Quadratic roots, narrow range

Percentage Accurate: 55.7% → 91.8%

Time: 11.7s

Alternatives: 21

Speedup: 4.6×

Specification

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

Math

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

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: 55.7% accurate, 1.0× speedup

Math

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

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: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(b, b + \left(1 + \frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{b}\right) \cdot b, \left(-4 \cdot c\right) \cdot a\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1200:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{t\_0} \cdot t\_0, \frac{1}{t\_1}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{t\_1}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-1, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, -0.25 \cdot \frac{a \cdot \mathsf{fma}\left(4, {c}^{4}, 16 \cdot {c}^{4}\right)}{{b}^{7}}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b)))
        (t_1
         (fma
          b
          (+ b (* (+ 1.0 (/ (sqrt (fma (* a c) -4.0 (* b b))) b)) b))
          (* (* -4.0 c) a))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1200.0)
     (/
      (fma (* (sqrt t_0) t_0) (/ 1.0 t_1) (/ (* (* (- b) b) b) t_1))
      (* 2.0 a))
     (fma
      -1.0
      (/ c b)
      (*
       a
       (fma
        -1.0
        (/ (pow c 2.0) (pow b 3.0))
        (*
         a
         (fma
          -2.0
          (/ (pow c 3.0) (pow b 5.0))
          (*
           -0.25
           (/
            (* a (fma 4.0 (pow c 4.0) (* 16.0 (pow c 4.0))))
            (pow b 7.0)))))))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma(b, (b + ((1.0 + (sqrt(fma((a * c), -4.0, (b * b))) / b)) * b)), ((-4.0 * c) * a));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1200.0) {
		tmp = fma((sqrt(t_0) * t_0), (1.0 / t_1), (((-b * b) * b) / t_1)) / (2.0 * a);
	} else {
		tmp = fma(-1.0, (c / b), (a * fma(-1.0, (pow(c, 2.0) / pow(b, 3.0)), (a * fma(-2.0, (pow(c, 3.0) / pow(b, 5.0)), (-0.25 * ((a * fma(4.0, pow(c, 4.0), (16.0 * pow(c, 4.0)))) / pow(b, 7.0))))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = fma(b, Float64(b + Float64(Float64(1.0 + Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) / b)) * b)), Float64(Float64(-4.0 * c) * a))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1200.0)
		tmp = Float64(fma(Float64(sqrt(t_0) * t_0), Float64(1.0 / t_1), Float64(Float64(Float64(Float64(-b) * b) * b) / t_1)) / Float64(2.0 * a));
	else
		tmp = fma(-1.0, Float64(c / b), Float64(a * fma(-1.0, Float64((c ^ 2.0) / (b ^ 3.0)), Float64(a * fma(-2.0, Float64((c ^ 3.0) / (b ^ 5.0)), Float64(-0.25 * Float64(Float64(a * fma(4.0, (c ^ 4.0), Float64(16.0 * (c ^ 4.0)))) / (b ^ 7.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b + N[(N[(1.0 + N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -1200.0], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision] + N[(N[(N[((-b) * b), $MachinePrecision] * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision] + N[(a * N[(-1.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(a * N[(-2.0 * N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(a * N[(4.0 * N[Power[c, 4.0], $MachinePrecision] + N[(16.0 * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1200

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 57.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 56.9%

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 56.9%

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

   7. Applied rewrites 56.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 57.1%

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 57.1%

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

   9. Applied rewrites 57.1%

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

   if -1200 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in a around 0

      \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}\right)\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{c}{\color{blue}{b}}, a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}\right)\right)\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot {c}^{4} + 16 \cdot {c}^{4}\right)}{{b}^{7}}\right)\right)\right) \]

   7. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(-1, \color{blue}{\frac{c}{b}}, a \cdot \mathsf{fma}\left(-1, \frac{{c}^{2}}{{b}^{3}}, a \cdot \mathsf{fma}\left(-2, \frac{{c}^{3}}{{b}^{5}}, -0.25 \cdot \frac{a \cdot \mathsf{fma}\left(4, {c}^{4}, 16 \cdot {c}^{4}\right)}{{b}^{7}}\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \mathsf{fma}\left(b, b + \left(1 + \frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{b}\right) \cdot b, \left(-4 \cdot c\right) \cdot a\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1200:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{t\_0} \cdot t\_0, \frac{1}{t\_1}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{t\_1}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot -0.25 - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b)))
        (t_1
         (fma
          b
          (+ b (* (+ 1.0 (/ (sqrt (fma (* a c) -4.0 (* b b))) b)) b))
          (* (* -4.0 c) a))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1200.0)
     (/
      (fma (* (sqrt t_0) t_0) (/ 1.0 t_1) (/ (* (* (- b) b) b) t_1))
      (* 2.0 a))
     (+
      (/ (- (* (* (* (* a a) c) (* c c)) (* (pow b -4.0) -2.0)) c) b)
      (/
       (-
        (* (* (pow (* c a) 4.0) (/ 20.0 (* (pow b 6.0) a))) -0.25)
        (* (* c c) (/ a (* b b))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = fma(b, (b + ((1.0 + (sqrt(fma((a * c), -4.0, (b * b))) / b)) * b)), ((-4.0 * c) * a));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1200.0) {
		tmp = fma((sqrt(t_0) * t_0), (1.0 / t_1), (((-b * b) * b) / t_1)) / (2.0 * a);
	} else {
		tmp = ((((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)) - c) / b) + ((((pow((c * a), 4.0) * (20.0 / (pow(b, 6.0) * a))) * -0.25) - ((c * c) * (a / (b * b)))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = fma(b, Float64(b + Float64(Float64(1.0 + Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) / b)) * b)), Float64(Float64(-4.0 * c) * a))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1200.0)
		tmp = Float64(fma(Float64(sqrt(t_0) * t_0), Float64(1.0 / t_1), Float64(Float64(Float64(Float64(-b) * b) * b) / t_1)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)) - c) / b) + Float64(Float64(Float64(Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64((b ^ 6.0) * a))) * -0.25) - Float64(Float64(c * c) * Float64(a / Float64(b * b)))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b + N[(N[(1.0 + N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -1200.0], N[(N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision] + N[(N[(N[((-b) * b), $MachinePrecision] * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1200

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 57.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 56.9%

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 56.9%

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

   7. Applied rewrites 56.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. sum-to-mult N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-+.f64 N/A

