -5.0 (* a c) (* -2.0 (* b b)))) (pow b 7.0)) c)
    (pow b -3.0))
   (* c c))
  a
  (/ (- c) b)))

C

double code(double a, double b, double c) {
	return fma((((((a * fma(-5.0, (a * c), (-2.0 * (b * b)))) / pow(b, 7.0)) * c) - pow(b, -3.0)) * (c * c)), a, (-c / b));
}

Julia

function code(a, b, c)
	return fma(Float64(Float64(Float64(Float64(Float64(a * fma(-5.0, Float64(a * c), Float64(-2.0 * Float64(b * b)))) / (b ^ 7.0)) * c) - (b ^ -3.0)) * Float64(c * c)), a, Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(a * N[(-5.0 * N[(a * c), $MachinePrecision] + N[(-2.0 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.9%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 94.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5}}\right) - \frac{1}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5}}\right) - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5}}\right) - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

8. Applied rewrites 94.7%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{a}{{b}^{5}}, -2, \frac{-5 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{7}}\right) \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

9. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\left(\frac{-5 \cdot \left({a}^{2} \cdot c\right) + -2 \cdot \left(a \cdot {b}^{2}\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]
10. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{-5 \cdot \left({a}^{2} \cdot c\right) + -2 \cdot \left(a \cdot {b}^{2}\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, {a}^{2} \cdot c, -2 \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    3. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    4. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    5. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    8. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    9. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    10. lift-pow.f64 94.7

        \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

11. Applied rewrites 94.7%

    \[\leadsto \mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(-5, \left(a \cdot a\right) \cdot c, -2 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

12. Taylor expanded in a around 0

    \[\leadsto \mathsf{fma}\left(\left(\frac{a \cdot \left(-5 \cdot \left(a \cdot c\right) + -2 \cdot {b}^{2}\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]
13. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{a \cdot \left(-5 \cdot \left(a \cdot c\right) + -2 \cdot {b}^{2}\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{a \cdot \mathsf{fma}\left(-5, a \cdot c, -2 \cdot {b}^{2}\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    3. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{a \cdot \mathsf{fma}\left(-5, a \cdot c, -2 \cdot {b}^{2}\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{a \cdot \mathsf{fma}\left(-5, a \cdot c, -2 \cdot {b}^{2}\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    5. pow2 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{a \cdot \mathsf{fma}\left(-5, a \cdot c, -2 \cdot \left(b \cdot b\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

    6. lift-*.f64 94.7

       \[\leadsto \mathsf{fma}\left(\left(\frac{a \cdot \mathsf{fma}\left(-5, a \cdot c, -2 \cdot \left(b \cdot b\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

14. Applied rewrites 94.7%

    \[\leadsto \mathsf{fma}\left(\left(\frac{a \cdot \mathsf{fma}\left(-5, a \cdot c, -2 \cdot \left(b \cdot b\right)\right)}{{b}^{7}} \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]
15. Add Preprocessing

Alternative 2: 93.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, 2, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right) + c}{-b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+
   (fma (/ (* (* a a) (pow c 3.0)) (pow b 4.0)) 2.0 (/ (* (* c c) a) (* b b)))
   c)
  (- b)))

C

double code(double a, double b, double c) {
	return (fma((((a * a) * pow(c, 3.0)) / pow(b, 4.0)), 2.0, (((c * c) * a) / (b * b))) + c) / -b;
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(Float64(Float64(Float64(a * a) * (c ^ 3.0)) / (b ^ 4.0)), 2.0, Float64(Float64(Float64(c * c) * a) / Float64(b * b))) + c) / Float64(-b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]

Derivation

1. Initial program 31.9%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 94.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{c + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}\right) \]

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

8. Applied rewrites 92.8%

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

9. Final simplification 92.8%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, 2, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right) + c}{-b} \]
10. Add Preprocessing

