Cubic critical, narrow range

Percentage Accurate: 55.0% → 92.5%

Time: 5.1s

Alternatives: 16

Speedup: 2.9×

Specification

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

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Initial Program: 55.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * 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) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Alternative 1: 92.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\\ t_1 := \mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\\ t_2 := {\left(a \cdot c\right)}^{2}\\ t_3 := \sqrt{t\_1}\\ t_4 := \mathsf{fma}\left(b, b, t\_3 \cdot t\_3 - \left(-b\right) \cdot t\_3\right)\\ t_5 := \mathsf{fma}\left(9, t\_2, 18 \cdot t\_2\right) - 0.25 \cdot {t\_0}^{2}\\ t_6 := -27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(t\_0 \cdot t\_5\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_1}^{1.5}\right)}{t\_4}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {t\_5}^{2}, 0.5 \cdot \left(t\_0 \cdot t\_6\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, t\_0, \mathsf{fma}\left(0.5, \frac{t\_6}{{b}^{4}}, 0.5 \cdot \frac{t\_5}{b \cdot b}\right)\right)\right)}{t\_4}}{3}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma -6.0 (* a c) (* -3.0 (* a c))))
        (t_1 (fma b b (* (* -3.0 a) c)))
        (t_2 (pow (* a c) 2.0))
        (t_3 (sqrt t_1))
        (t_4 (fma b b (- (* t_3 t_3) (* (- b) t_3))))
        (t_5 (- (fma 9.0 t_2 (* 18.0 t_2)) (* 0.25 (pow t_0 2.0))))
        (t_6 (- (* -27.0 (pow (* a c) 3.0)) (* 0.5 (* t_0 t_5)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.0)
     (/ (/ (/ (fma (* b b) (- b) (pow t_1 1.5)) t_4) 3.0) a)
     (/
      (/
       (/
        (*
         b
         (fma
          -0.5
          (/ (fma 0.25 (pow t_5 2.0) (* 0.5 (* t_0 t_6))) (pow b 6.0))
          (fma 0.5 t_0 (fma 0.5 (/ t_6 (pow b 4.0)) (* 0.5 (/ t_5 (* b b)))))))
        t_4)
       3.0)
      a))))

C

double code(double a, double b, double c) {
	double t_0 = fma(-6.0, (a * c), (-3.0 * (a * c)));
	double t_1 = fma(b, b, ((-3.0 * a) * c));
	double t_2 = pow((a * c), 2.0);
	double t_3 = sqrt(t_1);
	double t_4 = fma(b, b, ((t_3 * t_3) - (-b * t_3)));
	double t_5 = fma(9.0, t_2, (18.0 * t_2)) - (0.25 * pow(t_0, 2.0));
	double t_6 = (-27.0 * pow((a * c), 3.0)) - (0.5 * (t_0 * t_5));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -2.0) {
		tmp = ((fma((b * b), -b, pow(t_1, 1.5)) / t_4) / 3.0) / a;
	} else {
		tmp = (((b * fma(-0.5, (fma(0.25, pow(t_5, 2.0), (0.5 * (t_0 * t_6))) / pow(b, 6.0)), fma(0.5, t_0, fma(0.5, (t_6 / pow(b, 4.0)), (0.5 * (t_5 / (b * b))))))) / t_4) / 3.0) / a;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(-6.0, Float64(a * c), Float64(-3.0 * Float64(a * c)))
	t_1 = fma(b, b, Float64(Float64(-3.0 * a) * c))
	t_2 = Float64(a * c) ^ 2.0
	t_3 = sqrt(t_1)
	t_4 = fma(b, b, Float64(Float64(t_3 * t_3) - Float64(Float64(-b) * t_3)))
	t_5 = Float64(fma(9.0, t_2, Float64(18.0 * t_2)) - Float64(0.25 * (t_0 ^ 2.0)))
	t_6 = Float64(Float64(-27.0 * (Float64(a * c) ^ 3.0)) - Float64(0.5 * Float64(t_0 * t_5)))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -2.0)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), Float64(-b), (t_1 ^ 1.5)) / t_4) / 3.0) / a);
	else
		tmp = Float64(Float64(Float64(Float64(b * fma(-0.5, Float64(fma(0.25, (t_5 ^ 2.0), Float64(0.5 * Float64(t_0 * t_6))) / (b ^ 6.0)), fma(0.5, t_0, fma(0.5, Float64(t_6 / (b ^ 4.0)), Float64(0.5 * Float64(t_5 / Float64(b * b))))))) / t_4) / 3.0) / a);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-6.0 * N[(a * c), $MachinePrecision] + N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[(b * b + N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[((-b) * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(9.0 * t$95$2 + N[(18.0 * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(-27.0 * N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.0], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(b * N[(-0.5 * N[(N[(0.25 * N[Power[t$95$5, 2.0], $MachinePrecision] + N[(0.5 * N[(t$95$0 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * t$95$0 + N[(0.5 * N[(t$95$6 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] / 3.0), $MachinePrecision] / a), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip3-+ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 83.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   9. Applied rewrites 84.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}\right)}}{3}}{a} \]

