Cubic critical, narrow range

Percentage Accurate: 55.7% → 92.3%

Time: 7.9s

Alternatives: 24

Speedup: 3.3×

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: 55.7% 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: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)\\ t_2 := \frac{t\_1}{{b}^{4}}\\ t_3 := {\left(a \cdot c\right)}^{4}\\ t_4 := \frac{a \cdot c}{b \cdot b}\\ t_5 := \sqrt{t\_0}\\ t_6 := \left(-b\right) \cdot t\_5\\ t_7 := \left(a \cdot a\right) \cdot \left(c \cdot c\right)\\ t_8 := \frac{t\_7}{b \cdot b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, t\_5 \cdot t\_0\right)}{\mathsf{fma}\left(b, b, \frac{{t\_0}^{3} - {t\_6}^{3}}{\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(t\_6, t\_6, t\_0 \cdot t\_6\right)\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, t\_2, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, t\_8, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, t\_3, 5.0625 \cdot t\_3\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, t\_2, \mathsf{fma}\left(4.5, t\_8, 5.0625 \cdot \frac{t\_3}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \left(b \cdot b\right) \cdot \left(\left(2 + \mathsf{fma}\left(-3, t\_4, -1.6875 \cdot \frac{t\_1}{{b}^{6}}\right)\right) - \mathsf{fma}\left(1.125, \frac{t\_7}{{b}^{4}}, 1.5 \cdot t\_4\right)\right)\right)}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b)))
        (t_1 (* (* (* a a) a) (* (* c c) c)))
        (t_2 (/ t_1 (pow b 4.0)))
        (t_3 (pow (* a c) 4.0))
        (t_4 (/ (* a c) (* b b)))
        (t_5 (sqrt t_0))
        (t_6 (* (- b) t_5))
        (t_7 (* (* a a) (* c c)))
        (t_8 (/ t_7 (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/
      (/
       (fma (* b b) (- b) (* t_5 t_0))
       (fma
        b
        b
        (/
         (- (pow t_0 3.0) (pow t_6 3.0))
         (fma t_0 t_0 (fma t_6 t_6 (* t_0 t_6))))))
      (* 3.0 a))
     (/
      (/
       (*
        b
        (fma
         -3.0
         (* a c)
         (fma
          -1.6875
          t_2
          (fma
           -1.5
           (* a c)
           (fma
            -1.125
            t_8
            (fma
             -0.5
             (/ (fma 1.265625 t_3 (* 5.0625 t_3)) (pow b 6.0))
             (fma 3.375 t_2 (fma 4.5 t_8 (* 5.0625 (/ t_3 (pow b 6.0)))))))))))
       (fma
        b
        b
        (*
         (* b b)
         (-
          (+ 2.0 (fma -3.0 t_4 (* -1.6875 (/ t_1 (pow b 6.0)))))
          (fma 1.125 (/ t_7 (pow b 4.0)) (* 1.5 t_4))))))
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = ((a * a) * a) * ((c * c) * c);
	double t_2 = t_1 / pow(b, 4.0);
	double t_3 = pow((a * c), 4.0);
	double t_4 = (a * c) / (b * b);
	double t_5 = sqrt(t_0);
	double t_6 = -b * t_5;
	double t_7 = (a * a) * (c * c);
	double t_8 = t_7 / (b * b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (fma((b * b), -b, (t_5 * t_0)) / fma(b, b, ((pow(t_0, 3.0) - pow(t_6, 3.0)) / fma(t_0, t_0, fma(t_6, t_6, (t_0 * t_6)))))) / (3.0 * a);
	} else {
		tmp = ((b * fma(-3.0, (a * c), fma(-1.6875, t_2, fma(-1.5, (a * c), fma(-1.125, t_8, fma(-0.5, (fma(1.265625, t_3, (5.0625 * t_3)) / pow(b, 6.0)), fma(3.375, t_2, fma(4.5, t_8, (5.0625 * (t_3 / pow(b, 6.0))))))))))) / fma(b, b, ((b * b) * ((2.0 + fma(-3.0, t_4, (-1.6875 * (t_1 / pow(b, 6.0))))) - fma(1.125, (t_7 / pow(b, 4.0)), (1.5 * t_4)))))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = Float64(Float64(Float64(a * a) * a) * Float64(Float64(c * c) * c))
	t_2 = Float64(t_1 / (b ^ 4.0))
	t_3 = Float64(a * c) ^ 4.0
	t_4 = Float64(Float64(a * c) / Float64(b * b))
	t_5 = sqrt(t_0)
	t_6 = Float64(Float64(-b) * t_5)
	t_7 = Float64(Float64(a * a) * Float64(c * c))
	t_8 = Float64(t_7 / Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), Float64(t_5 * t_0)) / fma(b, b, Float64(Float64((t_0 ^ 3.0) - (t_6 ^ 3.0)) / fma(t_0, t_0, fma(t_6, t_6, Float64(t_0 * t_6)))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(b * fma(-3.0, Float64(a * c), fma(-1.6875, t_2, fma(-1.5, Float64(a * c), fma(-1.125, t_8, fma(-0.5, Float64(fma(1.265625, t_3, Float64(5.0625 * t_3)) / (b ^ 6.0)), fma(3.375, t_2, fma(4.5, t_8, Float64(5.0625 * Float64(t_3 / (b ^ 6.0))))))))))) / fma(b, b, Float64(Float64(b * b) * Float64(Float64(2.0 + fma(-3.0, t_4, Float64(-1.6875 * Float64(t_1 / (b ^ 6.0))))) - fma(1.125, Float64(t_7 / (b ^ 4.0)), Float64(1.5 * t_4)))))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$6 = N[((-b) * t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$7 / N[(b * b), $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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[(t$95$5 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[t$95$6, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(t$95$6 * t$95$6 + N[(t$95$0 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(-3.0 * N[(a * c), $MachinePrecision] + N[(-1.6875 * t$95$2 + N[(-1.5 * N[(a * c), $MachinePrecision] + N[(-1.125 * t$95$8 + N[(-0.5 * N[(N[(1.265625 * t$95$3 + N[(5.0625 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(3.375 * t$95$2 + N[(4.5 * t$95$8 + N[(5.0625 * N[(t$95$3 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(b * b), $MachinePrecision] * N[(N[(2.0 + N[(-3.0 * t$95$4 + N[(-1.6875 * N[(t$95$1 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.125 * N[(t$95$7 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $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)) < -1.5

   1. Initial program 55.7%

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

   3. Applied rewrites 57.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 57.5%

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

   if -1.5 < (/.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 55.7%

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

   3. Applied rewrites 57.5%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-3 \cdot \left(a \cdot c\right) + \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \left(\frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{27}{8} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{9}{2} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{81}{16} \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   6. Applied rewrites 91.5%

