Cubic critical

Percentage Accurate: 51.9% → 88.6%

Time: 4.6s

Alternatives: 6

Speedup: N/A×

Specification

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: 51.9% 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: 88.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ t_1 := \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, t\_1\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-196}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+266}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ (* -1.0 b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
        (t_1 (* (/ (fma -1.0 b (fabs b)) a) 0.3333333333333333)))
   (if (<= t_0 (- INFINITY))
     (fma (/ -0.5 (fabs b)) c t_1)
     (if (<= t_0 -1e-196)
       t_0
       (if (<= t_0 0.0)
         (* (/ c b) -0.5)
         (if (<= t_0 4e+266) t_0 (fma (/ c (fabs b)) -0.5 t_1)))))))

C

double code(double a, double b, double c) {
	double t_0 = ((-1.0 * b) + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double t_1 = (fma(-1.0, b, fabs(b)) / a) * 0.3333333333333333;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-0.5 / fabs(b)), c, t_1);
	} else if (t_0 <= -1e-196) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (c / b) * -0.5;
	} else if (t_0 <= 4e+266) {
		tmp = t_0;
	} else {
		tmp = fma((c / fabs(b)), -0.5, t_1);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(-1.0 * b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	t_1 = Float64(Float64(fma(-1.0, b, abs(b)) / a) * 0.3333333333333333)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(-0.5 / abs(b)), c, t_1);
	elseif (t_0 <= -1e-196)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(c / b) * -0.5);
	elseif (t_0 <= 4e+266)
		tmp = t_0;
	else
		tmp = fma(Float64(c / abs(b)), -0.5, t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(-1.0 * b), $MachinePrecision] + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-1.0 * b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.5 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * c + t$95$1), $MachinePrecision], If[LessEqual[t$95$0, -1e-196], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 4e+266], t$95$0, N[(N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision] * -0.5 + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 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)) < -inf.0

   1. Initial program 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 25.4%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, -1, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.5}\right)}}{3}}{a} \]

   7. Taylor expanded in c around 0

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

   9. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{\left(\left|b\right|\right)}^{3}}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{\left(\left|b\right|\right)}^{5}}\right), c, -0.5 \cdot {\left(\left|b\right|\right)}^{-1}\right), c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lift-fabs.f64 72.7

          \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \]

   12. Applied rewrites 72.7%

       \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \]

   if -inf.0 < (/.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)) < -1e-196 or 0.0 < (/.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)) < 4.0000000000000001e266

   1. Initial program 90.7%

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

   if -1e-196 < (/.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)) < 0.0

   1. Initial program 18.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. 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 99.9

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

   5. Applied rewrites 99.9%

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

   if 4.0000000000000001e266 < (/.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 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 35.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, -1, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.5}\right)}}{3}}{a} \]

   7. Taylor expanded in c around 0

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

   9. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 4 \cdot 10^{+266}:\\ \;\;\;\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ t_1 := \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\\ t_2 := \mathsf{fma}\left({a}^{-1} \cdot 0.3333333333333333, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, \left(c \cdot a\right) \cdot -3\right)\right)}^{0.5}, -0.3333333333333333 \cdot \frac{b}{a}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, t\_1\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{-196}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+266}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (+ (* -1.0 b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
        (t_1 (* (/ (fma -1.0 b (fabs b)) a) 0.3333333333333333))
        (t_2
         (fma
          (* (pow a -1.0) 0.3333333333333333)
          (pow (fma (pow b 1.0) (pow b 1.0) (* (* c a) -3.0)) 0.5)
          (* -0.3333333333333333 (/ b a)))))
   (if (<= t_0 (- INFINITY))
     (fma (/ -0.5 (fabs b)) c t_1)
     (if (<= t_0 -1e-196)
       t_2
       (if (<= t_0 0.0)
         (* (/ c b) -0.5)
         (if (<= t_0 4e+266) t_2 (fma (/ c (fabs b)) -0.5 t_1)))))))

