Quadratic roots, narrow range

Percentage Accurate: 55.4% → 91.0%

Time: 11.9s

Alternatives: 16

Speedup: 3.6×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.4% 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: 91.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 11.0)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (fma
      (/
       (fma
        (* -5.0 (* a a))
        (pow c 4.0)
        (* (* (fma (* -2.0 a) c (* (- b) b)) (* c c)) (* b b)))
       (pow b 7.0))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 11.0) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = fma((fma((-5.0 * (a * a)), pow(c, 4.0), ((fma((-2.0 * a), c, (-b * b)) * (c * c)) * (b * b))) / pow(b, 7.0)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 11.0)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = fma(Float64(fma(Float64(-5.0 * Float64(a * a)), (c ^ 4.0), Float64(Float64(fma(Float64(-2.0 * a), c, Float64(Float64(-b) * b)) * Float64(c * c)) * Float64(b * b))) / (b ^ 7.0)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 11.0], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-5.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision] + N[(N[(N[(N[(-2.0 * a), $MachinePrecision] * c + N[((-b) * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 11

   1. Initial program 84.9%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 86.0%

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

   if 11 < b

   1. Initial program 44.7%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{-5 \cdot \left({a}^{2} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(-2 \cdot \left(a \cdot {c}^{3}\right) + -1 \cdot \left({b}^{2} \cdot {c}^{2}\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \mathsf{fma}\left(-2 \cdot a, {c}^{3}, \left(\left(-b\right) \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left({c}^{2} \cdot \left(-2 \cdot \left(a \cdot c\right) + -1 \cdot {b}^{2}\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 95.6%

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(\mathsf{fma}\left(-2 \cdot a, c, -b \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 11:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-5 \cdot \left(a \cdot a\right), {c}^{4}, \left(\mathsf{fma}\left(-2 \cdot a, c, \left(-b\right) \cdot b\right) \cdot \left(c \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{7}}, a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left(a \cdot c\right) \cdot -2}{{b}^{5}} - {\left({b}^{3}\right)}^{-1}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 15.5)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (fma
      (* (- (/ (* (* a c) -2.0) (pow b 5.0)) (pow (pow b 3.0) -1.0)) (* c c))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 15.5) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = fma((((((a * c) * -2.0) / pow(b, 5.0)) - pow(pow(b, 3.0), -1.0)) * (c * c)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(a * c) * -2.0) / (b ^ 5.0)) - ((b ^ 3.0) ^ -1.0)) * Float64(c * c)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 15.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(a * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[Power[N[Power[b, 3.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 85.8%

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

   if 15.5 < b

   1. Initial program 44.3%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 94.1%

      \[\leadsto \mathsf{fma}\left(\left(\frac{\left(a \cdot c\right) \cdot -2}{{b}^{5}} - \frac{1}{{b}^{3}}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left(a \cdot c\right) \cdot -2}{{b}^{5}} - {\left({b}^{3}\right)}^{-1}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(c \cdot -2, a \cdot \frac{a}{{b}^{5}}, \frac{-a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 15.5)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (*
      (-
       (* (fma (* c -2.0) (* a (/ a (pow b 5.0))) (/ (- a) (pow b 3.0))) c)
       (pow b -1.0))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 15.5) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = ((fma((c * -2.0), (a * (a / pow(b, 5.0))), (-a / pow(b, 3.0))) * c) - pow(b, -1.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(fma(Float64(c * -2.0), Float64(a * Float64(a / (b ^ 5.0))), Float64(Float64(-a) / (b ^ 3.0))) * c) - (b ^ -1.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 15.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * -2.0), $MachinePrecision] * N[(a * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-a) / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 85.8%

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

   if 15.5 < b

   1. Initial program 44.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(c \cdot -2, a \cdot \frac{a}{{b}^{5}}, \frac{-a}{{b}^{3}}\right) \cdot c - \frac{1}{b}\right) \cdot c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(c \cdot -2, a \cdot \frac{a}{{b}^{5}}, \frac{-a}{{b}^{3}}\right) \cdot c - {b}^{-1}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(c \cdot -2, a \cdot \frac{a}{{b}^{5}}, \frac{\frac{a}{b}}{\left(-b\right) \cdot b}\right) \cdot c - {b}^{-1}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 15.5)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (*
      (-
       (* (fma (* c -2.0) (* a (/ a (pow b 5.0))) (/ (/ a b) (* (- b) b))) c)
       (pow b -1.0))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 15.5) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = ((fma((c * -2.0), (a * (a / pow(b, 5.0))), ((a / b) / (-b * b))) * c) - pow(b, -1.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(fma(Float64(c * -2.0), Float64(a * Float64(a / (b ^ 5.0))), Float64(Float64(a / b) / Float64(Float64(-b) * b))) * c) - (b ^ -1.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 15.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * -2.0), $MachinePrecision] * N[(a * N[(a / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / b), $MachinePrecision] / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[Power[b, -1.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 85.8%