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 5.0625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   7. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-27}{16}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{-3}{2}, a \cdot c, \mathsf{fma}\left(\frac{-9}{8}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{81}{64}, {\left(a \cdot c\right)}^{4}, \frac{81}{16} \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{27}{8}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{9}{2}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{81}{16} \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \color{blue}{{b}^{2} \cdot \left(\left(2 + \left(-3 \cdot \frac{a \cdot c}{{b}^{2}} + \frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)\right) - \left(\frac{9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \frac{3}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}\right)}}{3 \cdot a} \]

   9. Applied rewrites 91.6%

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 5.0625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \color{blue}{\left(b \cdot b\right) \cdot \left(\left(2 + \mathsf{fma}\left(-3, \frac{a \cdot c}{b \cdot b}, -1.6875 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{6}}\right)\right) - \mathsf{fma}\left(1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, 1.5 \cdot \frac{a \cdot c}{b \cdot b}\right)\right)}\right)}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 92.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)\\ t_2 := \frac{t\_1}{{b}^{4}}\\ t_3 := {\left(a \cdot c\right)}^{4}\\ t_4 := \sqrt{t\_0}\\ t_5 := \left(-b\right) \cdot t\_4\\ t_6 := \left(a \cdot a\right) \cdot \left(c \cdot c\right)\\ t_7 := \frac{t\_6}{b \cdot b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, t\_4 \cdot t\_0\right)}{\mathsf{fma}\left(b, b, \frac{{t\_0}^{3} - {t\_5}^{3}}{\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(t\_5, t\_5, t\_0 \cdot t\_5\right)\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, t\_2, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, t\_7, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, t\_3, 5.0625 \cdot t\_3\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, t\_2, \mathsf{fma}\left(4.5, t\_7, 5.0625 \cdot \frac{t\_3}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, t\_0 - \left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(1.125, \frac{t\_6}{{b}^{4}}, \mathsf{fma}\left(1.5, \frac{a \cdot c}{b \cdot b}, 1.6875 \cdot \frac{t\_1}{{b}^{6}}\right)\right) - 1\right)\right)}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b)))
        (t_1 (* (* (* a a) a) (* (* c c) c)))
        (t_2 (/ t_1 (pow b 4.0)))
        (t_3 (pow (* a c) 4.0))
        (t_4 (sqrt t_0))
        (t_5 (* (- b) t_4))
        (t_6 (* (* a a) (* c c)))
        (t_7 (/ t_6 (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/
      (/
       (fma (* b b) (- b) (* t_4 t_0))
       (fma
        b
        b
        (/
         (- (pow t_0 3.0) (pow t_5 3.0))
         (fma t_0 t_0 (fma t_5 t_5 (* t_0 t_5))))))
      (* 3.0 a))
     (/
      (/
       (*
        b
        (fma
         -3.0
         (* a c)
         (fma
          -1.6875
          t_2
          (fma
           -1.5
           (* a c)
           (fma
            -1.125
            t_7
            (fma
             -0.5
             (/ (fma 1.265625 t_3 (* 5.0625 t_3)) (pow b 6.0))
             (fma 3.375 t_2 (fma 4.5 t_7 (* 5.0625 (/ t_3 (pow b 6.0)))))))))))
       (fma
        b
        b
        (-
         t_0
         (*
          (* b b)
          (-
           (fma
            1.125
            (/ t_6 (pow b 4.0))
            (fma 1.5 (/ (* a c) (* b b)) (* 1.6875 (/ t_1 (pow b 6.0)))))
           1.0)))))
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = ((a * a) * a) * ((c * c) * c);
	double t_2 = t_1 / pow(b, 4.0);
	double t_3 = pow((a * c), 4.0);
	double t_4 = sqrt(t_0);
	double t_5 = -b * t_4;
	double t_6 = (a * a) * (c * c);
	double t_7 = t_6 / (b * b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (fma((b * b), -b, (t_4 * t_0)) / fma(b, b, ((pow(t_0, 3.0) - pow(t_5, 3.0)) / fma(t_0, t_0, fma(t_5, t_5, (t_0 * t_5)))))) / (3.0 * a);
	} else {
		tmp = ((b * fma(-3.0, (a * c), fma(-1.6875, t_2, fma(-1.5, (a * c), fma(-1.125, t_7, fma(-0.5, (fma(1.265625, t_3, (5.0625 * t_3)) / pow(b, 6.0)), fma(3.375, t_2, fma(4.5, t_7, (5.0625 * (t_3 / pow(b, 6.0))))))))))) / fma(b, b, (t_0 - ((b * b) * (fma(1.125, (t_6 / pow(b, 4.0)), fma(1.5, ((a * c) / (b * b)), (1.6875 * (t_1 / pow(b, 6.0))))) - 1.0))))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = Float64(Float64(Float64(a * a) * a) * Float64(Float64(c * c) * c))
	t_2 = Float64(t_1 / (b ^ 4.0))
	t_3 = Float64(a * c) ^ 4.0
	t_4 = sqrt(t_0)
	t_5 = Float64(Float64(-b) * t_4)
	t_6 = Float64(Float64(a * a) * Float64(c * c))
	t_7 = Float64(t_6 / Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), Float64(t_4 * t_0)) / fma(b, b, Float64(Float64((t_0 ^ 3.0) - (t_5 ^ 3.0)) / fma(t_0, t_0, fma(t_5, t_5, Float64(t_0 * t_5)))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(b * fma(-3.0, Float64(a * c), fma(-1.6875, t_2, fma(-1.5, Float64(a * c), fma(-1.125, t_7, fma(-0.5, Float64(fma(1.265625, t_3, Float64(5.0625 * t_3)) / (b ^ 6.0)), fma(3.375, t_2, fma(4.5, t_7, Float64(5.0625 * Float64(t_3 / (b ^ 6.0))))))))))) / fma(b, b, Float64(t_0 - Float64(Float64(b * b) * Float64(fma(1.125, Float64(t_6 / (b ^ 4.0)), fma(1.5, Float64(Float64(a * c) / Float64(b * b)), Float64(1.6875 * Float64(t_1 / (b ^ 6.0))))) - 1.0))))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$5 = N[((-b) * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 / N[(b * b), $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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(t$95$5 * t$95$5 + N[(t$95$0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(-3.0 * N[(a * c), $MachinePrecision] + N[(-1.6875 * t$95$2 + N[(-1.5 * N[(a * c), $MachinePrecision] + N[(-1.125 * t$95$7 + N[(-0.5 * N[(N[(1.265625 * t$95$3 + N[(5.0625 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(3.375 * t$95$2 + N[(4.5 * t$95$7 + N[(5.0625 * N[(t$95$3 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$0 - N[(N[(b * b), $MachinePrecision] * N[(N[(1.125 * N[(t$95$6 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(1.6875 * N[(t$95$1 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $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)) < -1.5