C

double code(double a, double b, double c) {
	double t_0 = ((-1.0 * b) + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double t_1 = (fma(-1.0, b, fabs(b)) / a) * 0.3333333333333333;
	double t_2 = fma((pow(a, -1.0) * 0.3333333333333333), pow(fma(pow(b, 1.0), pow(b, 1.0), ((c * a) * -3.0)), 0.5), (-0.3333333333333333 * (b / a)));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-0.5 / fabs(b)), c, t_1);
	} else if (t_0 <= -1e-196) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = (c / b) * -0.5;
	} else if (t_0 <= 4e+266) {
		tmp = t_2;
	} else {
		tmp = fma((c / fabs(b)), -0.5, t_1);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(Float64(-1.0 * b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	t_1 = Float64(Float64(fma(-1.0, b, abs(b)) / a) * 0.3333333333333333)
	t_2 = fma(Float64((a ^ -1.0) * 0.3333333333333333), (fma((b ^ 1.0), (b ^ 1.0), Float64(Float64(c * a) * -3.0)) ^ 0.5), Float64(-0.3333333333333333 * Float64(b / a)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(Float64(-0.5 / abs(b)), c, t_1);
	elseif (t_0 <= -1e-196)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(c / b) * -0.5);
	elseif (t_0 <= 4e+266)
		tmp = t_2;
	else
		tmp = fma(Float64(c / abs(b)), -0.5, t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[(-1.0 * b), $MachinePrecision] + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-1.0 * b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[a, -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[Power[N[(N[Power[b, 1.0], $MachinePrecision] * N[Power[b, 1.0], $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] + N[(-0.3333333333333333 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(-0.5 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * c + t$95$1), $MachinePrecision], If[LessEqual[t$95$0, -1e-196], t$95$2, If[LessEqual[t$95$0, 0.0], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 4e+266], t$95$2, N[(N[(c / N[Abs[b], $MachinePrecision]), $MachinePrecision] * -0.5 + t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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)) < -inf.0

   1. Initial program 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 25.4%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 25.4%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, -1, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.5}\right)}}{3}}{a} \]

   7. Taylor expanded in c around 0

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

   9. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{\left(\left|b\right|\right)}^{3}}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{\left(\left|b\right|\right)}^{5}}\right), c, -0.5 \cdot {\left(\left|b\right|\right)}^{-1}\right), c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lift-fabs.f64 72.7

          \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \]

   12. Applied rewrites 72.7%

       \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \]

   if -inf.0 < (/.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)) < -1e-196 or 0.0 < (/.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)) < 4.0000000000000001e266

   1. Initial program 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 90.6%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, -1, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.5}\right)}}{3}}{a} \]

   7. Taylor expanded in b around 0

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

   9. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{-1} \cdot 0.3333333333333333, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, \left(c \cdot a\right) \cdot -3\right)\right)}^{0.5}, -0.3333333333333333 \cdot \frac{b}{a}\right)} \]

   if -1e-196 < (/.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)) < 0.0

   1. Initial program 18.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. 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 99.9

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

   5. Applied rewrites 99.9%

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

   if 4.0000000000000001e266 < (/.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 35.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 35.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, -1, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.5}\right)}}{3}}{a} \]

   7. Taylor expanded in c around 0

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

   9. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\mathsf{fma}\left({a}^{-1} \cdot 0.3333333333333333, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, \left(c \cdot a\right) \cdot -3\right)\right)}^{0.5}, -0.3333333333333333 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 4 \cdot 10^{+266}:\\ \;\;\;\;\mathsf{fma}\left({a}^{-1} \cdot 0.3333333333333333, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, \left(c \cdot a\right) \cdot -3\right)\right)}^{0.5}, -0.3333333333333333 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\left|b\right|}, -0.5, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)\\ t_1 := \frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ t_2 := \mathsf{fma}\left({a}^{-1} \cdot 0.3333333333333333, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, \left(c \cdot a\right) \cdot -3\right)\right)}^{0.5}, -0.3333333333333333 \cdot \frac{b}{a}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-196}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+266}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0
         (fma
          (/ -0.5 (fabs b))
          c
          (* (/ (fma -1.0 b (fabs b)) a) 0.3333333333333333)))
        (t_1 (/ (+ (* -1.0 b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
        (t_2
         (fma
          (* (pow a -1.0) 0.3333333333333333)
          (pow (fma (pow b 1.0) (pow b 1.0) (* (* c a) -3.0)) 0.5)
          (* -0.3333333333333333 (/ b a)))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -1e-196)
       t_2
       (if (<= t_1 0.0) (* (/ c b) -0.5) (if (<= t_1 4e+266) t_2 t_0))))))

C

double code(double a, double b, double c) {
	double t_0 = fma((-0.5 / fabs(b)), c, ((fma(-1.0, b, fabs(b)) / a) * 0.3333333333333333));
	double t_1 = ((-1.0 * b) + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	double t_2 = fma((pow(a, -1.0) * 0.3333333333333333), pow(fma(pow(b, 1.0), pow(b, 1.0), ((c * a) * -3.0)), 0.5), (-0.3333333333333333 * (b / a)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -1e-196) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (c / b) * -0.5;
	} else if (t_1 <= 4e+266) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = fma(Float64(-0.5 / abs(b)), c, Float64(Float64(fma(-1.0, b, abs(b)) / a) * 0.3333333333333333))
	t_1 = Float64(Float64(Float64(-1.0 * b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
	t_2 = fma(Float64((a ^ -1.0) * 0.3333333333333333), (fma((b ^ 1.0), (b ^ 1.0), Float64(Float64(c * a) * -3.0)) ^ 0.5), Float64(-0.3333333333333333 * Float64(b / a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -1e-196)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(c / b) * -0.5);
	elseif (t_1 <= 4e+266)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-0.5 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * c + N[(N[(N[(-1.0 * b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(-1.0 * b), $MachinePrecision] + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[a, -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[Power[N[(N[Power[b, 1.0], $MachinePrecision] * N[Power[b, 1.0], $MachinePrecision] + N[(N[(c * a), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] + N[(-0.3333333333333333 * N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -1e-196], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$1, 4e+266], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 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)) < -inf.0 or 4.0000000000000001e266 < (/.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 31.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 31.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 31.9%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, -1, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.5}\right)}}{3}}{a} \]

