Cubic critical, narrow range

Percentage Accurate: 54.6% → 91.9%

Time: 5.7s

Alternatives: 13

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: 54.6% 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: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ t_1 := \frac{{a}^{4}}{{b}^{6}}\\ t_2 := \frac{{a}^{2}}{{b}^{3}}\\ t_3 := \mathsf{fma}\left(-0.75, t\_2, 0.375 \cdot t\_2\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{t\_0 \cdot t\_0 - b \cdot b}{t\_0 - \left(-b\right)} \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(1.5, \frac{a}{b}, c \cdot \mathsf{fma}\left(-3, c \cdot \mathsf{fma}\left(-0.75, \frac{a \cdot t\_3}{{b}^{2}}, \mathsf{fma}\left(-0.2222222222222222, \frac{b \cdot \mathsf{fma}\left(1.265625, t\_1, 5.0625 \cdot t\_1\right)}{a}, 0.5625 \cdot \frac{{a}^{3}}{{b}^{5}}\right)\right), -3 \cdot t\_3\right)\right)\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b))))
        (t_1 (/ (pow a 4.0) (pow b 6.0)))
        (t_2 (/ (pow a 2.0) (pow b 3.0)))
        (t_3 (fma -0.75 t_2 (* 0.375 t_2))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (* (/ (- (* t_0 t_0) (* b b)) (- t_0 (- b))) (/ 1.0 (* a 3.0)))
     (/
      1.0
      (/
       (fma
        -2.0
        b
        (*
         c
         (fma
          1.5
          (/ a b)
          (*
           c
           (fma
            -3.0
            (*
             c
             (fma
              -0.75
              (/ (* a t_3) (pow b 2.0))
              (fma
               -0.2222222222222222
               (/ (* b (fma 1.265625 t_1 (* 5.0625 t_1))) a)
               (* 0.5625 (/ (pow a 3.0) (pow b 5.0))))))
            (* -3.0 t_3))))))
       c)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double t_1 = pow(a, 4.0) / pow(b, 6.0);
	double t_2 = pow(a, 2.0) / pow(b, 3.0);
	double t_3 = fma(-0.75, t_2, (0.375 * t_2));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (((t_0 * t_0) - (b * b)) / (t_0 - -b)) * (1.0 / (a * 3.0));
	} else {
		tmp = 1.0 / (fma(-2.0, b, (c * fma(1.5, (a / b), (c * fma(-3.0, (c * fma(-0.75, ((a * t_3) / pow(b, 2.0)), fma(-0.2222222222222222, ((b * fma(1.265625, t_1, (5.0625 * t_1))) / a), (0.5625 * (pow(a, 3.0) / pow(b, 5.0)))))), (-3.0 * t_3)))))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	t_1 = Float64((a ^ 4.0) / (b ^ 6.0))
	t_2 = Float64((a ^ 2.0) / (b ^ 3.0))
	t_3 = fma(-0.75, t_2, Float64(0.375 * t_2))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) - Float64(b * b)) / Float64(t_0 - Float64(-b))) * Float64(1.0 / Float64(a * 3.0)));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(c * fma(1.5, Float64(a / b), Float64(c * fma(-3.0, Float64(c * fma(-0.75, Float64(Float64(a * t_3) / (b ^ 2.0)), fma(-0.2222222222222222, Float64(Float64(b * fma(1.265625, t_1, Float64(5.0625 * t_1))) / a), Float64(0.5625 * Float64((a ^ 3.0) / (b ^ 5.0)))))), Float64(-3.0 * t_3)))))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.75 * t$95$2 + N[(0.375 * t$95$2), $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], -0.27], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - (-b)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(c * N[(1.5 * N[(a / b), $MachinePrecision] + N[(c * N[(-3.0 * N[(c * N[(-0.75 * N[(N[(a * t$95$3), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.2222222222222222 * N[(N[(b * N[(1.265625 * t$95$1 + N[(5.0625 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] + N[(0.5625 * N[(N[Power[a, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-3.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $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)) < -0.27000000000000002

   1. Initial program 54.6%

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

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 54.6%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-special-/ N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 54.6%

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

   if -0.27000000000000002 < (/.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 54.6%