   1. Initial program 55.7%

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

   3. Applied rewrites 57.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 57.5%

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

   if -1.5 < (/.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 55.7%

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

   3. Applied rewrites 57.5%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-3 \cdot \left(a \cdot c\right) + \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \left(\frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{27}{8} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{9}{2} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{81}{16} \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   6. Applied rewrites 91.5%

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 5.0625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   7. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-27}{16}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{-3}{2}, a \cdot c, \mathsf{fma}\left(\frac{-9}{8}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{81}{64}, {\left(a \cdot c\right)}^{4}, \frac{81}{16} \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{27}{8}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{9}{2}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{81}{16} \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \color{blue}{{b}^{2} \cdot \left(\left(\frac{9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \left(\frac{3}{2} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)\right) - 1\right)}\right)}}{3 \cdot a} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-27}{16}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{-3}{2}, a \cdot c, \mathsf{fma}\left(\frac{-9}{8}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{81}{64}, {\left(a \cdot c\right)}^{4}, \frac{81}{16} \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{27}{8}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{9}{2}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{81}{16} \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - {b}^{2} \cdot \color{blue}{\left(\left(\frac{9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \left(\frac{3}{2} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)\right) - 1\right)}\right)}}{3 \cdot a} \]

      2. pow2 N/A

         \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-27}{16}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{-3}{2}, a \cdot c, \mathsf{fma}\left(\frac{-9}{8}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{81}{64}, {\left(a \cdot c\right)}^{4}, \frac{81}{16} \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{27}{8}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{9}{2}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{81}{16} \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot \left(\color{blue}{\left(\frac{9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \left(\frac{3}{2} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)\right)} - 1\right)\right)}}{3 \cdot a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-27}{16}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{-3}{2}, a \cdot c, \mathsf{fma}\left(\frac{-9}{8}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{81}{64}, {\left(a \cdot c\right)}^{4}, \frac{81}{16} \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{27}{8}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{9}{2}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{81}{16} \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot \left(\color{blue}{\left(\frac{9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \left(\frac{3}{2} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)\right)} - 1\right)\right)}}{3 \cdot a} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-27}{16}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{-3}{2}, a \cdot c, \mathsf{fma}\left(\frac{-9}{8}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{81}{64}, {\left(a \cdot c\right)}^{4}, \frac{81}{16} \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{27}{8}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{9}{2}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{81}{16} \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot \left(\left(\frac{9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \left(\frac{3}{2} \cdot \frac{a \cdot c}{{b}^{2}} + \frac{27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)\right) - \color{blue}{1}\right)\right)}}{3 \cdot a} \]

   9. Applied rewrites 91.6%

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 5.0625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \color{blue}{\left(b \cdot b\right) \cdot \left(\mathsf{fma}\left(1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(1.5, \frac{a \cdot c}{b \cdot b}, 1.6875 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{6}}\right)\right) - 1\right)}\right)}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := {\left(a \cdot c\right)}^{4}\\ t_2 := \sqrt{t\_0}\\ t_3 := \left(-b\right) \cdot t\_2\\ t_4 := \left(a \cdot a\right) \cdot \left(c \cdot c\right)\\ t_5 := \frac{t\_4}{b \cdot b}\\ t_6 := \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)\\ t_7 := \frac{t\_6}{{b}^{4}}\\ t_8 := \frac{a \cdot c}{b \cdot b}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, t\_2 \cdot t\_0\right)}{\mathsf{fma}\left(b, b, \frac{{t\_0}^{3} - {t\_3}^{3}}{\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(t\_3, t\_3, t\_0 \cdot t\_3\right)\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, t\_7, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, t\_5, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, t\_1, 5.0625 \cdot t\_1\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, t\_7, \mathsf{fma}\left(4.5, t\_5, 5.0625 \cdot \frac{t\_1}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(3 + \mathsf{fma}\left(-3, t\_8, -1.6875 \cdot \frac{t\_6}{{b}^{6}}\right)\right) - \mathsf{fma}\left(1.125, \frac{t\_4}{{b}^{4}}, 1.5 \cdot t\_8\right)\right)}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b)))
        (t_1 (pow (* a c) 4.0))
        (t_2 (sqrt t_0))
        (t_3 (* (- b) t_2))
        (t_4 (* (* a a) (* c c)))
        (t_5 (/ t_4 (* b b)))
        (t_6 (* (* (* a a) a) (* (* c c) c)))
        (t_7 (/ t_6 (pow b 4.0)))
        (t_8 (/ (* a c) (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/
      (/
       (fma (* b b) (- b) (* t_2 t_0))
       (fma
        b
        b
        (/
         (- (pow t_0 3.0) (pow t_3 3.0))
         (fma t_0 t_0 (fma t_3 t_3 (* t_0 t_3))))))
      (* 3.0 a))
     (/
      (/
       (*
        b
        (fma
         -3.0
         (* a c)
         (fma
          -1.6875
          t_7
          (fma
           -1.5
           (* a c)
           (fma
            -1.125
            t_5
            (fma
             -0.5
             (/ (fma 1.265625 t_1 (* 5.0625 t_1)) (pow b 6.0))
             (fma 3.375 t_7 (fma 4.5 t_5 (* 5.0625 (/ t_1 (pow b 6.0)))))))))))
       (*
        (* b b)
        (-
         (+ 3.0 (fma -3.0 t_8 (* -1.6875 (/ t_6 (pow b 6.0)))))
         (fma 1.125 (/ t_4 (pow b 4.0)) (* 1.5 t_8)))))
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = pow((a * c), 4.0);
	double t_2 = sqrt(t_0);
	double t_3 = -b * t_2;
	double t_4 = (a * a) * (c * c);
	double t_5 = t_4 / (b * b);
	double t_6 = ((a * a) * a) * ((c * c) * c);
	double t_7 = t_6 / pow(b, 4.0);
	double t_8 = (a * c) / (b * b);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (fma((b * b), -b, (t_2 * t_0)) / fma(b, b, ((pow(t_0, 3.0) - pow(t_3, 3.0)) / fma(t_0, t_0, fma(t_3, t_3, (t_0 * t_3)))))) / (3.0 * a);
	} else {
		tmp = ((b * fma(-3.0, (a * c), fma(-1.6875, t_7, fma(-1.5, (a * c), fma(-1.125, t_5, fma(-0.5, (fma(1.265625, t_1, (5.0625 * t_1)) / pow(b, 6.0)), fma(3.375, t_7, fma(4.5, t_5, (5.0625 * (t_1 / pow(b, 6.0))))))))))) / ((b * b) * ((3.0 + fma(-3.0, t_8, (-1.6875 * (t_6 / pow(b, 6.0))))) - fma(1.125, (t_4 / pow(b, 4.0)), (1.5 * t_8))))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = Float64(a * c) ^ 4.0
	t_2 = sqrt(t_0)
	t_3 = Float64(Float64(-b) * t_2)
	t_4 = Float64(Float64(a * a) * Float64(c * c))
	t_5 = Float64(t_4 / Float64(b * b))
	t_6 = Float64(Float64(Float64(a * a) * a) * Float64(Float64(c * c) * c))
	t_7 = Float64(t_6 / (b ^ 4.0))
	t_8 = Float64(Float64(a * c) / Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), Float64(t_2 * t_0)) / fma(b, b, Float64(Float64((t_0 ^ 3.0) - (t_3 ^ 3.0)) / fma(t_0, t_0, fma(t_3, t_3, Float64(t_0 * t_3)))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(b * fma(-3.0, Float64(a * c), fma(-1.6875, t_7, fma(-1.5, Float64(a * c), fma(-1.125, t_5, fma(-0.5, Float64(fma(1.265625, t_1, Float64(5.0625 * t_1)) / (b ^ 6.0)), fma(3.375, t_7, fma(4.5, t_5, Float64(5.0625 * Float64(t_1 / (b ^ 6.0))))))))))) / Float64(Float64(b * b) * Float64(Float64(3.0 + fma(-3.0, t_8, Float64(-1.6875 * Float64(t_6 / (b ^ 6.0))))) - fma(1.125, Float64(t_4 / (b ^ 4.0)), Float64(1.5 * t_8))))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[((-b) * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[(a * c), $MachinePrecision] / N[(b * b), $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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(t$95$3 * t$95$3 + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(-3.0 * N[(a * c), $MachinePrecision] + N[(-1.6875 * t$95$7 + N[(-1.5 * N[(a * c), $MachinePrecision] + N[(-1.125 * t$95$5 + N[(-0.5 * N[(N[(1.265625 * t$95$1 + N[(5.0625 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(3.375 * t$95$7 + N[(4.5 * t$95$5 + N[(5.0625 * N[(t$95$1 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(N[(3.0 + N[(-3.0 * t$95$8 + N[(-1.6875 * N[(t$95$6 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.125 * N[(t$95$4 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] + N[(1.5 * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $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)) < -1.5