   7. Taylor expanded in c around 0

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

   9. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{\left(\left|b\right|\right)}^{3}}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{\left(\left|b\right|\right)}^{5}}\right), c, -0.5 \cdot {\left(\left|b\right|\right)}^{-1}\right), c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

       2. lift-fabs.f64 76.0

          \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \]

   12. Applied rewrites 76.0%

       \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \]

   if -inf.0 < (/.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)) < -1e-196 or 0.0 < (/.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)) < 4.0000000000000001e266

   1. Initial program 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   4. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-pow.f64 N/A

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

   6. Applied rewrites 90.6%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, -1, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.5}\right)}}{3}}{a} \]

   7. Taylor expanded in b around 0

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

   9. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{-1} \cdot 0.3333333333333333, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, \left(c \cdot a\right) \cdot -3\right)\right)}^{0.5}, -0.3333333333333333 \cdot \frac{b}{a}\right)} \]

   if -1e-196 < (/.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)) < 0.0

   1. Initial program 18.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. 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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq -1 \cdot 10^{-196}:\\ \;\;\;\;\mathsf{fma}\left({a}^{-1} \cdot 0.3333333333333333, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, \left(c \cdot a\right) \cdot -3\right)\right)}^{0.5}, -0.3333333333333333 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 0:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \mathbf{elif}\;\frac{-1 \cdot b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \leq 4 \cdot 10^{+266}:\\ \;\;\;\;\mathsf{fma}\left({a}^{-1} \cdot 0.3333333333333333, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, \left(c \cdot a\right) \cdot -3\right)\right)}^{0.5}, -0.3333333333333333 \cdot \frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.4%

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

   3. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 67.3

         \[\leadsto -0.6666666666666666 \cdot \frac{b}{\color{blue}{a}} \]

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 39.1%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   4. 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 56.0

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

   5. Applied rewrites 56.0%

      \[\leadsto \color{blue}{\frac{c}{b} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 66.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma (/ -0.5 (fabs b)) c (* (/ (fma -1.0 b (fabs b)) a) 0.3333333333333333)))

C

double code(double a, double b, double c) {
	return fma((-0.5 / fabs(b)), c, ((fma(-1.0, b, fabs(b)) / a) * 0.3333333333333333));
}

Julia

function code(a, b, c)
	return fma(Float64(-0.5 / abs(b)), c, Float64(Float64(fma(-1.0, b, abs(b)) / a) * 0.3333333333333333))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 / N[Abs[b], $MachinePrecision]), $MachinePrecision] * c + N[(N[(N[(-1.0 * b + N[Abs[b], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 58.0%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, {\left(\mathsf{fma}\left({b}^{1}, {b}^{1}, -3 \cdot \left(c \cdot a\right)\right)\right)}^{0.5}\right)}{3}}{a}} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-pow.f64 N/A

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

6. Applied rewrites 58.1%

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(b, -1, {\left(\mathsf{fma}\left(\left|b\right|, \left|b\right|, \left(a \cdot -3\right) \cdot c\right)\right)}^{0.5}\right)}}{3}}{a} \]

7. Taylor expanded in c around 0

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

9. Applied rewrites 46.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{\left(\left|b\right|\right)}^{3}}, -0.375, \frac{-0.5625 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{{\left(\left|b\right|\right)}^{5}}\right), c, -0.5 \cdot {\left(\left|b\right|\right)}^{-1}\right), c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right)} \]

10. Taylor expanded in a around 0

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

    1. lower-/.f64 N/A

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

    2. lift-fabs.f64 62.0

       \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \]

12. Applied rewrites 62.0%

    \[\leadsto \mathsf{fma}\left(\frac{-0.5}{\left|b\right|}, c, \frac{\mathsf{fma}\left(-1, b, \left|b\right|\right)}{a} \cdot 0.3333333333333333\right) \]
13. Add Preprocessing

Alternative 6: 33.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ -0.6666666666666666 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.6666666666666666 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / 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 = (-0.6666666666666666d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Python

def code(a, b, c):
	return -0.6666666666666666 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.6666666666666666 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.6666666666666666 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.1%

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

3. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 37.4

      \[\leadsto -0.6666666666666666 \cdot \frac{b}{\color{blue}{a}} \]

5. Applied rewrites 37.4%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))