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

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in c around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + c \cdot \left(\frac{3}{2} \cdot \frac{a}{b} + c \cdot \left(-3 \cdot \left(c \cdot \left(\frac{-3}{4} \cdot \frac{a \cdot \left(\frac{-3}{4} \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{3}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{a}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a} + \frac{9}{16} \cdot \frac{{a}^{3}}{{b}^{5}}\right)\right)\right) + -3 \cdot \left(\frac{-3}{4} \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{3}{8} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right)}{c}}} \]

   6. Applied rewrites 91.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, c \cdot \mathsf{fma}\left(1.5, \frac{a}{b}, c \cdot \mathsf{fma}\left(-3, c \cdot \mathsf{fma}\left(-0.75, \frac{a \cdot \mathsf{fma}\left(-0.75, \frac{{a}^{2}}{{b}^{3}}, 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)}{{b}^{2}}, \mathsf{fma}\left(-0.2222222222222222, \frac{b \cdot \mathsf{fma}\left(1.265625, \frac{{a}^{4}}{{b}^{6}}, 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a}, 0.5625 \cdot \frac{{a}^{3}}{{b}^{5}}\right)\right), -3 \cdot \mathsf{fma}\left(-0.75, \frac{{a}^{2}}{{b}^{3}}, 0.375 \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right)\right)}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 91.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ t_1 := \frac{{c}^{4}}{{b}^{6}}\\ t_2 := \frac{c}{{b}^{3}}\\ t_3 := \mathsf{fma}\left(-0.75, t\_2, 0.375 \cdot t\_2\right)\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{t\_0 \cdot t\_0 - b \cdot b}{t\_0 - \left(-b\right)} \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-3, a \cdot \mathsf{fma}\left(-0.75, \frac{c \cdot t\_3}{{b}^{2}}, \mathsf{fma}\left(-0.2222222222222222, \frac{b \cdot \mathsf{fma}\left(1.265625, t\_1, 5.0625 \cdot t\_1\right)}{{c}^{2}}, 0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right), -3 \cdot t\_3\right), 1.5 \cdot \frac{1}{b}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b))))
        (t_1 (/ (pow c 4.0) (pow b 6.0)))
        (t_2 (/ c (pow b 3.0)))
        (t_3 (fma -0.75 t_2 (* 0.375 t_2))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (* (/ (- (* t_0 t_0) (* b b)) (- t_0 (- b))) (/ 1.0 (* a 3.0)))
     (/
      1.0
      (fma
       -2.0
       (/ b c)
       (*
        a
        (fma
         a
         (fma
          -3.0
          (*
           a
           (fma
            -0.75
            (/ (* c t_3) (pow b 2.0))
            (fma
             -0.2222222222222222
             (/ (* b (fma 1.265625 t_1 (* 5.0625 t_1))) (pow c 2.0))
             (* 0.5625 (/ (pow c 2.0) (pow b 5.0))))))
          (* -3.0 t_3))
         (* 1.5 (/ 1.0 b)))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double t_1 = pow(c, 4.0) / pow(b, 6.0);
	double t_2 = c / pow(b, 3.0);
	double t_3 = fma(-0.75, t_2, (0.375 * t_2));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (((t_0 * t_0) - (b * b)) / (t_0 - -b)) * (1.0 / (a * 3.0));
	} else {
		tmp = 1.0 / fma(-2.0, (b / c), (a * fma(a, fma(-3.0, (a * fma(-0.75, ((c * t_3) / pow(b, 2.0)), fma(-0.2222222222222222, ((b * fma(1.265625, t_1, (5.0625 * t_1))) / pow(c, 2.0)), (0.5625 * (pow(c, 2.0) / pow(b, 5.0)))))), (-3.0 * t_3)), (1.5 * (1.0 / b)))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-3.0 * a), c, Float64(b * b)))
	t_1 = Float64((c ^ 4.0) / (b ^ 6.0))
	t_2 = Float64(c / (b ^ 3.0))
	t_3 = fma(-0.75, t_2, Float64(0.375 * t_2))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a)) <= -0.27)
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) - Float64(b * b)) / Float64(t_0 - Float64(-b))) * Float64(1.0 / Float64(a * 3.0)));
	else
		tmp = Float64(1.0 / fma(-2.0, Float64(b / c), Float64(a * fma(a, fma(-3.0, Float64(a * fma(-0.75, Float64(Float64(c * t_3) / (b ^ 2.0)), fma(-0.2222222222222222, Float64(Float64(b * fma(1.265625, t_1, Float64(5.0625 * t_1))) / (c ^ 2.0)), Float64(0.5625 * Float64((c ^ 2.0) / (b ^ 5.0)))))), Float64(-3.0 * t_3)), Float64(1.5 * Float64(1.0 / b))))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[c, 4.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.75 * t$95$2 + N[(0.375 * t$95$2), $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], -0.27], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - (-b)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(a * N[(a * N[(-3.0 * N[(a * N[(-0.75 * N[(N[(c * t$95$3), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.2222222222222222 * N[(N[(b * N[(1.265625 * t$95$1 + N[(5.0625 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5625 * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-3.0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.27000000000000002