   1. Initial program 55.7%

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

   3. Applied rewrites 57.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 57.5%

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

   if -1.5 < (/.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 55.7%

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

   3. Applied rewrites 57.5%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-3 \cdot \left(a \cdot c\right) + \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \left(\frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{27}{8} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{9}{2} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{81}{16} \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   6. Applied rewrites 91.5%

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 5.0625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   7. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(\frac{-27}{16}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{-3}{2}, a \cdot c, \mathsf{fma}\left(\frac{-9}{8}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(\frac{81}{64}, {\left(a \cdot c\right)}^{4}, \frac{81}{16} \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(\frac{27}{8}, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(\frac{9}{2}, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \frac{81}{16} \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\color{blue}{{b}^{2} \cdot \left(\left(3 + \left(-3 \cdot \frac{a \cdot c}{{b}^{2}} + \frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)\right) - \left(\frac{9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + \frac{3}{2} \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}}}{3 \cdot a} \]

   9. Applied rewrites 91.6%

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 5.0625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}{\color{blue}{\left(b \cdot b\right) \cdot \left(\left(3 + \mathsf{fma}\left(-3, \frac{a \cdot c}{b \cdot b}, -1.6875 \cdot \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{6}}\right)\right) - \mathsf{fma}\left(1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{{b}^{4}}, 1.5 \cdot \frac{a \cdot c}{b \cdot b}\right)\right)}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \left(c \cdot c\right)\\ t_1 := \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)\\ t_2 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_3 := \sqrt{t\_2}\\ t_4 := {\left(a \cdot c\right)}^{4}\\ t_5 := \left(-b\right) \cdot t\_3\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, t\_3 \cdot t\_2\right)}{\mathsf{fma}\left(b, b, \frac{{t\_2}^{3} - {t\_5}^{3}}{\mathsf{fma}\left(t\_2, t\_2, \mathsf{fma}\left(t\_5, t\_5, t\_2 \cdot t\_5\right)\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(1.265625, t\_4, 5.0625 \cdot t\_4\right), \mathsf{fma}\left(5.0625, t\_4, \left(b \cdot b\right) \cdot \mathsf{fma}\left(-1.6875, t\_1, \mathsf{fma}\left(3.375, t\_1, \left(b \cdot b\right) \cdot \mathsf{fma}\left(-1.5, a \cdot \left(\left(b \cdot b\right) \cdot c\right), \mathsf{fma}\left(-1.125, t\_0, 4.5 \cdot t\_0\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right)}{\mathsf{fma}\left(b, b, t\_2 - t\_5\right)}}{3 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a a) (* c c)))
        (t_1 (* (* (* a a) a) (* (* c c) c)))
        (t_2 (fma (* -3.0 a) c (* b b)))
        (t_3 (sqrt t_2))
        (t_4 (pow (* a c) 4.0))
        (t_5 (* (- b) t_3)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/
      (/
       (fma (* b b) (- b) (* t_3 t_2))
       (fma
        b
        b
        (/
         (- (pow t_2 3.0) (pow t_5 3.0))
         (fma t_2 t_2 (fma t_5 t_5 (* t_2 t_5))))))
      (* 3.0 a))
     (/
      (/
       (*
        b
        (fma
         -3.0
         (* a c)
         (/
          (fma
           -0.5
           (fma 1.265625 t_4 (* 5.0625 t_4))
           (fma
            5.0625
            t_4
            (*
             (* b b)
             (fma
              -1.6875
              t_1
              (fma
               3.375
               t_1
               (*
                (* b b)
                (fma
                 -1.5
                 (* a (* (* b b) c))
                 (fma -1.125 t_0 (* 4.5 t_0)))))))))
          (pow b 6.0))))
       (fma b b (- t_2 t_5)))
      (* 3.0 a)))))