   if -2 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 51.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 51.3

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

   5. Applied rewrites 51.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip3-+ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 51.1%

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

   8. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-27 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 18 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-6 \cdot \left(a \cdot c\right) + -3 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}\right)}}{3}}{a} \]

   10. Applied rewrites 93.5%

       \[\leadsto \frac{\frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(0.25, {\left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)}^{2}, 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{6}}, \mathsf{fma}\left(0.5, \mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right), \mathsf{fma}\left(0.5, \frac{-27 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right) \cdot \left(\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}\right)\right)}{{b}^{4}}, 0.5 \cdot \frac{\mathsf{fma}\left(9, {\left(a \cdot c\right)}^{2}, 18 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(\mathsf{fma}\left(-6, a \cdot c, -3 \cdot \left(a \cdot c\right)\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}\right)}}{3}}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 - \left(-b\right) \cdot t\_1\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.375, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \mathsf{fma}\left(0.5, c, \mathsf{fma}\left(1.0546875, {a}^{3} \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{0.5625 \cdot \left(a \cdot \left({c}^{3} \cdot a\right)\right)}{{b}^{4}}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* (* -3.0 a) c))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.0)
     (/
      (/
       (/
        (fma (* b b) (- b) (pow t_0 1.5))
        (fma b b (- (* t_1 t_1) (* (- b) t_1))))
       3.0)
      a)
     (-
      (/
       (fma
        0.375
        (/ (* (* c c) a) (* b b))
        (fma
         0.5
         c
         (fma
          1.0546875
          (* (pow a 3.0) (/ (pow c 4.0) (pow b 6.0)))
          (/ (* 0.5625 (* a (* (pow c 3.0) a))) (pow b 4.0)))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, ((-3.0 * a) * c));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -2.0) {
		tmp = ((fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) - (-b * t_1)))) / 3.0) / a;
	} else {
		tmp = -(fma(0.375, (((c * c) * a) / (b * b)), fma(0.5, c, fma(1.0546875, (pow(a, 3.0) * (pow(c, 4.0) / pow(b, 6.0))), ((0.5625 * (a * (pow(c, 3.0) * a))) / pow(b, 4.0))))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(-3.0 * a) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -2.0)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) - Float64(Float64(-b) * t_1)))) / 3.0) / a);
	else
		tmp = Float64(-Float64(fma(0.375, Float64(Float64(Float64(c * c) * a) / Float64(b * b)), fma(0.5, c, fma(1.0546875, Float64((a ^ 3.0) * Float64((c ^ 4.0) / (b ^ 6.0))), Float64(Float64(0.5625 * Float64(a * Float64((c ^ 3.0) * a))) / (b ^ 4.0))))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $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[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.0], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], (-N[(N[(0.375 * N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(0.5 * c + N[(1.0546875 * N[(N[Power[a, 3.0], $MachinePrecision] * N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5625 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip3-+ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 83.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   9. Applied rewrites 84.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}\right)}}{3}}{a} \]

   if -2 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   5. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\frac{1}{2} \cdot c + \left(\frac{9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)}{b}} \]