   1. Initial program 54.6%

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

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 54.6%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-special-/ N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 54.6%

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

   if -0.27000000000000002 < (/.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 54.6%

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

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(a \cdot \left(-3 \cdot \left(a \cdot \left(\frac{-3}{4} \cdot \frac{c \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}} + \left(\frac{-2}{9} \cdot \frac{b \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}} + \frac{9}{16} \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right)\right) + -3 \cdot \left(\frac{-3}{4} \cdot \frac{c}{{b}^{3}} + \frac{3}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}} \]

   6. Applied rewrites 91.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(-3, a \cdot \mathsf{fma}\left(-0.75, \frac{c \cdot \mathsf{fma}\left(-0.75, \frac{c}{{b}^{3}}, 0.375 \cdot \frac{c}{{b}^{3}}\right)}{{b}^{2}}, \mathsf{fma}\left(-0.2222222222222222, \frac{b \cdot \mathsf{fma}\left(1.265625, \frac{{c}^{4}}{{b}^{6}}, 5.0625 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{{c}^{2}}, 0.5625 \cdot \frac{{c}^{2}}{{b}^{5}}\right)\right), -3 \cdot \mathsf{fma}\left(-0.75, \frac{c}{{b}^{3}}, 0.375 \cdot \frac{c}{{b}^{3}}\right)\right), 1.5 \cdot \frac{1}{b}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{t\_0 \cdot t\_0 - b \cdot b}{t\_0 - \left(-b\right)} \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-1.0546875, \left(a \cdot a\right) \cdot \left(c \cdot c\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(-0.5625, a \cdot c, -0.375 \cdot \left(b \cdot b\right)\right)\right)}{{b}^{7}} \cdot \left(c \cdot c\right), \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (* (/ (- (* t_0 t_0) (* b b)) (- t_0 (- b))) (/ 1.0 (* a 3.0)))
     (fma
      a
      (*
       (/
        (fma
         -1.0546875
         (* (* a a) (* c c))
         (* (* b b) (fma -0.5625 (* a c) (* -0.375 (* b b)))))
        (pow b 7.0))
       (* c c))
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (((t_0 * t_0) - (b * b)) / (t_0 - -b)) * (1.0 / (a * 3.0));
	} else {
		tmp = fma(a, ((fma(-1.0546875, ((a * a) * (c * c)), ((b * b) * fma(-0.5625, (a * c), (-0.375 * (b * b))))) / pow(b, 7.0)) * (c * c)), ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(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)) <= -0.27)
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) - Float64(b * b)) / Float64(t_0 - Float64(-b))) * Float64(1.0 / Float64(a * 3.0)));
	else
		tmp = fma(a, Float64(Float64(fma(-1.0546875, Float64(Float64(a * a) * Float64(c * c)), Float64(Float64(b * b) * fma(-0.5625, Float64(a * c), Float64(-0.375 * Float64(b * b))))) / (b ^ 7.0)) * Float64(c * c)), Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - (-b)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(-1.0546875 * N[(N[(a * a), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(-0.5625 * N[(a * c), $MachinePrecision] + N[(-0.375 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + 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)) < -0.27000000000000002

   1. Initial program 54.6%

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

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 54.6%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-special-/ N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 54.6%

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

   if -0.27000000000000002 < (/.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 54.6%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(a, {c}^{2} \cdot \color{blue}{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 91.3%