C

double code(double a, double b, double c) {
	double t_0 = (a * a) * (c * c);
	double t_1 = ((a * a) * a) * ((c * c) * c);
	double t_2 = fma((-3.0 * a), c, (b * b));
	double t_3 = sqrt(t_2);
	double t_4 = pow((a * c), 4.0);
	double t_5 = -b * t_3;
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (fma((b * b), -b, (t_3 * t_2)) / fma(b, b, ((pow(t_2, 3.0) - pow(t_5, 3.0)) / fma(t_2, t_2, fma(t_5, t_5, (t_2 * t_5)))))) / (3.0 * a);
	} else {
		tmp = ((b * fma(-3.0, (a * c), (fma(-0.5, fma(1.265625, t_4, (5.0625 * t_4)), fma(5.0625, t_4, ((b * b) * fma(-1.6875, t_1, fma(3.375, t_1, ((b * b) * fma(-1.5, (a * ((b * b) * c)), fma(-1.125, t_0, (4.5 * t_0))))))))) / pow(b, 6.0)))) / fma(b, b, (t_2 - t_5))) / (3.0 * a);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * a) * Float64(c * c))
	t_1 = Float64(Float64(Float64(a * a) * a) * Float64(Float64(c * c) * c))
	t_2 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_3 = sqrt(t_2)
	t_4 = Float64(a * c) ^ 4.0
	t_5 = Float64(Float64(-b) * t_3)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), Float64(t_3 * t_2)) / fma(b, b, Float64(Float64((t_2 ^ 3.0) - (t_5 ^ 3.0)) / fma(t_2, t_2, fma(t_5, t_5, Float64(t_2 * t_5)))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(b * fma(-3.0, Float64(a * c), Float64(fma(-0.5, fma(1.265625, t_4, Float64(5.0625 * t_4)), fma(5.0625, t_4, Float64(Float64(b * b) * fma(-1.6875, t_1, fma(3.375, t_1, Float64(Float64(b * b) * fma(-1.5, Float64(a * Float64(Float64(b * b) * c)), fma(-1.125, t_0, Float64(4.5 * t_0))))))))) / (b ^ 6.0)))) / fma(b, b, Float64(t_2 - t_5))) / Float64(3.0 * a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$5 = N[((-b) * t$95$3), $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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] - N[Power[t$95$5, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2 + N[(t$95$5 * t$95$5 + N[(t$95$2 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(-3.0 * N[(a * c), $MachinePrecision] + N[(N[(-0.5 * N[(1.265625 * t$95$4 + N[(5.0625 * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(5.0625 * t$95$4 + N[(N[(b * b), $MachinePrecision] * N[(-1.6875 * t$95$1 + N[(3.375 * t$95$1 + N[(N[(b * b), $MachinePrecision] * N[(-1.5 * N[(a * N[(N[(b * b), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] + N[(-1.125 * t$95$0 + N[(4.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$2 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $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)) < -1.5

   1. Initial program 55.7%

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

   3. Applied rewrites 57.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 57.5%

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

   if -1.5 < (/.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 55.7%

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

   3. Applied rewrites 57.5%

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

   4. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-3 \cdot \left(a \cdot c\right) + \left(\frac{-27}{16} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-3}{2} \cdot \left(a \cdot c\right) + \left(\frac{-9}{8} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \left(\frac{-1}{2} \cdot \frac{\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)}{{b}^{6}} + \left(\frac{27}{8} \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{9}{2} \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + \frac{81}{16} \cdot \frac{{a}^{4} \cdot {c}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   6. Applied rewrites 91.5%

      \[\leadsto \frac{\frac{\color{blue}{b \cdot \mathsf{fma}\left(-3, a \cdot c, \mathsf{fma}\left(-1.6875, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(-1.5, a \cdot c, \mathsf{fma}\left(-1.125, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, \mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right)}{{b}^{6}}, \mathsf{fma}\left(3.375, \frac{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}}, \mathsf{fma}\left(4.5, \frac{\left(a \cdot a\right) \cdot \left(c \cdot c\right)}{b \cdot b}, 5.0625 \cdot \frac{{\left(a \cdot c\right)}^{4}}{{b}^{6}}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   7. Taylor expanded in b around 0

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \frac{\frac{-1}{2} \cdot \left(\frac{81}{64} \cdot \left({a}^{4} \cdot {c}^{4}\right) + \frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right)\right) + \left(\frac{81}{16} \cdot \left({a}^{4} \cdot {c}^{4}\right) + {b}^{2} \cdot \left(\frac{-27}{16} \cdot \left({a}^{3} \cdot {c}^{3}\right) + \left(\frac{27}{8} \cdot \left({a}^{3} \cdot {c}^{3}\right) + {b}^{2} \cdot \left(\frac{-3}{2} \cdot \left(a \cdot \left({b}^{2} \cdot c\right)\right) + \left(\frac{-9}{8} \cdot \left({a}^{2} \cdot {c}^{2}\right) + \frac{9}{2} \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]

   9. Applied rewrites 91.5%

      \[\leadsto \frac{\frac{b \cdot \mathsf{fma}\left(-3, a \cdot c, \frac{\mathsf{fma}\left(-0.5, \mathsf{fma}\left(1.265625, {\left(a \cdot c\right)}^{4}, 5.0625 \cdot {\left(a \cdot c\right)}^{4}\right), \mathsf{fma}\left(5.0625, {\left(a \cdot c\right)}^{4}, \left(b \cdot b\right) \cdot \mathsf{fma}\left(-1.6875, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), \mathsf{fma}\left(3.375, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-1.5, a \cdot \left(\left(b \cdot b\right) \cdot c\right), \mathsf{fma}\left(-1.125, \left(a \cdot a\right) \cdot \left(c \cdot c\right), 4.5 \cdot \left(\left(a \cdot a\right) \cdot \left(c \cdot c\right)\right)\right)\right)\right)\right)\right)\right)}{{b}^{6}}\right)}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right) - \left(-b\right) \cdot \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\right)}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ t_2 := \left(-b\right) \cdot t\_1\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, t\_1 \cdot t\_0\right)}{\mathsf{fma}\left(b, b, \frac{{t\_0}^{3} - {t\_2}^{3}}{\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(t\_2, t\_2, t\_0 \cdot t\_2\right)\right)}\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \mathsf{fma}\left(\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(-1.0546875 \cdot \frac{a \cdot c}{{b}^{6}} - 0.5625 \cdot {b}^{-4}\right), a, \frac{-0.375 \cdot \left(c \cdot c\right)}{b \cdot b}\right), -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)) (t_2 (* (- b) t_1)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/
      (/
       (fma (* b b) (- b) (* t_1 t_0))
       (fma
        b
        b
        (/
         (- (pow t_0 3.0) (pow t_2 3.0))
         (fma t_0 t_0 (fma t_2 t_2 (* t_0 t_2))))))
      (* 3.0 a))
     (/
      (fma
       a
       (fma
        (*
         (* (* c c) c)
         (- (* -1.0546875 (/ (* a c) (pow b 6.0))) (* 0.5625 (pow b -4.0))))
        a
        (/ (* -0.375 (* c c)) (* b b)))
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double t_2 = -b * t_1;
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (fma((b * b), -b, (t_1 * t_0)) / fma(b, b, ((pow(t_0, 3.0) - pow(t_2, 3.0)) / fma(t_0, t_0, fma(t_2, t_2, (t_0 * t_2)))))) / (3.0 * a);
	} else {
		tmp = fma(a, fma((((c * c) * c) * ((-1.0546875 * ((a * c) / pow(b, 6.0))) - (0.5625 * pow(b, -4.0)))), a, ((-0.375 * (c * c)) / (b * b))), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	t_1 = sqrt(t_0)
	t_2 = Float64(Float64(-b) * t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), Float64(t_1 * t_0)) / fma(b, b, Float64(Float64((t_0 ^ 3.0) - (t_2 ^ 3.0)) / fma(t_0, t_0, fma(t_2, t_2, Float64(t_0 * t_2)))))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(a, fma(Float64(Float64(Float64(c * c) * c) * Float64(Float64(-1.0546875 * Float64(Float64(a * c) / (b ^ 6.0))) - Float64(0.5625 * (b ^ -4.0)))), a, Float64(Float64(-0.375 * Float64(c * c)) / Float64(b * b))), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[((-b) * t$95$1), $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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(t$95$2 * t$95$2 + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(-1.0546875 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5625 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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)) < -1.5