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

      6. lower-unsound-/.f64 57.1%

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 57.1%

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

   9. Applied rewrites 57.1%

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

   if -1200 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   6. Applied rewrites 90.9%

      \[\leadsto \frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \color{blue}{\frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot -0.25 - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -1200:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{1.5}, \frac{1}{t\_0}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{t\_0}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot -0.25 - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (fma
          b
          (+ b (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
          (* (* -4.0 c) a))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -1200.0)
     (/
      (fma
       (pow (fma (* a c) -4.0 (* b b)) 1.5)
       (/ 1.0 t_0)
       (/ (* (* (- b) b) b) t_0))
      (* 2.0 a))
     (+
      (/ (- (* (* (* (* a a) c) (* c c)) (* (pow b -4.0) -2.0)) c) b)
      (/
       (-
        (* (* (pow (* c a) 4.0) (/ 20.0 (* (pow b 6.0) a))) -0.25)
        (* (* c c) (/ a (* b b))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, (b + (sqrt(fma((-4.0 * c), a, (b * b))) + b)), ((-4.0 * c) * a));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -1200.0) {
		tmp = fma(pow(fma((a * c), -4.0, (b * b)), 1.5), (1.0 / t_0), (((-b * b) * b) / t_0)) / (2.0 * a);
	} else {
		tmp = ((((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)) - c) / b) + ((((pow((c * a), 4.0) * (20.0 / (pow(b, 6.0) * a))) * -0.25) - ((c * c) * (a / (b * b)))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, Float64(b + Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)), Float64(Float64(-4.0 * c) * a))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -1200.0)
		tmp = Float64(fma((fma(Float64(a * c), -4.0, Float64(b * b)) ^ 1.5), Float64(1.0 / t_0), Float64(Float64(Float64(Float64(-b) * b) * b) / t_0)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)) - c) / b) + Float64(Float64(Float64(Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64((b ^ 6.0) * a))) * -0.25) - Float64(Float64(c * c) * Float64(a / Float64(b * b)))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -1200.0], N[(N[(N[Power[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision] + N[(N[(N[((-b) * b), $MachinePrecision] * b), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -1200

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. metadata-eval 57.2%

          \[\leadsto \frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{\color{blue}{1.5}}, \frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}\right)}{2 \cdot a} \]

   7. Applied rewrites 57.2%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{1.5}}, \frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}\right)}{2 \cdot a} \]

   if -1200 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   6. Applied rewrites 90.9%

      \[\leadsto \frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \color{blue}{\frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot -0.25 - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := {\left(a \cdot c\right)}^{4}\\ t_1 := t\_0 \cdot 20\\ t_2 := \left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\\ t_3 := t\_0 \cdot {b}^{-6}\\ t_4 := t\_2 \cdot {b}^{-4}\\ t_5 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_6 := \frac{t\_5}{b \cdot b}\\ t_7 := t\_5 \cdot 0\\ \frac{\mathsf{fma}\left(t\_4, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_6, -4, \mathsf{fma}\left(t\_4, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_2 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(t\_7 \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(t\_7 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_6, \frac{\mathsf{fma}\left(-1, t\_1, -0.5 \cdot t\_1\right)}{{b}^{6}} + \mathsf{fma}\left(t\_6, 4, \mathsf{fma}\left(t\_3, 4, \mathsf{fma}\left(8, t\_6, \mathsf{fma}\left(16, t\_4, t\_3 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(b, b + \left(\mathsf{fma}\left(t\_2 \cdot {b}^{-6}, -4, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, t\_5 \cdot {b}^{-4}\right)\right) + 1\right) \cdot b, \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right) \cdot \left(a + a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* a c) 4.0))
        (t_1 (* t_0 20.0))
        (t_2 (* (* (* c c) c) (* (* a a) a)))
        (t_3 (* t_0 (pow b -6.0)))
        (t_4 (* t_2 (pow b -4.0)))
        (t_5 (* (* c c) (* a a)))
        (t_6 (/ t_5 (* b b)))
        (t_7 (* t_5 0.0)))
   (/
    (*
     (fma
      t_4
      -8.0
      (fma
       (* -4.0 a)
       c
       (fma
        t_6
        -4.0
        (fma
         t_4
         -4.0
         (fma
          (* -2.0 a)
          c
          (fma
           (* (* (* (* t_2 0.0) c) a) (pow b -6.0))
           -2.0
           (fma
            (* (* (* t_7 c) a) (pow b -4.0))
            -2.0
            (fma
             (* (* (* t_7 (* c c)) (* a a)) (pow b -6.0))
             -2.0
             (fma
              -2.0
              t_6
              (+
               (/ (fma -1.0 t_1 (* -0.5 t_1)) (pow b 6.0))
               (fma
                t_6
                4.0
                (fma
                 t_3
                 4.0
                 (fma 8.0 t_6 (fma 16.0 t_4 (* t_3 32.0)))))))))))))))
     b)
    (*
     (fma
      b
      (+
       b
       (*
        (+
         (fma
          (* t_2 (pow b -6.0))
          -4.0
          (* -2.0 (fma a (/ c (* b b)) (* t_5 (pow b -4.0)))))
         1.0)
        b))
      (fma (* a c) -4.0 (* b b)))
     (+ a a)))))

C

double code(double a, double b, double c) {
	double t_0 = pow((a * c), 4.0);
	double t_1 = t_0 * 20.0;
	double t_2 = ((c * c) * c) * ((a * a) * a);
	double t_3 = t_0 * pow(b, -6.0);
	double t_4 = t_2 * pow(b, -4.0);
	double t_5 = (c * c) * (a * a);
	double t_6 = t_5 / (b * b);
	double t_7 = t_5 * 0.0;
	return (fma(t_4, -8.0, fma((-4.0 * a), c, fma(t_6, -4.0, fma(t_4, -4.0, fma((-2.0 * a), c, fma(((((t_2 * 0.0) * c) * a) * pow(b, -6.0)), -2.0, fma((((t_7 * c) * a) * pow(b, -4.0)), -2.0, fma((((t_7 * (c * c)) * (a * a)) * pow(b, -6.0)), -2.0, fma(-2.0, t_6, ((fma(-1.0, t_1, (-0.5 * t_1)) / pow(b, 6.0)) + fma(t_6, 4.0, fma(t_3, 4.0, fma(8.0, t_6, fma(16.0, t_4, (t_3 * 32.0))))))))))))))) * b) / (fma(b, (b + ((fma((t_2 * pow(b, -6.0)), -4.0, (-2.0 * fma(a, (c / (b * b)), (t_5 * pow(b, -4.0))))) + 1.0) * b)), fma((a * c), -4.0, (b * b))) * (a + a));
}