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

Alternative 4: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{t\_0 \cdot t\_0 - b \cdot b}{t\_0 - \left(-b\right)} \cdot \frac{1}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{\left(b \cdot b\right) \cdot b} \cdot \left(c \cdot c\right), \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (* (/ (- (* t_0 t_0) (* b b)) (- t_0 (- b))) (/ 1.0 (* a 3.0)))
     (fma
      a
      (* (/ (fma -0.5625 (* a (/ c (* b b))) -0.375) (* (* b b) b)) (* c c))
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (((t_0 * t_0) - (b * b)) / (t_0 - -b)) * (1.0 / (a * 3.0));
	} else {
		tmp = fma(a, ((fma(-0.5625, (a * (c / (b * b))), -0.375) / ((b * b) * b)) * (c * c)), ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(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)) <= -0.27)
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) - Float64(b * b)) / Float64(t_0 - Float64(-b))) * Float64(1.0 / Float64(a * 3.0)));
	else
		tmp = fma(a, Float64(Float64(fma(-0.5625, Float64(a * Float64(c / Float64(b * b))), -0.375) / Float64(Float64(b * b) * b)) * Float64(c * c)), Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(b * b), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - (-b)), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(-0.5625 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.375), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + 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)) < -0.27000000000000002

   1. Initial program 54.6%

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

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 54.6%

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

      1. lift-+.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. flip-+ N/A

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

      8. lower-special-/ N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 54.6%

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

   if -0.27000000000000002 < (/.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 54.6%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(a, {c}^{2} \cdot \color{blue}{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

Alternative 5: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{\left(b \cdot b\right) \cdot b} \cdot \left(c \cdot c\right), \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -3.0 a) c (* b b)))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 3.0 a))
     (fma
      a
      (* (/ (fma -0.5625 (* a (/ c (* b b))) -0.375) (* (* b b) b)) (* c c))
      (* (/ c b) -0.5)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-3.0 * a), c, (b * b)));
	double tmp;
	if (((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)) <= -0.27) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (3.0 * a);
	} else {
		tmp = fma(a, ((fma(-0.5625, (a * (c / (b * b))), -0.375) / ((b * b) * b)) * (c * c)), ((c / b) * -0.5));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(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)) <= -0.27)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(3.0 * a));
	else
		tmp = fma(a, Float64(Float64(fma(-0.5625, Float64(a * Float64(c / Float64(b * b))), -0.375) / Float64(Float64(b * b) * b)) * Float64(c * c)), Float64(Float64(c / b) * -0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.27], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(-0.5625 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.375), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + 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)) < -0.27000000000000002

   1. Initial program 54.6%

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

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

      9. 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 54.6%

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

   if -0.27000000000000002 < (/.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 54.6%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(a, {c}^{2} \cdot \color{blue}{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

Alternative 6: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -0.27:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-a \cdot 3\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, a \cdot \frac{c}{b \cdot b}, -0.375\right)}{\left(b \cdot b\right) \cdot b} \cdot \left(c \cdot c\right), \frac{c}{b} \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)) -0.27)
   (/ (+ (- b) (sqrt (fma b b (* (- (* a 3.0)) c)))) (* 3.0 a))
   (fma
    a
    (* (/ (fma -0.5625 (* a (/ c (* b b))) -0.375) (* (* b b) b)) (* c c))
    (* (/ 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)) <= -0.27) {
		tmp = (-b + sqrt(fma(b, b, (-(a * 3.0) * c)))) / (3.0 * a);
	} else {
		tmp = fma(a, ((fma(-0.5625, (a * (c / (b * b))), -0.375) / ((b * b) * b)) * (c * c)), ((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)) <= -0.27)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-Float64(a * 3.0)) * c)))) / Float64(3.0 * a));
	else
		tmp = fma(a, Float64(Float64(fma(-0.5625, Float64(a * Float64(c / Float64(b * b))), -0.375) / Float64(Float64(b * b) * b)) * Float64(c * c)), 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], -0.27], N[(N[((-b) + N[Sqrt[N[(b * b + N[((-N[(a * 3.0), $MachinePrecision]) * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(-0.5625 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.375), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] + 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)) < -0.27000000000000002

   1. Initial program 54.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. fp-cancel-sub-sign-inv N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.7

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

   3. Applied rewrites 54.7%

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

   if -0.27000000000000002 < (/.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 54.6%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(a, {c}^{2} \cdot \color{blue}{\left(c \cdot \left(\frac{-135}{128} \cdot \frac{{a}^{2} \cdot c}{{b}^{7}} + \frac{-9}{16} \cdot \frac{a}{{b}^{5}}\right) - \frac{3}{8} \cdot \frac{1}{{b}^{3}}\right)}, \frac{c}{b} \cdot \frac{-1}{2}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 91.3%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. sub-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