   1. Initial program 55.7%

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

   3. Applied rewrites 57.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 57.5%

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

   if -1.5 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval 91.1

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

   10. Applied rewrites 91.1%

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

Alternative 6: 92.0% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] + N[(t$95$0 - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(-1.0546875 * N[(N[(a * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5625 * N[Power[b, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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)) < -1.5

   1. Initial program 55.7%

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

   3. Applied rewrites 57.5%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   5. Applied rewrites 57.5%

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

   if -1.5 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval 91.1

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

   10. Applied rewrites 91.1%

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

Alternative 7: 91.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, t\_1 \cdot t\_0\right)}{b \cdot b + \left(t\_0 - \left(-b\right) \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-1.0546875, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)}{{b}^{7}} \cdot c - \frac{0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/
      (/ (fma (* b b) (- b) (* t_1 t_0)) (+ (* b b) (- t_0 (* (- b) t_1))))
      (* 3.0 a))
     (*
      (-
       (*
        (/
         (fma
          -1.0546875
          (* (* (* a a) a) (* c c))
          (* (* b b) (fma -0.5625 (* (* a a) c) (* -0.375 (* a (* b b))))))
         (pow b 7.0))
        c)
       (/ 0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (fma((b * b), -b, (t_1 * t_0)) / ((b * b) + (t_0 - (-b * t_1)))) / (3.0 * a);
	} else {
		tmp = (((fma(-1.0546875, (((a * a) * a) * (c * c)), ((b * b) * fma(-0.5625, ((a * a) * c), (-0.375 * (a * (b * b)))))) / pow(b, 7.0)) * c) - (0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), Float64(t_1 * t_0)) / Float64(Float64(b * b) + Float64(t_0 - Float64(Float64(-b) * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(fma(-1.0546875, Float64(Float64(Float64(a * a) * a) * Float64(c * c)), Float64(Float64(b * b) * fma(-0.5625, Float64(Float64(a * a) * c), Float64(-0.375 * Float64(a * Float64(b * b)))))) / (b ^ 7.0)) * c) - Float64(0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] + N[(t$95$0 - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-1.0546875 * N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] + N[(-0.375 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $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)) < -1.5

   1. Initial program 55.7%

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

   3. Applied rewrites 57.5%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 N/A

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

   5. Applied rewrites 57.5%

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

   if -1.5 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

   7. Applied rewrites 90.9%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{5}}, -0.5625, \frac{-1.0546875 \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot c\right)}{{b}^{7}}\right), c, \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right) \cdot c - \frac{0.5}{b}\right) \cdot \color{blue}{c} \]

   8. Taylor expanded in b around 0

      \[\leadsto \left(\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

   10. Applied rewrites 90.9%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-1.0546875, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)}{{b}^{7}} \cdot c - \frac{0.5}{b}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b \cdot b, -b, t\_1 \cdot t\_0\right)}{\mathsf{fma}\left(b, b, t\_0 - \left(-b\right) \cdot t\_1\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-1.0546875, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)}{{b}^{7}} \cdot c - \frac{0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))) (t_1 (sqrt t_0)))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/
      (/ (fma (* b b) (- b) (* t_1 t_0)) (fma b b (- t_0 (* (- b) t_1))))
      (* 3.0 a))
     (*
      (-
       (*
        (/
         (fma
          -1.0546875
          (* (* (* a a) a) (* c c))
          (* (* b b) (fma -0.5625 (* (* a a) c) (* -0.375 (* a (* b b))))))
         (pow b 7.0))
        c)
       (/ 0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double t_1 = sqrt(t_0);
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (fma((b * b), -b, (t_1 * t_0)) / fma(b, b, (t_0 - (-b * t_1)))) / (3.0 * a);
	} else {
		tmp = (((fma(-1.0546875, (((a * a) * a) * (c * c)), ((b * b) * fma(-0.5625, ((a * a) * c), (-0.375 * (a * (b * b)))))) / pow(b, 7.0)) * c) - (0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	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)) <= -1.5)
		tmp = Float64(Float64(fma(Float64(b * b), Float64(-b), Float64(t_1 * t_0)) / fma(b, b, Float64(t_0 - Float64(Float64(-b) * t_1)))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(fma(-1.0546875, Float64(Float64(Float64(a * a) * a) * Float64(c * c)), Float64(Float64(b * b) * fma(-0.5625, Float64(Float64(a * a) * c), Float64(-0.375 * Float64(a * Float64(b * b)))))) / (b ^ 7.0)) * c) - Float64(0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] * (-b) + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(b * b + N[(t$95$0 - N[((-b) * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-1.0546875 * N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] + N[(-0.375 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $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)) < -1.5

   1. Initial program 55.7%

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

   3. Applied rewrites 57.5%

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

   if -1.5 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

   7. Applied rewrites 90.9%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{5}}, -0.5625, \frac{-1.0546875 \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot c\right)}{{b}^{7}}\right), c, \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right) \cdot c - \frac{0.5}{b}\right) \cdot \color{blue}{c} \]