   7. Applied rewrites 93.4%

      \[\leadsto -\frac{\mathsf{fma}\left(0.375, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \mathsf{fma}\left(0.5, c, \mathsf{fma}\left(1.0546875, {a}^{3} \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{0.5625 \cdot \left(a \cdot \left({c}^{3} \cdot a\right)\right)}{{b}^{4}}\right)\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 - \left(-b\right) \cdot t\_1\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* (* -3.0 a) c))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.0)
     (/
      (/
       (/
        (fma (* b b) (- b) (pow t_0 1.5))
        (fma b b (- (* t_1 t_1) (* (- b) t_1))))
       3.0)
      a)
     (/
      (fma
       (/ (* (* a a) (* (* c c) c)) (pow b 4.0))
       -0.5625
       (fma
        -0.5
        c
        (fma
         (/ (/ (* (pow (* c a) 4.0) 6.328125) a) (pow b 6.0))
         -0.16666666666666666
         (/ (* -0.375 (* (* c c) a)) (* b b)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, ((-3.0 * a) * c));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -2.0) {
		tmp = ((fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) - (-b * t_1)))) / 3.0) / a;
	} else {
		tmp = fma((((a * a) * ((c * c) * c)) / pow(b, 4.0)), -0.5625, fma(-0.5, c, fma((((pow((c * a), 4.0) * 6.328125) / a) / pow(b, 6.0)), -0.16666666666666666, ((-0.375 * ((c * c) * a)) / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(-3.0 * a) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -2.0)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) - Float64(Float64(-b) * t_1)))) / 3.0) / a);
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / (b ^ 4.0)), -0.5625, fma(-0.5, c, fma(Float64(Float64(Float64((Float64(c * a) ^ 4.0) * 6.328125) / a) / (b ^ 6.0)), -0.16666666666666666, Float64(Float64(-0.375 * Float64(Float64(c * c) * a)) / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $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[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.0], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(N[(N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / a), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + N[(N[(-0.375 * N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip3-+ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 83.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   9. Applied rewrites 84.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}\right)}}{3}}{a} \]

   if -2 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      6. lift-*.f64 93.4

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

   6. Applied rewrites 93.4%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 - \left(-b\right) \cdot t\_1\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(0.375, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(0.5 + \left(c \cdot c\right) \cdot \mathsf{fma}\left(0.5625, \frac{a \cdot a}{{b}^{4}}, 1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* (* -3.0 a) c))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.0)
     (/
      (/
       (/
        (fma (* b b) (- b) (pow t_0 1.5))
        (fma b b (- (* t_1 t_1) (* (- b) t_1))))
       3.0)
      a)
     (-
      (/
       (fma
        0.375
        (/ (* (* c c) a) (* b b))
        (*
         c
         (+
          0.5
          (*
           (* c c)
           (fma
            0.5625
            (/ (* a a) (pow b 4.0))
            (* 1.0546875 (/ (* (pow a 3.0) c) (pow b 6.0))))))))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, ((-3.0 * a) * c));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -2.0) {
		tmp = ((fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) - (-b * t_1)))) / 3.0) / a;
	} else {
		tmp = -(fma(0.375, (((c * c) * a) / (b * b)), (c * (0.5 + ((c * c) * fma(0.5625, ((a * a) / pow(b, 4.0)), (1.0546875 * ((pow(a, 3.0) * c) / pow(b, 6.0)))))))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(-3.0 * a) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -2.0)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) - Float64(Float64(-b) * t_1)))) / 3.0) / a);
	else
		tmp = Float64(-Float64(fma(0.375, Float64(Float64(Float64(c * c) * a) / Float64(b * b)), Float64(c * Float64(0.5 + Float64(Float64(c * c) * fma(0.5625, Float64(Float64(a * a) / (b ^ 4.0)), Float64(1.0546875 * Float64(Float64((a ^ 3.0) * c) / (b ^ 6.0)))))))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $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[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.0], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], (-N[(N[(0.375 * N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(c * N[(0.5 + N[(N[(c * c), $MachinePrecision] * N[(0.5625 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.0546875 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip3-+ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 83.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   9. Applied rewrites 84.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}\right)}}{3}}{a} \]

   if -2 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   5. Taylor expanded in b around -inf