Alternative 7: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 470:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-a \cdot 3\right) \cdot c\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 470.0)
   (/ (+ (- b) (sqrt (fma b b (* (- (* a 3.0)) c)))) (* 3.0 a))
   (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 470.0) {
		tmp = (-b + sqrt(fma(b, b, (-(a * 3.0) * c)))) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 470.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-Float64(a * 3.0)) * c)))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 470.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[((-N[(a * 3.0), $MachinePrecision]) * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 470

   1. Initial program 54.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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. fp-cancel-sub-sign-inv N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 54.7

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

   3. Applied rewrites 54.7%

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

   if 470 < b

   1. Initial program 54.6%

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

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 82.8

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

   6. Applied rewrites 82.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 470:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 470.0)
   (/ (+ (- b) (sqrt (fma b b (* -3.0 (* c a))))) (* 3.0 a))
   (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 470.0) {
		tmp = (-b + sqrt(fma(b, b, (-3.0 * (c * a))))) / (3.0 * a);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 470.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-3.0 * Float64(c * a))))) / Float64(3.0 * a));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 470.0], 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[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 470

   1. Initial program 54.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   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 54.7

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

   3. Applied rewrites 54.7%

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

   if 470 < b

   1. Initial program 54.6%

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

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 82.8

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

   6. Applied rewrites 82.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 470:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 470.0)
   (* (+ (sqrt (fma (* -3.0 a) c (* b b))) (- b)) (/ 0.3333333333333333 a))
   (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 470.0) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) + -b) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 470.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) + Float64(-b)) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 470.0], N[(N[(N[Sqrt[N[(N[(-3.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 470

   1. Initial program 54.6%

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

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 54.6

         \[\leadsto \left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)\right) \cdot \frac{0.3333333333333333}{\color{blue}{a}} \]

   6. Applied rewrites 54.6%

      \[\leadsto \left(\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} + \left(-b\right)\right) \cdot \color{blue}{\frac{0.3333333333333333}{a}} \]

   if 470 < b

   1. Initial program 54.6%

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

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 82.8

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

   6. Applied rewrites 82.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 470:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot a, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 470.0)
   (/ (- (sqrt (fma (* -3.0 a) c (* b b))) b) (* a 3.0))
   (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 470.0) {
		tmp = (sqrt(fma((-3.0 * a), c, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 470.0)
		tmp = Float64(Float64(sqrt(fma(Float64(-3.0 * a), c, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 470.0], 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[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 470

   1. Initial program 54.6%

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

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 54.6%

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

   if 470 < b

   1. Initial program 54.6%

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

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-special-/ N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 54.6%

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

   4. Taylor expanded in c around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 82.8

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

   6. Applied rewrites 82.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 82.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (fma -2.0 b (* 1.5 (/ (* a c) b))) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (fma(-2.0, b, (1.5 * ((a * c) / b))) / c);
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(fma(-2.0, b, Float64(1.5 * Float64(Float64(a * c) / b))) / c))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * b + N[(1.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

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

   11. lower-special-/ N/A

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

   12. lower-/.f64 N/A

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

   13. lower-special-/ N/A

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

   14. lower-/.f64 N/A

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

3. Applied rewrites 54.6%

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

4. Taylor expanded in c around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lift-*.f64 82.8

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}} \]

6. Applied rewrites 82.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(-2, b, 1.5 \cdot \frac{a \cdot c}{b}\right)}{c}}} \]
7. Add Preprocessing

Alternative 12: 82.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (fma -2.0 (/ b c) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(-2.0, (b / c), (1.5 * (a / b)));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(-2.0, Float64(b / c), Float64(1.5 * Float64(a / b))))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(-2.0 * N[(b / c), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

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

   11. lower-special-/ N/A

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

   12. lower-/.f64 N/A

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

   13. lower-special-/ N/A

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

   14. lower-/.f64 N/A

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

3. Applied rewrites 54.6%

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

4. Taylor expanded in a around 0

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

   1. lower-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 82.8

      \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)} \]

6. Applied rewrites 82.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(-2, \frac{b}{c}, 1.5 \cdot \frac{a}{b}\right)}} \]
7. Add Preprocessing

Alternative 13: 65.1% 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 54.6%

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

2. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 65.1

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

4. Applied rewrites 65.1%

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

Reproduce

herbie shell --seed 2025132 
(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)))