   8. Taylor expanded in b around 0

      \[\leadsto \left(\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

   10. Applied rewrites 90.9%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-1.0546875, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)}{{b}^{7}} \cdot c - \frac{0.5}{b}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1.5:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-1.0546875, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)}{{b}^{7}} \cdot c - \frac{0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -1.5)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (*
      (-
       (*
        (/
         (fma
          -1.0546875
          (* (* (* a a) a) (* c c))
          (* (* b b) (fma -0.5625 (* (* a a) c) (* -0.375 (* a (* b b))))))
         (pow b 7.0))
        c)
       (/ 0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -1.5) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (((fma(-1.0546875, (((a * a) * a) * (c * c)), ((b * b) * fma(-0.5625, ((a * a) * c), (-0.375 * (a * (b * b)))))) / pow(b, 7.0)) * c) - (0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -1.5)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(fma(-1.0546875, Float64(Float64(Float64(a * a) * a) * Float64(c * c)), Float64(Float64(b * b) * fma(-0.5625, Float64(Float64(a * a) * c), Float64(-0.375 * Float64(a * Float64(b * b)))))) / (b ^ 7.0)) * c) - Float64(0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $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], -1.5], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-1.0546875 * N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(-0.5625 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] + N[(-0.375 * N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $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)) < -1.5

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 57.2%

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

   if -1.5 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto c \cdot \color{blue}{\left(c \cdot \left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}} + c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{3} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{{a}^{2}}{{b}^{5}}\right)\right) - \frac{1}{2} \cdot \frac{1}{b}\right)} \]

   7. Applied rewrites 90.9%

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a \cdot a}{{b}^{5}}, -0.5625, \frac{-1.0546875 \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot c\right)}{{b}^{7}}\right), c, \frac{a}{\left(b \cdot b\right) \cdot b} \cdot -0.375\right) \cdot c - \frac{0.5}{b}\right) \cdot \color{blue}{c} \]

   8. Taylor expanded in b around 0

      \[\leadsto \left(\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{\frac{-135}{128} \cdot \left({a}^{3} \cdot {c}^{2}\right) + {b}^{2} \cdot \left(\frac{-9}{16} \cdot \left({a}^{2} \cdot c\right) + \frac{-3}{8} \cdot \left(a \cdot {b}^{2}\right)\right)}{{b}^{7}} \cdot c - \frac{\frac{1}{2}}{b}\right) \cdot c \]

   10. Applied rewrites 90.9%

       \[\leadsto \left(\frac{\mathsf{fma}\left(-1.0546875, \left(\left(a \cdot a\right) \cdot a\right) \cdot \left(c \cdot c\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625, \left(a \cdot a\right) \cdot c, -0.375 \cdot \left(a \cdot \left(b \cdot b\right)\right)\right)\right)}{{b}^{7}} \cdot c - \frac{0.5}{b}\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0315:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}, -0.375, \frac{-0.5625 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot a\right)}{{b}^{5}}\right), a, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0315)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (fma
      (fma
       (/ (* c c) (* (* b b) b))
       -0.375
       (/ (* -0.5625 (* (* (* c c) c) a)) (pow b 5.0)))
      a
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0315) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(fma(((c * c) / ((b * b) * b)), -0.375, ((-0.5625 * (((c * c) * c) * a)) / pow(b, 5.0))), a, ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0315)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = fma(fma(Float64(Float64(c * c) / Float64(Float64(b * b) * b)), -0.375, Float64(Float64(-0.5625 * Float64(Float64(Float64(c * c) * c) * a)) / (b ^ 5.0))), a, Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $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], -0.0315], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(-0.5625 * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + 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)) < -0.0315

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 57.2%

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

   if -0.0315 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 88.0%

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

Alternative 11: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0315:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot a\right)\right) \cdot -0.5625}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b}}{b} + \frac{\mathsf{fma}\left(\left(c \cdot c\right) \cdot \frac{a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0315)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (+
      (/ (/ (* (* a (* (* (* c c) c) a)) -0.5625) (* (* (* b b) b) b)) b)
      (/ (fma (* (* c c) (/ a (* b b))) -0.375 (* -0.5 c)) b)))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0315) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = ((((a * (((c * c) * c) * a)) * -0.5625) / (((b * b) * b) * b)) / b) + (fma(((c * c) * (a / (b * b))), -0.375, (-0.5 * c)) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0315)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(a * Float64(Float64(Float64(c * c) * c) * a)) * -0.5625) / Float64(Float64(Float64(b * b) * b) * b)) / b) + Float64(fma(Float64(Float64(c * c) * Float64(a / Float64(b * b))), -0.375, Float64(-0.5 * c)) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $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], -0.0315], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-0.5 * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.0315

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 57.2%

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

   if -0.0315 < (/.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 55.7%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 87.9%

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

   6. Applied rewrites 87.9%

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

Alternative 12: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0315:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(\left(c \cdot c\right) \cdot c\right)}{b \cdot b}, -0.375 \cdot \left(c \cdot c\right)\right)}{b \cdot b}, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0315)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      (fma
       a
       (/
        (fma -0.5625 (/ (* a (* (* c c) c)) (* b b)) (* -0.375 (* c c)))
        (* b b))
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0315) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(a, (fma(-0.5625, ((a * ((c * c) * c)) / (b * b)), (-0.375 * (c * c))) / (b * b)), (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0315)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(a, Float64(fma(-0.5625, Float64(Float64(a * Float64(Float64(c * c) * c)) / Float64(b * b)), Float64(-0.375 * Float64(c * c))) / Float64(b * b)), Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $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], -0.0315], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[(-0.5625 * N[(N[(a * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + 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)) < -0.0315

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 57.2%

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

   if -0.0315 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow3 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 88.0

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

   10. Applied rewrites 88.0%

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

Alternative 13: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0315:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c \cdot c}{b \cdot b}, -0.375, \frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot a}{\left(\left(b \cdot b\right) \cdot b\right) \cdot b} \cdot -0.5625\right), a, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0315)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (/
      (fma
       (fma
        (/ (* c c) (* b b))
        -0.375
        (* (/ (* (* (* c c) c) a) (* (* (* b b) b) b)) -0.5625))
       a
       (* -0.5 c))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0315) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = fma(fma(((c * c) / (b * b)), -0.375, (((((c * c) * c) * a) / (((b * b) * b) * b)) * -0.5625)), a, (-0.5 * c)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0315)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(fma(fma(Float64(Float64(c * c) / Float64(b * b)), -0.375, Float64(Float64(Float64(Float64(Float64(c * c) * c) * a) / Float64(Float64(Float64(b * b) * b) * b)) * -0.5625)), a, Float64(-0.5 * c)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $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], -0.0315], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * -0.5625), $MachinePrecision]), $MachinePrecision] * a + 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)) < -0.0315