Julia

function code(a, b, c)
	t_0 = Float64(a * c) ^ 4.0
	t_1 = Float64(t_0 * 20.0)
	t_2 = Float64(Float64(Float64(c * c) * c) * Float64(Float64(a * a) * a))
	t_3 = Float64(t_0 * (b ^ -6.0))
	t_4 = Float64(t_2 * (b ^ -4.0))
	t_5 = Float64(Float64(c * c) * Float64(a * a))
	t_6 = Float64(t_5 / Float64(b * b))
	t_7 = Float64(t_5 * 0.0)
	return Float64(Float64(fma(t_4, -8.0, fma(Float64(-4.0 * a), c, fma(t_6, -4.0, fma(t_4, -4.0, fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_2 * 0.0) * c) * a) * (b ^ -6.0)), -2.0, fma(Float64(Float64(Float64(t_7 * c) * a) * (b ^ -4.0)), -2.0, fma(Float64(Float64(Float64(t_7 * Float64(c * c)) * Float64(a * a)) * (b ^ -6.0)), -2.0, fma(-2.0, t_6, Float64(Float64(fma(-1.0, t_1, Float64(-0.5 * t_1)) / (b ^ 6.0)) + fma(t_6, 4.0, fma(t_3, 4.0, fma(8.0, t_6, fma(16.0, t_4, Float64(t_3 * 32.0))))))))))))))) * b) / Float64(fma(b, Float64(b + Float64(Float64(fma(Float64(t_2 * (b ^ -6.0)), -4.0, Float64(-2.0 * fma(a, Float64(c / Float64(b * b)), Float64(t_5 * (b ^ -4.0))))) + 1.0) * b)), fma(Float64(a * c), -4.0, Float64(b * b))) * Float64(a + a)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 20.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$5 * 0.0), $MachinePrecision]}, N[(N[(N[(t$95$4 * -8.0 + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$6 * -4.0 + N[(t$95$4 * -4.0 + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$2 * 0.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$7 * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(N[(N[(t$95$7 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$6 + N[(N[(N[(-1.0 * t$95$1 + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * 4.0 + N[(t$95$3 * 4.0 + N[(8.0 * t$95$6 + N[(16.0 * t$95$4 + N[(t$95$3 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(N[(b * N[(b + N[(N[(N[(N[(t$95$2 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -4.0 + N[(-2.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Initial program 55.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. flip3-- N/A

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

   5. lower-unsound-/.f64 N/A

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

3. Applied rewrites 55.6%

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

4. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-8 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-4 \cdot \left(a \cdot c\right) + \left(-4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-2 \cdot \left(a \cdot c\right) + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-8 \cdot \left({a}^{3} \cdot {c}^{3}\right) + 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}} + \left(-2 \cdot \frac{{a}^{2} \cdot \left({c}^{2} \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-1 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{-1}{2} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(4 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}} + \left(8 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot b\right)}}{2 \cdot a} \]

6. Applied rewrites 91.4%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot b\right)}}{2 \cdot a} \]

7. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \color{blue}{\left(b \cdot \left(1 + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)} \cdot b\right)}}{2 \cdot a} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \left(b \cdot \color{blue}{\left(1 + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)}\right) \cdot b\right)}}{2 \cdot a} \]

   2. lower-+.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \left(b \cdot \left(1 + \color{blue}{\left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)}\right)\right) \cdot b\right)}}{2 \cdot a} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \left(b \cdot \left(1 + \mathsf{fma}\left(-4, \color{blue}{\frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}}, -2 \cdot \frac{a \cdot c}{{b}^{2}} + -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right) \cdot b\right)}}{2 \cdot a} \]

9. Applied rewrites 91.6%

   \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \color{blue}{\left(b \cdot \left(1 + \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot c}{{b}^{2}}, -2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}}\right)\right)\right)\right)} \cdot b\right)}}{2 \cdot a} \]

11. Applied rewrites 91.6%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, -8, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, -4, \mathsf{fma}\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, -4, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-4}, -2, \mathsf{fma}\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, -0.5 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{\mathsf{fma}\left(b, b + \left(\mathsf{fma}\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-6}, -4, -2 \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot b}, \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-4}\right)\right) + 1\right) \cdot b, \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right) \cdot \left(a + a\right)}} \]
12. Add Preprocessing