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \left(\frac{1}{2} \cdot c + \left(\frac{9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}}\right)\right)}{b}} \]

   7. Applied rewrites 93.4%

      \[\leadsto -\frac{\mathsf{fma}\left(0.375, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \mathsf{fma}\left(0.5, c, \mathsf{fma}\left(1.0546875, {a}^{3} \cdot \frac{{c}^{4}}{{b}^{6}}, \frac{0.5625 \cdot \left(a \cdot \left({c}^{3} \cdot a\right)\right)}{{b}^{4}}\right)\right)\right)}{b} \]

   8. Taylor expanded in c around 0

      \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + {c}^{2} \cdot \left(\frac{9}{16} \cdot \frac{{a}^{2}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + {c}^{2} \cdot \left(\frac{9}{16} \cdot \frac{{a}^{2}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      2. lower-+.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + {c}^{2} \cdot \left(\frac{9}{16} \cdot \frac{{a}^{2}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      3. lower-*.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + {c}^{2} \cdot \left(\frac{9}{16} \cdot \frac{{a}^{2}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      4. pow2 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \left(\frac{9}{16} \cdot \frac{{a}^{2}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      5. lift-*.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \left(\frac{9}{16} \cdot \frac{{a}^{2}}{{b}^{4}} + \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      6. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{9}{16}, \frac{{a}^{2}}{{b}^{4}}, \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      7. lower-/.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{9}{16}, \frac{{a}^{2}}{{b}^{4}}, \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      8. pow2 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{9}{16}, \frac{a \cdot a}{{b}^{4}}, \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      9. lift-*.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{9}{16}, \frac{a \cdot a}{{b}^{4}}, \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      10. lift-pow.f64 N/A

          \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{9}{16}, \frac{a \cdot a}{{b}^{4}}, \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      11. lower-*.f64 N/A

          \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{9}{16}, \frac{a \cdot a}{{b}^{4}}, \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

      12. lower-/.f64 N/A

          \[\leadsto -\frac{\mathsf{fma}\left(\frac{3}{8}, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(\frac{1}{2} + \left(c \cdot c\right) \cdot \mathsf{fma}\left(\frac{9}{16}, \frac{a \cdot a}{{b}^{4}}, \frac{135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]

   10. Applied rewrites 93.3%

       \[\leadsto -\frac{\mathsf{fma}\left(0.375, \frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, c \cdot \left(0.5 + \left(c \cdot c\right) \cdot \mathsf{fma}\left(0.5625, \frac{a \cdot a}{{b}^{4}}, 1.0546875 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}}\right)\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -2:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 - \left(-b\right) \cdot t\_1\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}\right), c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* (* -3.0 a) c))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -2.0)
     (/
      (/
       (/
        (fma (* b b) (- b) (pow t_0 1.5))
        (fma b b (- (* t_1 t_1) (* (- b) t_1))))
       3.0)
      a)
     (/
      (*
       (-
        (*
         (fma
          (fma
           (/ (* a a) (pow b 4.0))
           -0.5625
           (/ (* -1.0546875 (* (pow a 3.0) c)) (pow b 6.0)))
          c
          (* (/ a (* b b)) -0.375))
         c)
        0.5)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, ((-3.0 * a) * c));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -2.0) {
		tmp = ((fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) - (-b * t_1)))) / 3.0) / a;
	} else {
		tmp = (((fma(fma(((a * a) / pow(b, 4.0)), -0.5625, ((-1.0546875 * (pow(a, 3.0) * c)) / pow(b, 6.0))), c, ((a / (b * b)) * -0.375)) * c) - 0.5) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(-3.0 * a) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -2.0)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) - Float64(Float64(-b) * t_1)))) / 3.0) / a);
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(Float64(Float64(a * a) / (b ^ 4.0)), -0.5625, Float64(Float64(-1.0546875 * Float64((a ^ 3.0) * c)) / (b ^ 6.0))), c, Float64(Float64(a / Float64(b * b)) * -0.375)) * c) - 0.5) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $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[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -2.0], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(N[(-1.0546875 * N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 0.5), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -2