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

   if 15.5 < b

   1. Initial program 44.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

   7. Applied rewrites 94.0%

      \[\leadsto \left(\mathsf{fma}\left(c \cdot -2, a \cdot \frac{a}{{b}^{5}}, \frac{\frac{-a}{b}}{b \cdot b}\right) \cdot c - \frac{1}{b}\right) \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(c \cdot -2, a \cdot \frac{a}{{b}^{5}}, \frac{\frac{a}{b}}{\left(-b\right) \cdot b}\right) \cdot c - {b}^{-1}\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, 2, \frac{a}{b} \cdot \frac{c}{b}\right) + 1}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 15.5)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (*
      (/
       (+
        (fma (/ (* (* a a) (* c c)) (pow b 4.0)) 2.0 (* (/ a b) (/ c b)))
        1.0)
       (- b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 15.5) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = ((fma((((a * a) * (c * c)) / pow(b, 4.0)), 2.0, ((a / b) * (c / b))) + 1.0) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(a * a) * Float64(c * c)) / (b ^ 4.0)), 2.0, Float64(Float64(a / b) * Float64(c / b))) + 1.0) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 15.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / (-b)), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 85.8%

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

   if 15.5 < b

   1. Initial program 44.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 74.0%

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

   9. Taylor expanded in b around -inf

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

   11. Applied rewrites 93.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, 2, \frac{a}{b} \cdot \frac{c}{b}\right) + 1}{-b} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, 2, \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)\right)}{-b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 15.5)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (*
      (/
       (fma (/ (* (* a a) (* c c)) (pow b 4.0)) 2.0 (fma a (/ c (* b b)) 1.0))
       (- b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 15.5) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = (fma((((a * a) * (c * c)) / pow(b, 4.0)), 2.0, fma(a, (c / (b * b)), 1.0)) / -b) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(Float64(a * a) * Float64(c * c)) / (b ^ 4.0)), 2.0, fma(a, Float64(c / Float64(b * b)), 1.0)) / Float64(-b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 15.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * 2.0 + N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / (-b)), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 85.8%

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

   if 15.5 < b

   1. Initial program 44.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 93.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, 2, \mathsf{fma}\left(a, \frac{c}{b \cdot b}, 1\right)\right)}{-b} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -2, c \cdot c, \left(-\mathsf{fma}\left(b, b, a \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 15.5)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (*
      (/
       (fma (* (* a a) -2.0) (* c c) (* (- (fma b b (* a c))) (* b b)))
       (pow b 5.0))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 15.5) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = (fma(((a * a) * -2.0), (c * c), (-fma(b, b, (a * c)) * (b * b))) / pow(b, 5.0)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = Float64(Float64(fma(Float64(Float64(a * a) * -2.0), Float64(c * c), Float64(Float64(-fma(b, b, Float64(a * c))) * Float64(b * b))) / (b ^ 5.0)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 15.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * -2.0), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[((-N[(b * b + N[(a * c), $MachinePrecision]), $MachinePrecision]) * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 85.8%

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

   if 15.5 < b

   1. Initial program 44.3%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 93.7%

      \[\leadsto \frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -2, c \cdot c, \left(-\mathsf{fma}\left(b, b, a \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot -2, c \cdot c, \left(-\mathsf{fma}\left(b, b, a \cdot c\right)\right) \cdot \left(b \cdot b\right)\right)}{{b}^{5}} \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{{b}^{3}}, \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 15.5)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (- (fma a (/ (* c c) (pow b 3.0)) (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 15.5) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = -fma(a, ((c * c) / pow(b, 3.0)), (c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = Float64(-fma(a, Float64(Float64(c * c) / (b ^ 3.0)), Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 15.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], (-N[(a * N[(N[(c * c), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 85.8%

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

   if 15.5 < b

   1. Initial program 44.3%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-out N/A

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

      5. lower-neg.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-/.f64 90.4

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

   8. Applied rewrites 90.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(a, \frac{c \cdot c}{{b}^{3}}, \frac{c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c\\ \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - t\_0}{\left(b + \sqrt{t\_0}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (fma -4.0 a (/ (* b b) c)) c)))
   (if (<= b 15.5)
     (/ (- (* b b) t_0) (* (+ b (sqrt t_0)) (- (* 2.0 a))))
     (/ (- (fma (/ (* c c) b) (/ a b) c)) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-4.0, a, ((b * b) / c)) * c;
	double tmp;
	if (b <= 15.5) {
		tmp = ((b * b) - t_0) / ((b + sqrt(t_0)) * -(2.0 * a));
	} else {
		tmp = -fma(((c * c) / b), (a / b), c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(b + sqrt(t_0)) * Float64(-Float64(2.0 * a))));
	else
		tmp = Float64(Float64(-fma(Float64(Float64(c * c) / b), Float64(a / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 15.5], N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(b + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * (-N[(2.0 * a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + c), $MachinePrecision]) / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   7. Applied rewrites 85.8%