Alternative 5: 91.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} t_0 := \left(c \cdot c\right) \cdot \left(a \cdot a\right)\\ t_1 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ t_2 := \left(a \cdot a\right) \cdot a\\ t_3 := \left(c \cdot c\right) \cdot c\\ t_4 := \frac{t\_0}{b \cdot b}\\ t_5 := {\left(a \cdot c\right)}^{4}\\ t_6 := t\_5 \cdot {b}^{-6}\\ t_7 := t\_5 \cdot 20\\ t_8 := t\_3 \cdot t\_2\\ t_9 := t\_0 \cdot 0\\ \frac{\frac{\mathsf{fma}\left(\left(-8 \cdot t\_2\right) \cdot t\_3, {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(t\_4, -4, \mathsf{fma}\left(-4 \cdot t\_8, {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(t\_8 \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(t\_9 \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(t\_9 \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, t\_4, \frac{\mathsf{fma}\left(-1, t\_7, -0.5 \cdot t\_7\right)}{{b}^{6}} + \mathsf{fma}\left(t\_4, 4, \mathsf{fma}\left(t\_6, 4, \mathsf{fma}\left(8, t\_4, \mathsf{fma}\left(16, t\_8 \cdot {b}^{-4}, t\_6 \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}{t\_1 + \mathsf{fma}\left(b, b, \sqrt{t\_1} \cdot b\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) (* a a)))
        (t_1 (fma (* c -4.0) a (* b b)))
        (t_2 (* (* a a) a))
        (t_3 (* (* c c) c))
        (t_4 (/ t_0 (* b b)))
        (t_5 (pow (* a c) 4.0))
        (t_6 (* t_5 (pow b -6.0)))
        (t_7 (* t_5 20.0))
        (t_8 (* t_3 t_2))
        (t_9 (* t_0 0.0)))
   (/
    (/
     (*
      (fma
       (* (* -8.0 t_2) t_3)
       (pow b -4.0)
       (fma
        (* -4.0 a)
        c
        (fma
         t_4
         -4.0
         (fma
          (* -4.0 t_8)
          (pow b -4.0)
          (fma
           (* -2.0 a)
           c
           (fma
            (* (* (* (* t_8 0.0) c) a) (pow b -6.0))
            -2.0
            (fma
             (* -2.0 (* (* t_9 c) a))
             (pow b -4.0)
             (fma
              (* (* (* t_9 (* c c)) (* a a)) (pow b -6.0))
              -2.0
              (fma
               -2.0
               t_4
               (+
                (/ (fma -1.0 t_7 (* -0.5 t_7)) (pow b 6.0))
                (fma
                 t_4
                 4.0
                 (fma
                  t_6
                  4.0
                  (fma
                   8.0
                   t_4
                   (fma 16.0 (* t_8 (pow b -4.0)) (* t_6 32.0)))))))))))))))
      b)
     (+ t_1 (fma b b (* (sqrt t_1) b))))
    (* 2.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * (a * a);
	double t_1 = fma((c * -4.0), a, (b * b));
	double t_2 = (a * a) * a;
	double t_3 = (c * c) * c;
	double t_4 = t_0 / (b * b);
	double t_5 = pow((a * c), 4.0);
	double t_6 = t_5 * pow(b, -6.0);
	double t_7 = t_5 * 20.0;
	double t_8 = t_3 * t_2;
	double t_9 = t_0 * 0.0;
	return ((fma(((-8.0 * t_2) * t_3), pow(b, -4.0), fma((-4.0 * a), c, fma(t_4, -4.0, fma((-4.0 * t_8), pow(b, -4.0), fma((-2.0 * a), c, fma(((((t_8 * 0.0) * c) * a) * pow(b, -6.0)), -2.0, fma((-2.0 * ((t_9 * c) * a)), pow(b, -4.0), fma((((t_9 * (c * c)) * (a * a)) * pow(b, -6.0)), -2.0, fma(-2.0, t_4, ((fma(-1.0, t_7, (-0.5 * t_7)) / pow(b, 6.0)) + fma(t_4, 4.0, fma(t_6, 4.0, fma(8.0, t_4, fma(16.0, (t_8 * pow(b, -4.0)), (t_6 * 32.0))))))))))))))) * b) / (t_1 + fma(b, b, (sqrt(t_1) * b)))) / (2.0 * a);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * Float64(a * a))
	t_1 = fma(Float64(c * -4.0), a, Float64(b * b))
	t_2 = Float64(Float64(a * a) * a)
	t_3 = Float64(Float64(c * c) * c)
	t_4 = Float64(t_0 / Float64(b * b))
	t_5 = Float64(a * c) ^ 4.0
	t_6 = Float64(t_5 * (b ^ -6.0))
	t_7 = Float64(t_5 * 20.0)
	t_8 = Float64(t_3 * t_2)
	t_9 = Float64(t_0 * 0.0)
	return Float64(Float64(Float64(fma(Float64(Float64(-8.0 * t_2) * t_3), (b ^ -4.0), fma(Float64(-4.0 * a), c, fma(t_4, -4.0, fma(Float64(-4.0 * t_8), (b ^ -4.0), fma(Float64(-2.0 * a), c, fma(Float64(Float64(Float64(Float64(t_8 * 0.0) * c) * a) * (b ^ -6.0)), -2.0, fma(Float64(-2.0 * Float64(Float64(t_9 * c) * a)), (b ^ -4.0), fma(Float64(Float64(Float64(t_9 * Float64(c * c)) * Float64(a * a)) * (b ^ -6.0)), -2.0, fma(-2.0, t_4, Float64(Float64(fma(-1.0, t_7, Float64(-0.5 * t_7)) / (b ^ 6.0)) + fma(t_4, 4.0, fma(t_6, 4.0, fma(8.0, t_4, fma(16.0, Float64(t_8 * (b ^ -4.0)), Float64(t_6 * 32.0))))))))))))))) * b) / Float64(t_1 + fma(b, b, Float64(sqrt(t_1) * b)))) / Float64(2.0 * a))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$3 = N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$5 * 20.0), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$3 * t$95$2), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$0 * 0.0), $MachinePrecision]}, N[(N[(N[(N[(N[(N[(-8.0 * t$95$2), $MachinePrecision] * t$95$3), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(-4.0 * a), $MachinePrecision] * c + N[(t$95$4 * -4.0 + N[(N[(-4.0 * t$95$8), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(-2.0 * a), $MachinePrecision] * c + N[(N[(N[(N[(N[(t$95$8 * 0.0), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(-2.0 * N[(N[(t$95$9 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -4.0], $MachinePrecision] + N[(N[(N[(N[(t$95$9 * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Power[b, -6.0], $MachinePrecision]), $MachinePrecision] * -2.0 + N[(-2.0 * t$95$4 + N[(N[(N[(-1.0 * t$95$7 + N[(-0.5 * t$95$7), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * 4.0 + N[(t$95$6 * 4.0 + N[(8.0 * t$95$4 + N[(16.0 * N[(t$95$8 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$6 * 32.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] / N[(t$95$1 + N[(b * b + N[(N[Sqrt[t$95$1], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Initial program 55.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. flip3-- N/A

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

   5. lower-unsound-/.f64 N/A

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

3. Applied rewrites 55.6%

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

4. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-8 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-4 \cdot \left(a \cdot c\right) + \left(-4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(-2 \cdot \left(a \cdot c\right) + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-8 \cdot \left({a}^{3} \cdot {c}^{3}\right) + 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{a \cdot \left(c \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}} + \left(-2 \cdot \frac{{a}^{2} \cdot \left({c}^{2} \cdot \left(-4 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}} + \left(-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(-1 \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{-1}{2} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(4 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(4 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}} + \left(8 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(16 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot b\right)}}{2 \cdot a} \]

6. Applied rewrites 91.4%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-8, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-4, a \cdot c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-4, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, a \cdot c, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-8, {a}^{3} \cdot {c}^{3}, 8 \cdot \left({a}^{3} \cdot {c}^{3}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{a \cdot \left(c \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{4}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot \left({c}^{2} \cdot \mathsf{fma}\left(-4, {a}^{2} \cdot {c}^{2}, 4 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{{b}^{6}}, \mathsf{fma}\left(4, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(4, \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(16, \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}}, 32 \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot b\right)}}{2 \cdot a} \]

8. Applied rewrites 91.5%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(-8 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), {b}^{-4}, \mathsf{fma}\left(-4 \cdot a, c, \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, -4, \mathsf{fma}\left(-4 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right), {b}^{-4}, \mathsf{fma}\left(-2 \cdot a, c, \mathsf{fma}\left(\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2 \cdot \left(\left(\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot 0\right) \cdot c\right) \cdot a\right), {b}^{-4}, \mathsf{fma}\left(\left(\left(\left(\left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right) \cdot 0\right) \cdot \left(c \cdot c\right)\right) \cdot \left(a \cdot a\right)\right) \cdot {b}^{-6}, -2, \mathsf{fma}\left(-2, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, \frac{\mathsf{fma}\left(-1, {\left(a \cdot c\right)}^{4} \cdot 20, -0.5 \cdot \left({\left(a \cdot c\right)}^{4} \cdot 20\right)\right)}{{b}^{6}} + \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, 4, \mathsf{fma}\left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}, 4, \mathsf{fma}\left(8, \frac{\left(c \cdot c\right) \cdot \left(a \cdot a\right)}{b \cdot b}, \mathsf{fma}\left(16, \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot {b}^{-4}, \left({\left(a \cdot c\right)}^{4} \cdot {b}^{-6}\right) \cdot 32\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot b}}{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right) + \mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot b\right)}}{2 \cdot a} \]
9. Add Preprocessing