   1. Initial program 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 83.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip3-+ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 83.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   9. Applied rewrites 84.8%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}\right)}}{3}}{a} \]

   if -2 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 51.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   5. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - \frac{1}{2}\right)}{b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - \frac{1}{2}\right) \cdot c}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{2}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{4}}\right)\right) - \frac{1}{2}\right) \cdot c}{b} \]

   7. Applied rewrites 93.3%

      \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{4}}, -0.5625, \frac{-1.0546875 \cdot \left({a}^{3} \cdot c\right)}{{b}^{6}}\right), c, \frac{a}{b \cdot b} \cdot -0.375\right) \cdot c - 0.5\right) \cdot c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.85:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, {t\_0}^{1.5}\right)}{\mathsf{fma}\left(b, b, t\_1 \cdot t\_1 - \left(-b\right) \cdot t\_1\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma b b (* (* -3.0 a) c))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.85)
     (/
      (/
       (/
        (fma (* b b) (- b) (pow t_0 1.5))
        (fma b b (- (* t_1 t_1) (* (- b) t_1))))
       3.0)
      a)
     (/
      (fma
       (/ (* (* a a) (* (* c c) c)) (pow b 4.0))
       -0.5625
       (fma -0.5 c (* -0.375 (/ (* a (* c c)) (* b b)))))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma(b, b, ((-3.0 * a) * c));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.85) {
		tmp = ((fma((b * b), -b, pow(t_0, 1.5)) / fma(b, b, ((t_1 * t_1) - (-b * t_1)))) / 3.0) / a;
	} else {
		tmp = fma((((a * a) * ((c * c) * c)) / pow(b, 4.0)), -0.5625, fma(-0.5, c, (-0.375 * ((a * (c * c)) / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(b, b, Float64(Float64(-3.0 * a) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.85)
		tmp = Float64(Float64(Float64(fma(Float64(b * b), Float64(-b), (t_0 ^ 1.5)) / fma(b, b, Float64(Float64(t_1 * t_1) - Float64(Float64(-b) * t_1)))) / 3.0) / a);
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / (b ^ 4.0)), -0.5625, fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $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[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.85], N[(N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.849999999999999978

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. flip3-+ N/A

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

      7. lower-/.f64 N/A

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

   7. Applied rewrites 82.5%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow3 N/A

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

      5. sqr-neg-rev N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. sqrt-pow2 N/A

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

      17. metadata-eval N/A

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

      18. lower-pow.f64 N/A

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

   9. Applied rewrites 83.6%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(b \cdot b, -b, {\left(\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)\right)}^{1.5}\right)}}{\mathsf{fma}\left(b, b, \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}\right)}}{3}}{a} \]

   if -0.849999999999999978 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      6. lift-*.f64 93.7

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

   6. Applied rewrites 93.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{2}}\right)\right)}{b} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{2}}\right)\right)}{b} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b} \]

      7. lift-*.f64 91.3

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b} \]

   9. Applied rewrites 91.3%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 90.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.85:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.85)
   (/ (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) 3.0) a)
   (/
    (fma
     (/ (* (* a a) (* (* c c) c)) (pow b 4.0))
     -0.5625
     (fma -0.5 c (* -0.375 (/ (* a (* c c)) (* b b)))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.85) {
		tmp = ((-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / 3.0) / a;
	} else {
		tmp = fma((((a * a) * ((c * c) * c)) / pow(b, 4.0)), -0.5625, fma(-0.5, c, (-0.375 * ((a * (c * c)) / (b * b))))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.85)
		tmp = Float64(Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / 3.0) / a);
	else
		tmp = Float64(fma(Float64(Float64(Float64(a * a) * Float64(Float64(c * c) * c)) / (b ^ 4.0)), -0.5625, fma(-0.5, c, Float64(-0.375 * Float64(Float64(a * Float64(c * c)) / Float64(b * b))))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.85], N[(N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] * -0.5625 + N[(-0.5 * c + N[(-0.375 * N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.849999999999999978

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-sqrt.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if -0.849999999999999978 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot \frac{405}{64}}{a}}{{b}^{6}}, \frac{-1}{6}, \frac{\frac{-3}{8} \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