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

   if 15.5 < b

   1. Initial program 44.3%

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

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. div-add N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 90.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{b \cdot b - \mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}{\left(b + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}\right) \cdot \left(-2 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(\frac{b}{c}, b, -4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.5)
   (/ (+ (- b) (sqrt (* (fma (/ b c) b (* -4.0 a)) c))) (* 2.0 a))
   (/ (- (fma (/ (* c c) b) (/ a b) c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.5) {
		tmp = (-b + sqrt((fma((b / c), b, (-4.0 * a)) * c))) / (2.0 * a);
	} else {
		tmp = -fma(((c * c) / b), (a / b), c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(fma(Float64(b / c), b, Float64(-4.0 * a)) * c))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-fma(Float64(Float64(c * c) / b), Float64(a / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.5], N[(N[((-b) + N[Sqrt[N[(N[(N[(b / c), $MachinePrecision] * b + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + c), $MachinePrecision]) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

   7. Applied rewrites 84.7%

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

   if 15.5 < b

   1. Initial program 44.3%

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

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. div-add N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 90.4%

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

Alternative 11: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.5)
   (/ (+ (- b) (sqrt (fma (* c -4.0) a (* b b)))) (* 2.0 a))
   (/ (- (fma (/ (* c c) b) (/ a b) c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.5) {
		tmp = (-b + sqrt(fma((c * -4.0), a, (b * b)))) / (2.0 * a);
	} else {
		tmp = -fma(((c * c) / b), (a / b), c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-fma(Float64(Float64(c * c) / b), Float64(a / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.5], N[(N[((-b) + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + c), $MachinePrecision]) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

   4. Applied rewrites 84.7%

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

   if 15.5 < b

   1. Initial program 44.3%

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

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. div-add N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 90.4%

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

Alternative 12: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 15.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{c \cdot c}{b}, \frac{a}{b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 15.5)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/ (- (fma (/ (* c c) b) (/ a b) c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 15.5) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = -fma(((c * c) / b), (a / b), c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 15.5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-fma(Float64(Float64(c * c) / b), Float64(a / b), c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 15.5], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(N[(c * c), $MachinePrecision] / b), $MachinePrecision] * N[(a / b), $MachinePrecision] + c), $MachinePrecision]) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 15.5

   1. Initial program 84.7%

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

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

      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. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-abs-rev N/A

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

      10. fabs-fabs N/A

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

      11. fabs-fabs N/A

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

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

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. associate-*l* N/A

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

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

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

      19. *-commutative N/A

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

      20. lift-*.f64 N/A

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

   4. Applied rewrites 84.7%

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

   if 15.5 < b

   1. Initial program 44.3%

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

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

      2. unpow3 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l* N/A

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

      6. div-add N/A

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

      7. lower-/.f64 N/A

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

   5. Applied rewrites 90.4%

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

Alternative 13: 81.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

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

   2. unpow3 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-/l* N/A

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

   6. div-add N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 83.1%

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

Alternative 14: 81.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.5%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 88.5%

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

6. Taylor expanded in b around -inf

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

8. Applied rewrites 83.0%

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

Alternative 15: 64.4% 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 52.5%

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

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 66.7

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

5. Applied rewrites 66.7%

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

Alternative 16: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 52.5%

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

3. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 52.5

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

5. Applied rewrites 52.5%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. div-add N/A

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

   4. lower-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 51.9

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

7. Applied rewrites 51.9%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f64 N/A

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

   6. frac-add N/A

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

   7. lower-/.f64 N/A

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

9. Applied rewrites 53.1%

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

10. Taylor expanded in a around 0

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

    1. div-add N/A

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

    2. associate-*r/ N/A

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

    3. associate-*r/ N/A

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

    4. distribute-rgt-out N/A

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

    5. metadata-eval N/A

       \[\leadsto \frac{-1}{4} \cdot \left(\frac{b}{a} \cdot \color{blue}{0}\right) \]

    6. mul0-rgt N/A

       \[\leadsto \frac{-1}{4} \cdot \color{blue}{0} \]

    7. metadata-eval 3.2

       \[\leadsto \color{blue}{0} \]

12. Applied rewrites 3.2%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))