Alternative 6: 91.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{a \cdot {c}^{2}}{{b}^{2}}\\ t_1 := \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}\\ t_2 := {a}^{4} \cdot {c}^{4}\\ \left(b \cdot \mathsf{fma}\left(-4, c, \mathsf{fma}\left(-4, t\_1, \mathsf{fma}\left(-2, c, \mathsf{fma}\left(-2, t\_0, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, t\_2, 16 \cdot t\_2\right)}{a \cdot {b}^{6}}, \mathsf{fma}\left(8, t\_0, \mathsf{fma}\left(8, t\_1, 16 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a (pow c 2.0)) (pow b 2.0)))
        (t_1 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 4.0)))
        (t_2 (* (pow a 4.0) (pow c 4.0))))
   (*
    (*
     b
     (fma
      -4.0
      c
      (fma
       -4.0
       t_1
       (fma
        -2.0
        c
        (fma
         -2.0
         t_0
         (fma
          -0.5
          (/ (fma 4.0 t_2 (* 16.0 t_2)) (* a (pow b 6.0)))
          (fma
           8.0
           t_0
           (fma
            8.0
            t_1
            (* 16.0 (/ (* (pow a 3.0) (pow c 4.0)) (pow b 6.0)))))))))))
    (/
     (/
      1.0
      (fma b (+ b (+ (sqrt (fma (* -4.0 c) a (* b b))) b)) (* (* -4.0 c) a)))
     2.0))))

C

double code(double a, double b, double c) {
	double t_0 = (a * pow(c, 2.0)) / pow(b, 2.0);
	double t_1 = (pow(a, 2.0) * pow(c, 3.0)) / pow(b, 4.0);
	double t_2 = pow(a, 4.0) * pow(c, 4.0);
	return (b * fma(-4.0, c, fma(-4.0, t_1, fma(-2.0, c, fma(-2.0, t_0, fma(-0.5, (fma(4.0, t_2, (16.0 * t_2)) / (a * pow(b, 6.0))), fma(8.0, t_0, fma(8.0, t_1, (16.0 * ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 6.0))))))))))) * ((1.0 / fma(b, (b + (sqrt(fma((-4.0 * c), a, (b * b))) + b)), ((-4.0 * c) * a))) / 2.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))
	t_1 = Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 4.0))
	t_2 = Float64((a ^ 4.0) * (c ^ 4.0))
	return Float64(Float64(b * fma(-4.0, c, fma(-4.0, t_1, fma(-2.0, c, fma(-2.0, t_0, fma(-0.5, Float64(fma(4.0, t_2, Float64(16.0 * t_2)) / Float64(a * (b ^ 6.0))), fma(8.0, t_0, fma(8.0, t_1, Float64(16.0 * Float64(Float64((a ^ 3.0) * (c ^ 4.0)) / (b ^ 6.0))))))))))) * Float64(Float64(1.0 / fma(b, Float64(b + Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)), Float64(Float64(-4.0 * c) * a))) / 2.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[a, 4.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(b * N[(-4.0 * c + N[(-4.0 * t$95$1 + N[(-2.0 * c + N[(-2.0 * t$95$0 + N[(-0.5 * N[(N[(4.0 * t$95$2 + N[(16.0 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(8.0 * t$95$0 + N[(8.0 * t$95$1 + N[(16.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(b * N[(b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 55.7%

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

   1. lift-*.f64 N/A

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

   2. pow2 N/A

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

   3. remove-double-neg N/A

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

   4. lift-neg.f64 N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-neg N/A

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

   8. lower-unsound-/.f64 N/A

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

   9. lower-unsound-pow.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. remove-double-neg N/A

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

   12. metadata-eval 55.6%

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

3. Applied rewrites 55.6%

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

5. Applied rewrites 56.6%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\left(b \cdot \left(-4 \cdot c + \left(-4 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(-2 \cdot c + \left(-2 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{4 \cdot \left({a}^{4} \cdot {c}^{4}\right) + 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}} + \left(8 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(8 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + 16 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2} \]

8. Applied rewrites 91.5%

   \[\leadsto \color{blue}{\left(b \cdot \mathsf{fma}\left(-4, c, \mathsf{fma}\left(-4, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-2, c, \mathsf{fma}\left(-2, \frac{a \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}, \mathsf{fma}\left(8, \frac{a \cdot {c}^{2}}{{b}^{2}}, \mathsf{fma}\left(8, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, 16 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)\right)} \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2} \]
9. Add Preprocessing

Alternative 7: 90.9% accurate, 0.2× speedup

Math

\[\frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot -0.25 - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (/ (- (* (* (* (* a a) c) (* c c)) (* (pow b -4.0) -2.0)) c) b)
  (/
   (-
    (* (* (pow (* c a) 4.0) (/ 20.0 (* (pow b 6.0) a))) -0.25)
    (* (* c c) (/ a (* b b))))
   b)))

C

double code(double a, double b, double c) {
	return ((((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)) - c) / b) + ((((pow((c * a), 4.0) * (20.0 / (pow(b, 6.0) * a))) * -0.25) - ((c * c) * (a / (b * b)))) / 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 = ((((((a * a) * c) * (c * c)) * ((b ** (-4.0d0)) * (-2.0d0))) - c) / b) + ((((((c * a) ** 4.0d0) * (20.0d0 / ((b ** 6.0d0) * a))) * (-0.25d0)) - ((c * c) * (a / (b * b)))) / b)
end function

Java

public static double code(double a, double b, double c) {
	return ((((((a * a) * c) * (c * c)) * (Math.pow(b, -4.0) * -2.0)) - c) / b) + ((((Math.pow((c * a), 4.0) * (20.0 / (Math.pow(b, 6.0) * a))) * -0.25) - ((c * c) * (a / (b * b)))) / b);
}