      6. lift-*.f64 93.7

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

   6. Applied rewrites 93.7%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b} \]

   7. Taylor expanded in a around 0

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b} \]

      4. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{2}}\right)\right)}{b} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{2}}\right)\right)}{b} \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \frac{-9}{16}, \mathsf{fma}\left(\frac{-1}{2}, c, \frac{-3}{8} \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b} \]

      7. lift-*.f64 91.3

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b} \]

   9. Applied rewrites 91.3%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, -0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{b \cdot b}\right)\right)}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.85:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{5}} \cdot c - \frac{0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.85)
   (/ (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) 3.0) a)
   (*
    (-
     (* (/ (fma -0.5625 (* (* a a) c) (* -0.375 (* a (* b b)))) (pow b 5.0)) c)
     (/ 0.5 b))
    c)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.85) {
		tmp = ((-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / 3.0) / a;
	} else {
		tmp = (((fma(-0.5625, ((a * a) * c), (-0.375 * (a * (b * b)))) / pow(b, 5.0)) * c) - (0.5 / 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(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.85)
		tmp = Float64(Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / 3.0) / a);
	else
		tmp = Float64(Float64(Float64(Float64(fma(-0.5625, Float64(Float64(a * a) * c), Float64(-0.375 * Float64(a * Float64(b * b)))) / (b ^ 5.0)) * c) - Float64(0.5 / 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[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.85], N[(N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] + N[(-0.375 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(0.5 / 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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.849999999999999978

   1. Initial program 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-sqrt.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if -0.849999999999999978 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 50.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \color{blue}{c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \color{blue}{c} \]

   4. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{a}{{b}^{3}}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{5}}\right) \cdot c - \frac{0.5}{b}\right) \cdot c} \]

   5. Taylor expanded in b around 0

      \[\leadsto \left(\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-9}{16}, {a}^{2} \cdot c, \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      3. pow2 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-9}{16}, \left(a \cdot a\right) \cdot c, \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-9}{16}, \left(a \cdot a\right) \cdot c, \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-9}{16}, \left(a \cdot a\right) \cdot c, \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-9}{16}, \left(a \cdot a\right) \cdot c, \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-9}{16}, \left(a \cdot a\right) \cdot c, \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      8. pow2 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-9}{16}, \left(a \cdot a\right) \cdot c, \frac{-3}{8} \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{\mathsf{fma}\left(\frac{-9}{16}, \left(a \cdot a\right) \cdot c, \frac{-3}{8} \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{5}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

      10. lift-pow.f64 91.1

          \[\leadsto \left(\frac{\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{5}} \cdot c - \frac{0.5}{b}\right) \cdot c \]

   7. Applied rewrites 91.1%

      \[\leadsto \left(\frac{\mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)}{{b}^{5}} \cdot c - \frac{0.5}{b}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00018:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)}}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -0.375, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00018)
   (/ (/ (+ (- b) (sqrt (fma b b (* (* -3.0 a) c)))) 3.0) a)
   (fma (/ (* (* c c) a) (* (* b b) b)) -0.375 (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00018) {
		tmp = ((-b + sqrt(fma(b, b, ((-3.0 * a) * c)))) / 3.0) / a;
	} else {
		tmp = fma((((c * c) * a) / ((b * b) * b)), -0.375, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00018)
		tmp = Float64(Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c)))) / 3.0) / a);
	else
		tmp = fma(Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * b)), -0.375, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00018], N[(N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(c / b), $MachinePrecision] * -0.5), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.80000000000000011e-4

   1. Initial program 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 75.0

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

   5. Applied rewrites 75.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-sqrt.f64 75.0

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

   7. Applied rewrites 75.0%

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

   if -1.80000000000000011e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \frac{a \cdot {c}^{2}}{{b}^{3}} \cdot \frac{-3}{8} + \color{blue}{\frac{-1}{2}} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      12. lower-/.f64 91.1

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -0.375, \frac{c}{b} \cdot -0.5\right) \]

   4. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -0.375, \frac{c}{b} \cdot -0.5\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      2. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2} \cdot b}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2} \cdot b}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      6. lift-*.f64 91.1