Alternative 3: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -600:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -1, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -600.0)
   (/ (+ (- b) (sqrt (* (fma -4.0 a (/ (* b b) c)) c))) (* 2.0 a))
   (fma (/ (* (* c c) a) (* (* b b) b)) -1.0 (/ (- 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)) <= -600.0) {
		tmp = (-b + sqrt((fma(-4.0, a, ((b * b) / c)) * c))) / (2.0 * a);
	} else {
		tmp = fma((((c * c) * a) / ((b * b) * b)), -1.0, (-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)) <= -600.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * b)), -1.0, 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], -600.0], N[(N[((-b) + N[Sqrt[N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -1.0 + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -600

   1. Initial program 84.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -600 < (/.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 28.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 92.0

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

   5. Applied rewrites 92.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 92.0

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

   7. Applied rewrites 92.0%

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

Alternative 4: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -600:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -1, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -600.0)
   (/ (+ (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (+ a a))
   (fma (/ (* (* c c) a) (* (* b b) b)) -1.0 (/ (- 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)) <= -600.0) {
		tmp = (-b + sqrt(fma((c * a), -4.0, (b * b)))) / (a + a);
	} else {
		tmp = fma((((c * c) * a) / ((b * b) * b)), -1.0, (-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)) <= -600.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(a + a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * b)), -1.0, 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], -600.0], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -1.0 + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -600

   1. Initial program 84.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 84.6

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

   6. Applied rewrites 84.6%

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

   if -600 < (/.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 28.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 92.0

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

   5. Applied rewrites 92.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 92.0

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

   7. Applied rewrites 92.0%

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

Alternative 5: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -600:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -600.0)
   (/ (+ (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (+ a a))
   (/ (fma a (/ (* c c) (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -600.0) {
		tmp = (-b + sqrt(fma((c * a), -4.0, (b * b)))) / (a + a);
	} else {
		tmp = fma(a, ((c * c) / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -600.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(fma(a, Float64(Float64(c * c) / Float64(b * b)), c) / Float64(-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], -600.0], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -600

   1. Initial program 84.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 84.6

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

   6. Applied rewrites 84.6%

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

   if -600 < (/.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 28.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 92.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -600:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

5. Applied rewrites 94.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot \frac{\frac{{c}^{4}}{{b}^{6}} \cdot 20}{b}, -0.25, \frac{-2 \cdot {c}^{3}}{{b}^{5}}\right), a, -\frac{c \cdot c}{{b}^{3}}\right), a, \frac{-c}{b}\right)} \]

6. Taylor expanded in c around 0

   \[\leadsto \mathsf{fma}\left({c}^{2} \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5}}\right) - \frac{1}{{b}^{3}}\right), a, \frac{-c}{b}\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5}}\right) - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(c \cdot \left(-5 \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + -2 \cdot \frac{a}{{b}^{5}}\right) - \frac{1}{{b}^{3}}\right) \cdot {c}^{2}, a, \frac{-c}{b}\right) \]

8. Applied rewrites 94.7%

   \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{a}{{b}^{5}}, -2, \frac{-5 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{7}}\right) \cdot c - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]

9. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. lower--.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. pow2 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-pow.f64 92.8

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

11. Applied rewrites 92.8%

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

    1. lift-pow.f64 N/A

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

    2. unpow3 N/A

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

    3. pow2 N/A

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

    4. lower-*.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 92.8

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

13. Applied rewrites 92.8%

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

Alternative 7: 90.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. associate-*r/ N/A

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

   11. mul-1-neg N/A

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

   12. lower-/.f64 N/A

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

   13. lower-neg.f64 89.5

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

5. Applied rewrites 89.5%

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

6. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 89.5%

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

9. Final simplification 89.5%

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

Alternative 8: 81.0% 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 31.9%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
4. 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 80.5

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

5. Applied rewrites 80.5%

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

Reproduce

herbie shell --seed 2025037 
(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)))