Python

def code(a, b, c):
	return ((((((a * a) * c) * (c * c)) * (math.pow(b, -4.0) * -2.0)) - c) / b) + ((((math.pow((c * a), 4.0) * (20.0 / (math.pow(b, 6.0) * a))) * -0.25) - ((c * c) * (a / (b * b)))) / b)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)) - c) / b) + Float64(Float64(Float64(Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64((b ^ 6.0) * a))) * -0.25) - Float64(Float64(c * c) * Float64(a / Float64(b * b)))) / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((((a * a) * c) * (c * c)) * ((b ^ -4.0) * -2.0)) - c) / b) + ((((((c * a) ^ 4.0) * (20.0 / ((b ^ 6.0) * a))) * -0.25) - ((c * c) * (a / (b * b)))) / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 90.9%

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

6. Applied rewrites 90.9%

   \[\leadsto \frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \color{blue}{\frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot -0.25 - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b}} \]
7. Add Preprocessing

Alternative 8: 90.9% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (- c)
   (-
    (fma
     0.25
     (* (pow (* c a) 4.0) (/ 20.0 (* (pow b 6.0) a)))
     (* (* c c) (/ a (* b b))))
    (* (* (* (* a a) c) (* c c)) (* (pow b -4.0) -2.0))))
  b))

C

double code(double a, double b, double c) {
	return (-c - (fma(0.25, (pow((c * a), 4.0) * (20.0 / (pow(b, 6.0) * a))), ((c * c) * (a / (b * b)))) - ((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)))) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) - Float64(fma(0.25, Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64((b ^ 6.0) * a))), Float64(Float64(c * c) * Float64(a / Float64(b * b)))) - Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[((-c) - N[(N[(0.25 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 90.9%

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

6. Applied rewrites 90.9%

   \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]
7. Add Preprocessing

Alternative 9: 90.9% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (-
   (- (* (* (* (* a a) c) (* c c)) (* (pow b -4.0) -2.0)) c)
   (fma
    0.25
    (* (pow (* c a) 4.0) (/ 20.0 (* (pow b 6.0) a)))
    (* (* c c) (/ a (* b b)))))
  b))

C

double code(double a, double b, double c) {
	return ((((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)) - c) - fma(0.25, (pow((c * a), 4.0) * (20.0 / (pow(b, 6.0) * a))), ((c * c) * (a / (b * b))))) / b;
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)) - c) - fma(0.25, Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64((b ^ 6.0) * a))), Float64(Float64(c * c) * Float64(a / Float64(b * b))))) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4.0], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] - N[(0.25 * N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(20.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.7%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 90.9%

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

6. Applied rewrites 90.9%

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

Alternative 10: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -110:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{1.5}, \frac{1}{a}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{a}\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{c}{b}, a \cdot \mathsf{fma}\left(-2, \frac{a \cdot {c}^{3}}{{b}^{5}}, -1 \cdot \frac{{c}^{2}}{{b}^{3}}\right)\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -110.0)
   (*
    (fma
     (pow (fma (* a c) -4.0 (* b b)) 1.5)
     (/ 1.0 a)
     (/ (* (* (- b) b) b) a))
    (/
     (/
      1.0
      (fma b (+ b (+ (sqrt (fma (* -4.0 c) a (* b b))) b)) (* (* -4.0 c) a)))
     2.0))
   (fma
    -1.0
    (/ c b)
    (*
     a
     (fma
      -2.0
      (/ (* a (pow c 3.0)) (pow b 5.0))
      (* -1.0 (/ (pow c 2.0) (pow b 3.0))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -110.0) {
		tmp = fma(pow(fma((a * c), -4.0, (b * b)), 1.5), (1.0 / a), (((-b * b) * b) / a)) * ((1.0 / fma(b, (b + (sqrt(fma((-4.0 * c), a, (b * b))) + b)), ((-4.0 * c) * a))) / 2.0);
	} else {
		tmp = fma(-1.0, (c / b), (a * fma(-2.0, ((a * pow(c, 3.0)) / pow(b, 5.0)), (-1.0 * (pow(c, 2.0) / pow(b, 3.0))))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -110.0)
		tmp = Float64(fma((fma(Float64(a * c), -4.0, Float64(b * b)) ^ 1.5), Float64(1.0 / a), Float64(Float64(Float64(Float64(-b) * b) * b) / a)) * Float64(Float64(1.0 / fma(b, Float64(b + Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)), Float64(Float64(-4.0 * c) * a))) / 2.0));
	else
		tmp = fma(-1.0, Float64(c / b), Float64(a * fma(-2.0, Float64(Float64(a * (c ^ 3.0)) / (b ^ 5.0)), Float64(-1.0 * Float64((c ^ 2.0) / (b ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -110.0], N[(N[(N[Power[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] * N[(1.0 / a), $MachinePrecision] + N[(N[(N[((-b) * b), $MachinePrecision] * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(b * N[(b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision] + N[(a * N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -110

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 56.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-add N/A

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

      11. mult-flip N/A

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

   9. Applied rewrites 57.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{1.5}}, \frac{1}{a}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{a}\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2} \]

   if -110 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 87.8%

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

Alternative 11: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -110:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{1.5}, \frac{1}{a}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{a}\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot c}{{b}^{5}}, -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -110.0)
   (*
    (fma
     (pow (fma (* a c) -4.0 (* b b)) 1.5)
     (/ 1.0 a)
     (/ (* (* (- b) b) b) a))
    (/
     (/
      1.0
      (fma b (+ b (+ (sqrt (fma (* -4.0 c) a (* b b))) b)) (* (* -4.0 c) a)))
     2.0))
   (*
    c
    (-
     (*
      c
      (fma -2.0 (/ (* (pow a 2.0) c) (pow b 5.0)) (* -1.0 (/ a (pow b 3.0)))))
     (/ 1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -110.0) {
		tmp = fma(pow(fma((a * c), -4.0, (b * b)), 1.5), (1.0 / a), (((-b * b) * b) / a)) * ((1.0 / fma(b, (b + (sqrt(fma((-4.0 * c), a, (b * b))) + b)), ((-4.0 * c) * a))) / 2.0);
	} else {
		tmp = c * ((c * fma(-2.0, ((pow(a, 2.0) * c) / pow(b, 5.0)), (-1.0 * (a / pow(b, 3.0))))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -110.0)
		tmp = Float64(fma((fma(Float64(a * c), -4.0, Float64(b * b)) ^ 1.5), Float64(1.0 / a), Float64(Float64(Float64(Float64(-b) * b) * b) / a)) * Float64(Float64(1.0 / fma(b, Float64(b + Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)), Float64(Float64(-4.0 * c) * a))) / 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * fma(-2.0, Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0)), Float64(-1.0 * Float64(a / (b ^ 3.0))))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -110.0], N[(N[(N[Power[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] * N[(1.0 / a), $MachinePrecision] + N[(N[(N[((-b) * b), $MachinePrecision] * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(b * N[(b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -110