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -0.375, \frac{c}{b} \cdot -0.5\right) \]

   6. Applied rewrites 91.1%

      \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -0.375, \frac{c}{b} \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00018:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -0.375, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00018)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (fma (/ (* (* c c) a) (* (* b b) b)) -0.375 (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00018) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / ((b * b) * b)), -0.375, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00018)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * a) / Float64(Float64(b * b) * b)), -0.375, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00018], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(c / b), $MachinePrecision] * -0.5), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.80000000000000011e-4

   1. Initial program 74.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.0

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

   3. Applied rewrites 75.0%

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

   if -1.80000000000000011e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]

      2. *-commutative N/A

         \[\leadsto \frac{a \cdot {c}^{2}}{{b}^{3}} \cdot \frac{-3}{8} + \color{blue}{\frac{-1}{2}} \cdot \frac{c}{b} \]

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      12. lower-/.f64 91.1

          \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -0.375, \frac{c}{b} \cdot -0.5\right) \]

   4. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -0.375, \frac{c}{b} \cdot -0.5\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      2. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2} \cdot b}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2} \cdot b}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, \frac{-3}{8}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]

      6. lift-*.f64 91.1

         \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -0.375, \frac{c}{b} \cdot -0.5\right) \]

   6. Applied rewrites 91.1%

      \[\leadsto \mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -0.375, \frac{c}{b} \cdot -0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 84.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00018:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00018)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (fma (/ (* (* c c) a) (* b b)) -0.375 (* -0.5 c)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00018) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = fma((((c * c) * a) / (b * b)), -0.375, (-0.5 * 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(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00018)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(Float64(Float64(Float64(c * c) * a) / Float64(b * b)), -0.375, Float64(-0.5 * 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[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00018], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.80000000000000011e-4

   1. Initial program 74.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.0

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

   3. Applied rewrites 75.0%

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

   if -1.80000000000000011e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{\color{blue}{b}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{2} \cdot c}{b} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{a \cdot {c}^{2}}{{b}^{2}} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot c}{b} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{{c}^{2} \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{2}}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      10. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, \frac{-3}{8}, \frac{-1}{2} \cdot c\right)}{b} \]

      12. lower-*.f64 91.1

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b} \]

   4. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.00018:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.00018)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* (- (/ (* -0.375 (* c a)) (* b b)) 0.5) c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.00018) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = ((((-0.375 * (c * a)) / (b * b)) - 0.5) * 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(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.00018)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b)) - 0.5) * 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[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.00018], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.5), $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 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -1.80000000000000011e-4

   1. Initial program 74.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 75.0

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

   3. Applied rewrites 75.0%

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

   if -1.80000000000000011e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 42.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

   4. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

   5. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      3. lower--.f64 N/A

         \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(c \cdot a\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(c \cdot a\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(c \cdot a\right)}{b \cdot b} - \frac{1}{2}\right) \cdot c}{b} \]

      10. lift-*.f64 91.0

          \[\leadsto \frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b} \]

   7. Applied rewrites 91.0%

      \[\leadsto \frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (- (/ (* -0.375 (* c a)) (* b b)) 0.5) c) b))

C

double code(double a, double b, double c) {
	return ((((-0.375 * (c * a)) / (b * b)) - 0.5) * 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 = (((((-0.375d0) * (c * a)) / (b * b)) - 0.5d0) * c) / b
end function

Java

public static double code(double a, double b, double c) {
	return ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b;
}

Python

def code(a, b, c):
	return ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(-0.375 * Float64(c * a)) / Float64(b * b)) - 0.5) * c) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((-0.375 * (c * a)) / (b * b)) - 0.5) * c) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \left(\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}} + \frac{-1}{6} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{a \cdot {b}^{6}}\right)\right)}{b}} \]

4. Applied rewrites 91.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}}, -0.5625, \mathsf{fma}\left(-0.5, c, \mathsf{fma}\left(\frac{\frac{{\left(c \cdot a\right)}^{4} \cdot 6.328125}{a}}{{b}^{6}}, -0.16666666666666666, \frac{-0.375 \cdot \left(\left(c \cdot c\right) \cdot a\right)}{b \cdot b}\right)\right)\right)}{b}} \]