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 57.2%

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

   if -0.0315 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 88.0%

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

Alternative 14: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0315:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0}{\left(-b\right) - \sqrt{t\_0}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -0.375 \cdot a\right)}{\left(b \cdot b\right) \cdot b} \cdot c - \frac{0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fma (* -3.0 a) c (* b b))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0315)
     (/ (/ (- (* b b) t_0) (- (- b) (sqrt t_0))) (* 3.0 a))
     (*
      (-
       (*
        (/ (fma -0.5625 (/ (* (* a a) c) (* b b)) (* -0.375 a)) (* (* b b) b))
        c)
       (/ 0.5 b))
      c))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-3.0 * a), c, (b * b));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0315) {
		tmp = (((b * b) - t_0) / (-b - sqrt(t_0))) / (3.0 * a);
	} else {
		tmp = (((fma(-0.5625, (((a * a) * c) / (b * b)), (-0.375 * a)) / ((b * b) * b)) * c) - (0.5 / b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-3.0 * a), c, Float64(b * b))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.0315)
		tmp = Float64(Float64(Float64(Float64(b * b) - t_0) / Float64(Float64(-b) - sqrt(t_0))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(fma(-0.5625, Float64(Float64(Float64(a * a) * c) / Float64(b * b)), Float64(-0.375 * a)) / Float64(Float64(b * b) * b)) * c) - Float64(0.5 / b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $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], -0.0315], N[(N[(N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision] / N[((-b) - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $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)) < -0.0315

   1. Initial program 55.7%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 57.2%

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

   if -0.0315 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 87.8

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

   10. Applied rewrites 87.8%

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

Alternative 15: 89.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0315:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(-0.5625, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -0.375 \cdot a\right)}{\left(b \cdot b\right) \cdot b} \cdot c - \frac{0.5}{b}\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0315)
   (/ (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) 3.0) a)
   (*
    (-
     (*
      (/ (fma -0.5625 (/ (* (* a a) c) (* b b)) (* -0.375 a)) (* (* b b) b))
      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)) <= -0.0315) {
		tmp = ((sqrt(fma((-3.0 * a), c, (b * b))) + -b) / 3.0) / a;
	} else {
		tmp = (((fma(-0.5625, (((a * a) * c) / (b * b)), (-0.375 * a)) / ((b * b) * b)) * 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)) <= -0.0315)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / 3.0) / a);
	else
		tmp = Float64(Float64(Float64(Float64(fma(-0.5625, Float64(Float64(Float64(a * a) * c) / Float64(b * b)), Float64(-0.375 * a)) / Float64(Float64(b * b) * b)) * 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], -0.0315], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(N[(N[(N[(-0.5625 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $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)) < -0.0315

   1. Initial program 55.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      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 55.7%

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

   if -0.0315 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 87.8

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

   10. Applied rewrites 87.8%

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

Alternative 16: 85.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0085:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{\left(b \cdot b\right) \cdot b}, -0.375, \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0085)
   (/ (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) 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)) <= -0.0085) {
		tmp = ((sqrt(fma((-3.0 * a), c, (b * b))) + -b) / 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)) <= -0.0085)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / 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], -0.0085], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $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)) < -0.0085000000000000006

   1. Initial program 55.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      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 55.7%

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

   if -0.0085000000000000006 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. sqr-pow N/A

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

      4. unpow-prod-down N/A

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

      5. pow2 N/A

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

      6. sqrt-pow2 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   7. Applied rewrites 81.6%

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

Alternative 17: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0085:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}, -0.375, -0.5 \cdot c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0085)
   (/ (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) 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)) <= -0.0085) {
		tmp = ((sqrt(fma((-3.0 * a), c, (b * b))) + -b) / 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)) <= -0.0085)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / 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], -0.0085], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / 3.0), $MachinePrecision] / a), $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)) < -0.0085000000000000006

   1. Initial program 55.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      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 55.7%

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

   if -0.0085000000000000006 < (/.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 55.7%

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

   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 81.6

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

   4. Applied rewrites 81.6%

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

Alternative 18: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0085:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0085)
   (/ (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) 3.0) a)
   (/ (* c (- (* -0.375 (/ (* a c) (* b b))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0085) {
		tmp = ((sqrt(fma((-3.0 * a), c, (b * b))) + -b) / 3.0) / a;
	} else {
		tmp = (c * ((-0.375 * ((a * c) / (b * b))) - 0.5)) / 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)) <= -0.0085)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / 3.0) / a);
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / Float64(b * b))) - 0.5)) / 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], -0.0085], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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)) < -0.0085000000000000006

   1. Initial program 55.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      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 55.7%

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

   if -0.0085000000000000006 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 81.5

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

   10. Applied rewrites 81.5%

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

Alternative 19: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0085:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0085)
   (/ (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) (* a 3.0))
   (/ (* c (- (* -0.375 (/ (* a c) (* b b))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0085) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) + -b) / (a * 3.0);
	} else {
		tmp = (c * ((-0.375 * ((a * c) / (b * b))) - 0.5)) / 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)) <= -0.0085)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / Float64(b * b))) - 0.5)) / 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], -0.0085], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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)) < -0.0085000000000000006

   1. Initial program 55.7%

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

   3. Applied rewrites 55.7%

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

   if -0.0085000000000000006 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 81.5

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

   10. Applied rewrites 81.5%

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

Alternative 20: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.0085:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{b \cdot b} - 0.5\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.0085)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ (* c (- (* -0.375 (/ (* a c) (* b b))) 0.5)) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.0085) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = (c * ((-0.375 * ((a * c) / (b * b))) - 0.5)) / 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)) <= -0.0085)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(-0.375 * Float64(Float64(a * c) / Float64(b * b))) - 0.5)) / 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], -0.0085], 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[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $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)) < -0.0085000000000000006

   1. Initial program 55.7%

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

      1. lift--.f64 N/A

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

      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. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* 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 55.8

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

   3. Applied rewrites 55.8%

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

   if -0.0085000000000000006 < (/.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 55.7%

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

   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 91.0%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 91.1%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 81.5

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

   10. Applied rewrites 81.5%

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

Alternative 21: 81.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

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 91.0%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 91.1%

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

8. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-*.f64 81.5

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

10. Applied rewrites 81.5%

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

Alternative 22: 81.4% 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 55.7%

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

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 91.0%

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

5. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 87.8%

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

8. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b} \cdot c \]
9. 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. lift-*.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 81.4

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

10. Applied rewrites 81.4%

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

Alternative 23: 64.2% accurate, 3.3× 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 55.7%

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

2. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
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 64.2

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

4. Applied rewrites 64.2%

   \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
5. Add Preprocessing

Alternative 24: 64.1% accurate, 3.3× 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 55.7%

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

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 91.0%

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

5. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 87.8%

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

8. Taylor expanded in a around 0

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

   1. lower-/.f64 64.1

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

10. Applied rewrites 64.1%

    \[\leadsto \frac{-0.5}{b} \cdot c \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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)))