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 56.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-add N/A

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

      11. mult-flip N/A

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

   7. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-*l* N/A

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

      19. lift-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 N/A

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

   9. Applied rewrites 57.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{1.5}}, \frac{1}{a}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{a}\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2} \]

   if -110 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 87.6%

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

Alternative 12: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -110:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot t\_0, \frac{1}{a}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{a}\right) \cdot \frac{0.5}{\mathsf{fma}\left(\left(t\_1 + b\right) + b, b, \left(-4 \cdot c\right) \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot c}{{b}^{5}}, -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -4.0 (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -110.0)
     (*
      (fma (* t_1 t_0) (/ 1.0 a) (/ (* (* (- b) b) b) a))
      (/ 0.5 (fma (+ (+ t_1 b) b) b (* (* -4.0 c) a))))
     (*
      c
      (-
       (*
        c
        (fma
         -2.0
         (/ (* (pow a 2.0) c) (pow b 5.0))
         (* -1.0 (/ a (pow b 3.0)))))
       (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -4.0, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -110.0) {
		tmp = fma((t_1 * t_0), (1.0 / a), (((-b * b) * b) / a)) * (0.5 / fma(((t_1 + b) + b), b, ((-4.0 * c) * a)));
	} else {
		tmp = c * ((c * fma(-2.0, ((pow(a, 2.0) * c) / pow(b, 5.0)), (-1.0 * (a / pow(b, 3.0))))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -4.0, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -110.0)
		tmp = Float64(fma(Float64(t_1 * t_0), Float64(1.0 / a), Float64(Float64(Float64(Float64(-b) * b) * b) / a)) * Float64(0.5 / fma(Float64(Float64(t_1 + b) + b), b, Float64(Float64(-4.0 * c) * a))));
	else
		tmp = Float64(c * Float64(Float64(c * fma(-2.0, Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0)), Float64(-1.0 * Float64(a / (b ^ 3.0))))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -110.0], N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] * N[(1.0 / a), $MachinePrecision] + N[(N[(N[((-b) * b), $MachinePrecision] * b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(N[(N[(t$95$1 + b), $MachinePrecision] + b), $MachinePrecision] * b + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -110

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 56.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-*.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. div-add N/A

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

      11. mult-flip N/A

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

   7. Applied rewrites 57.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

   9. Applied rewrites 57.1%

      \[\leadsto \mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right), \frac{1}{a}, \frac{\left(\left(-b\right) \cdot b\right) \cdot b}{a}\right) \cdot \color{blue}{\frac{0.5}{\mathsf{fma}\left(\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right) + b, b, \left(-4 \cdot c\right) \cdot a\right)}} \]

   if -110 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 87.6%

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

Alternative 13: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -110:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \sqrt{t\_0}, \left(\left(-b\right) \cdot b\right) \cdot b\right)}{a} \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot c}{{b}^{5}}, -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* a c) -4.0 (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -110.0)
     (*
      (/ (fma t_0 (sqrt t_0) (* (* (- b) b) b)) a)
      (/
       (/
        1.0
        (fma b (+ b (+ (sqrt (fma (* -4.0 c) a (* b b))) b)) (* (* -4.0 c) a)))
       2.0))
     (*
      c
      (-
       (*
        c
        (fma
         -2.0
         (/ (* (pow a 2.0) c) (pow b 5.0))
         (* -1.0 (/ a (pow b 3.0)))))
       (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((a * c), -4.0, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -110.0) {
		tmp = (fma(t_0, sqrt(t_0), ((-b * b) * b)) / a) * ((1.0 / fma(b, (b + (sqrt(fma((-4.0 * c), a, (b * b))) + b)), ((-4.0 * c) * a))) / 2.0);
	} else {
		tmp = c * ((c * fma(-2.0, ((pow(a, 2.0) * c) / pow(b, 5.0)), (-1.0 * (a / pow(b, 3.0))))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(a * c), -4.0, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -110.0)
		tmp = Float64(Float64(fma(t_0, sqrt(t_0), Float64(Float64(Float64(-b) * b) * b)) / a) * Float64(Float64(1.0 / fma(b, Float64(b + Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)), Float64(Float64(-4.0 * c) * a))) / 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * fma(-2.0, Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0)), Float64(-1.0 * Float64(a / (b ^ 3.0))))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -110.0], N[(N[(N[(t$95$0 * N[Sqrt[t$95$0], $MachinePrecision] + N[(N[((-b) * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(N[(1.0 / N[(b * N[(b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -110

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 57.6%

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

   7. Applied rewrites 57.5%

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

   if -110 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 87.6%

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

Alternative 14: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -110:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{1.5} - \left(b \cdot b\right) \cdot b}{a} \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot c}{{b}^{5}}, -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -110.0)
   (*
    (/ (- (pow (fma (* a c) -4.0 (* b b)) 1.5) (* (* b b) b)) a)
    (/
     (/
      1.0
      (fma b (+ b (+ (sqrt (fma (* -4.0 c) a (* b b))) b)) (* (* -4.0 c) a)))
     2.0))
   (*
    c
    (-
     (*
      c
      (fma -2.0 (/ (* (pow a 2.0) c) (pow b 5.0)) (* -1.0 (/ a (pow b 3.0)))))
     (/ 1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -110.0) {
		tmp = ((pow(fma((a * c), -4.0, (b * b)), 1.5) - ((b * b) * b)) / a) * ((1.0 / fma(b, (b + (sqrt(fma((-4.0 * c), a, (b * b))) + b)), ((-4.0 * c) * a))) / 2.0);
	} else {
		tmp = c * ((c * fma(-2.0, ((pow(a, 2.0) * c) / pow(b, 5.0)), (-1.0 * (a / pow(b, 3.0))))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -110.0)
		tmp = Float64(Float64(Float64((fma(Float64(a * c), -4.0, Float64(b * b)) ^ 1.5) - Float64(Float64(b * b) * b)) / a) * Float64(Float64(1.0 / fma(b, Float64(b + Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)), Float64(Float64(-4.0 * c) * a))) / 2.0));
	else
		tmp = Float64(c * Float64(Float64(c * fma(-2.0, Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0)), Float64(-1.0 * Float64(a / (b ^ 3.0))))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -110.0], N[(N[(N[(N[Power[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision] - N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(N[(1.0 / N[(b * N[(b + N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -110

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-fma.f64 N/A

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

      22. metadata-eval 56.9%

          \[\leadsto \frac{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{\color{blue}{1.5}} - \left(b \cdot b\right) \cdot b}{a} \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2} \]