5. Taylor expanded in c around 0

   \[\leadsto \frac{c \cdot \left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right)}{b} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   3. lower--.f64 N/A

      \[\leadsto \frac{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(c \cdot a\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(c \cdot a\right)}{{b}^{2}} - \frac{1}{2}\right) \cdot c}{b} \]

   9. pow2 N/A

      \[\leadsto \frac{\left(\frac{\frac{-3}{8} \cdot \left(c \cdot a\right)}{b \cdot b} - \frac{1}{2}\right) \cdot c}{b} \]

   10. lift-*.f64 82.0

       \[\leadsto \frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b} \]

7. Applied rewrites 82.0%

   \[\leadsto \frac{\left(\frac{-0.375 \cdot \left(c \cdot a\right)}{b \cdot b} - 0.5\right) \cdot c}{b} \]
8. Add Preprocessing

Alternative 14: 81.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5}{b} \cdot c \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ (- (* -0.375 (/ (* a c) (* b b))) 0.5) b) c))

C

double code(double a, double b, double c) {
	return (((-0.375 * ((a * c) / (b * b))) - 0.5) / b) * c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((-0.375d0) * ((a * c) / (b * b))) - 0.5d0) / b) * c
end function

Java

public static double code(double a, double b, double c) {
	return (((-0.375 * ((a * c) / (b * b))) - 0.5) / b) * c;
}

Python

def code(a, b, c):
	return (((-0.375 * ((a * c) / (b * b))) - 0.5) / b) * c

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(-0.375 * Float64(Float64(a * c) / Float64(b * b))) - 0.5) / b) * c)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((-0.375 * ((a * c) / (b * b))) - 0.5) / b) * c;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / b), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 55.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \color{blue}{c} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \color{blue}{c} \]

4. Applied rewrites 88.1%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{a}{{b}^{3}}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{5}}\right) \cdot c - \frac{0.5}{b}\right) \cdot c} \]

5. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b} \cdot c \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b} \cdot c \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b} \cdot c \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b} \cdot c \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b} \cdot c \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b} \cdot c \]

   6. pow2 N/A

      \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{b \cdot b} - \frac{1}{2}}{b} \cdot c \]

   7. lift-*.f64 81.9

      \[\leadsto \frac{-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5}{b} \cdot c \]

7. Applied rewrites 81.9%

   \[\leadsto \frac{-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5}{b} \cdot c \]
8. Add Preprocessing

Alternative 15: 64.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \cdot -0.5 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ c b) -0.5))

C

double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

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) * (-0.5d0)
end function

Java

public static double code(double a, double b, double c) {
	return (c / b) * -0.5;
}

Python

def code(a, b, c):
	return (c / b) * -0.5

Julia

function code(a, b, c)
	return Float64(Float64(c / b) * -0.5)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c / b) * -0.5;
end

Wolfram

code[a_, b_, c_] := N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 55.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 64.8

      \[\leadsto \frac{c}{b} \cdot -0.5 \]

4. Applied rewrites 64.8%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
5. Add Preprocessing

Alternative 16: 64.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{b} \cdot c \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* (/ -0.5 b) c))

C

double code(double a, double b, double c) {
	return (-0.5 / b) * c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) / b) * c
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 / b) * c;
}

Python

def code(a, b, c):
	return (-0.5 / b) * c

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 / b) * c)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 / b) * c;
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / b), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 55.0%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

2. Taylor expanded in c around 0

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \color{blue}{c} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(c \cdot \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + \frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{2} \cdot \frac{1}{b}\right) \cdot \color{blue}{c} \]

4. Applied rewrites 88.1%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{a}{{b}^{3}}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{b}^{5}}\right) \cdot c - \frac{0.5}{b}\right) \cdot c} \]

5. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{-1}{2}}{b} \cdot c \]
6. Step-by-step derivation

   1. lower-/.f64 64.8

      \[\leadsto \frac{-0.5}{b} \cdot c \]

7. Applied rewrites 64.8%

   \[\leadsto \frac{-0.5}{b} \cdot c \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025105 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))