   7. Applied rewrites 56.9%

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}^{1.5}} - \left(b \cdot b\right) \cdot b}{a} \cdot \frac{\frac{1}{\mathsf{fma}\left(b, b + \left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right), \left(-4 \cdot c\right) \cdot a\right)}}{2} \]

   if -110 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 87.6%

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

Alternative 15: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -110:\\ \;\;\;\;\left(t\_1 \cdot t\_0 - \left(b \cdot b\right) \cdot b\right) \cdot \frac{0.5}{\mathsf{fma}\left(b, b + \left(t\_1 + b\right), \left(-4 \cdot c\right) \cdot a\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot c}{{b}^{5}}, -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -4.0 c) a (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -110.0)
     (*
      (- (* t_1 t_0) (* (* b b) b))
      (/ 0.5 (* (fma b (+ b (+ t_1 b)) (* (* -4.0 c) a)) a)))
     (*
      c
      (-
       (*
        c
        (fma
         -2.0
         (/ (* (pow a 2.0) c) (pow b 5.0))
         (* -1.0 (/ a (pow b 3.0)))))
       (/ 1.0 b))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-4.0 * c), a, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -110.0) {
		tmp = ((t_1 * t_0) - ((b * b) * b)) * (0.5 / (fma(b, (b + (t_1 + b)), ((-4.0 * c) * a)) * a));
	} else {
		tmp = c * ((c * fma(-2.0, ((pow(a, 2.0) * c) / pow(b, 5.0)), (-1.0 * (a / pow(b, 3.0))))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-4.0 * c), a, Float64(b * b))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -110.0)
		tmp = Float64(Float64(Float64(t_1 * t_0) - Float64(Float64(b * b) * b)) * Float64(0.5 / Float64(fma(b, Float64(b + Float64(t_1 + b)), Float64(Float64(-4.0 * c) * a)) * a)));
	else
		tmp = Float64(c * Float64(Float64(c * fma(-2.0, Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0)), Float64(-1.0 * Float64(a / (b ^ 3.0))))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -110.0], N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] - N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(0.5 / N[(N[(b * N[(b + N[(t$95$1 + b), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -110

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 56.6%

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

   if -110 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 87.6%

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

Alternative 16: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0415:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} - c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0415)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (/ (- (* -1.0 (/ (* a (pow c 2.0)) (pow b 2.0))) c) b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.0415) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = ((-1.0 * ((a * pow(c, 2.0)) / pow(b, 2.0))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.0415)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-1.0 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 2.0))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0415], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0415000000000000022

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.2%

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

   if -0.0415000000000000022 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-frac-neg2 N/A

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

   3. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{b}{-2 \cdot a} - \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot \frac{-0.5}{a}} \]

   4. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 81.4%

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

   6. Applied rewrites 81.4%

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

Alternative 17: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0415:\\ \;\;\;\;\frac{\frac{t\_0 - b \cdot b}{\sqrt{t\_0} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right) - \frac{1}{b}\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* c -4.0) a (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0415)
     (/ (/ (- t_0 (* b b)) (+ (sqrt t_0) b)) (* 2.0 a))
     (* (- (- (* a (/ c (* (* b b) b)))) (/ 1.0 b)) c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((c * -4.0), a, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.0415) {
		tmp = ((t_0 - (b * b)) / (sqrt(t_0) + b)) / (2.0 * a);
	} else {
		tmp = (-(a * (c / ((b * b) * b))) - (1.0 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(c * -4.0), a, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.0415)
		tmp = Float64(Float64(Float64(t_0 - Float64(b * b)) / Float64(sqrt(t_0) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-Float64(a * Float64(c / Float64(Float64(b * b) * b)))) - Float64(1.0 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0415], N[(N[(N[(t$95$0 - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(a * N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - N[(1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0415000000000000022

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. flip-- N/A

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

      5. lower-unsound-/.f64 N/A

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

   3. Applied rewrites 57.2%

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

   if -0.0415000000000000022 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 81.3%

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

   7. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 81.3%

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.3%

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 81.3%

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 81.3%

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

   9. Applied rewrites 81.3%

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

Alternative 18: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0415:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right) - \frac{1}{b}\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0415)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (* (- (- (* a (/ c (* (* b b) b)))) (/ 1.0 b)) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.0415) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (-(a * (c / ((b * b) * b))) - (1.0 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.0415)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-Float64(a * Float64(c / Float64(Float64(b * b) * b)))) - Float64(1.0 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0415], 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[(a * N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - N[(1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0415000000000000022

   1. Initial program 55.7%

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

      1. lift--.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqr-neg-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 55.8%

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

   3. Applied rewrites 55.8%

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

   if -0.0415000000000000022 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 81.3%

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

   7. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 81.3%

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.3%

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 81.3%

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 81.3%

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

   9. Applied rewrites 81.3%

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

Alternative 19: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0415:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right) - \frac{1}{b}\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0415)
   (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (+ a a))
   (* (- (- (* a (/ c (* (* b b) b)))) (/ 1.0 b)) c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.0415) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (a + a);
	} else {
		tmp = (-(a * (c / ((b * b) * b))) - (1.0 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.0415)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(-Float64(a * Float64(c / Float64(Float64(b * b) * b)))) - Float64(1.0 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.0415], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(a * N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - N[(1.0 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.0415000000000000022

   1. Initial program 55.7%

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

   3. Applied rewrites 55.7%

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

   if -0.0415000000000000022 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 81.3%

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

   7. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 81.3%

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 81.3%

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 81.3%

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

      12. lift-pow.f64 N/A

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

      13. unpow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 81.3%

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

   9. Applied rewrites 81.3%

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

Alternative 20: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -6 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -6e-5)
   (/ (- (sqrt (fma (* a c) -4.0 (* b b))) b) (+ a a))
   (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -6e-5) {
		tmp = (sqrt(fma((a * c), -4.0, (b * b))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -6e-5)
		tmp = Float64(Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -6e-5], N[(N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -6.00000000000000015e-5

   1. Initial program 55.7%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. remove-double-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. metadata-eval 55.6%

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

   3. Applied rewrites 55.6%

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

   5. Applied rewrites 56.6%

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

   7. Applied rewrites 55.7%

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

   if -6.00000000000000015e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 64.2%

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

   4. Applied rewrites 64.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 64.2%

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

   6. Applied rewrites 64.2%

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

Alternative 21: 64.2% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 64.2%

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

4. Applied rewrites 64.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 64.2%

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

6. Applied rewrites 64.2%

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

Reproduce

herbie shell --seed 2025187 
(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)))