Quadratic roots, narrow range

Percentage Accurate: 54.8% → 92.2%

Time: 14.9s

Alternatives: 15

Speedup: 3.6×

Specification

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

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Initial Program: 54.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

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

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

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

Alternative 1: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot 4\right) \cdot c\\ t_1 := \sqrt{b \cdot b - t\_0}\\ \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_1}^{3}}{b \cdot b + \left(t\_1 \cdot \sqrt{e^{\log b \cdot 2} - t\_0} + b \cdot t\_1\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a 4.0) c)) (t_1 (sqrt (- (* b b) t_0))))
   (if (<= b 0.075)
     (/
      (/
       (+ (pow (- b) 3.0) (pow t_1 3.0))
       (+ (* b b) (+ (* t_1 (sqrt (- (exp (* (log b) 2.0)) t_0))) (* b t_1))))
      (* 2.0 a))
     (/
      (-
       (*
        a
        (-
         (*
          a
          (+
           (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
           (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
         (ratio-of-squares c b)))
       c)
      b))))

Derivation

1. Split input into 2 regimes

2. if b < 0.0749999999999999972

   1. Initial program 91.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 92.5

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

   6. Applied rewrites 92.5%

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

   if 0.0749999999999999972 < b

   1. Initial program 51.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b}} \]

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]

   8. Applied rewrites 94.3%

      \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}^{3}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{e^{\log b \cdot 2} - \left(a \cdot 4\right) \cdot c} + b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c}{b \cdot b}\\ t_1 := {\left(a \cdot c\right)}^{2}\\ t_2 := {\left(a \cdot c\right)}^{3}\\ t_3 := -8 \cdot a + -4 \cdot a\\ t_4 := -8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\\ t_5 := \left(16 \cdot \left(a \cdot a\right) + 32 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {t\_3}^{2}\\ t_6 := \left(16 \cdot t\_1 + 32 \cdot t\_1\right) - 0.25 \cdot {t\_4}^{2}\\ \frac{\frac{b \cdot \left(-0.5 \cdot \frac{0.25 \cdot {t\_6}^{2} + 0.5 \cdot \left(t\_4 \cdot \left(-64 \cdot t\_2 - 0.5 \cdot \left(t\_4 \cdot t\_6\right)\right)\right)}{{b}^{6}} + \left(0.5 \cdot t\_4 + \left(c \cdot c\right) \cdot \left(0.5 \cdot \frac{c \cdot \left(-64 \cdot {a}^{3} - 0.5 \cdot \left(t\_3 \cdot t\_5\right)\right)}{{b}^{4}} + 0.5 \cdot \frac{t\_5}{b \cdot b}\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{t\_2}{{b}^{6}}\right) - \left(2 \cdot t\_0 + \left(2 \cdot \frac{t\_1}{{b}^{4}} + 4 \cdot t\_0\right)\right)\right)}}{2 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a c) (* b b)))
        (t_1 (pow (* a c) 2.0))
        (t_2 (pow (* a c) 3.0))
        (t_3 (+ (* -8.0 a) (* -4.0 a)))
        (t_4 (+ (* -8.0 (* a c)) (* -4.0 (* a c))))
        (t_5 (- (+ (* 16.0 (* a a)) (* 32.0 (* a a))) (* 0.25 (pow t_3 2.0))))
        (t_6 (- (+ (* 16.0 t_1) (* 32.0 t_1)) (* 0.25 (pow t_4 2.0)))))
   (/
    (/
     (*
      b
      (+
       (*
        -0.5
        (/
         (+
          (* 0.25 (pow t_6 2.0))
          (* 0.5 (* t_4 (- (* -64.0 t_2) (* 0.5 (* t_4 t_6))))))
         (pow b 6.0)))
       (+
        (* 0.5 t_4)
        (*
         (* c c)
         (+
          (*
           0.5
           (/ (* c (- (* -64.0 (pow a 3.0)) (* 0.5 (* t_3 t_5)))) (pow b 4.0)))
          (* 0.5 (/ t_5 (* b b))))))))
     (*
      (* b b)
      (-
       (+ 3.0 (* -4.0 (/ t_2 (pow b 6.0))))
       (+ (* 2.0 t_0) (+ (* 2.0 (/ t_1 (pow b 4.0))) (* 4.0 t_0))))))
    (* 2.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) / (b * b);
	double t_1 = pow((a * c), 2.0);
	double t_2 = pow((a * c), 3.0);
	double t_3 = (-8.0 * a) + (-4.0 * a);
	double t_4 = (-8.0 * (a * c)) + (-4.0 * (a * c));
	double t_5 = ((16.0 * (a * a)) + (32.0 * (a * a))) - (0.25 * pow(t_3, 2.0));
	double t_6 = ((16.0 * t_1) + (32.0 * t_1)) - (0.25 * pow(t_4, 2.0));
	return ((b * ((-0.5 * (((0.25 * pow(t_6, 2.0)) + (0.5 * (t_4 * ((-64.0 * t_2) - (0.5 * (t_4 * t_6)))))) / pow(b, 6.0))) + ((0.5 * t_4) + ((c * c) * ((0.5 * ((c * ((-64.0 * pow(a, 3.0)) - (0.5 * (t_3 * t_5)))) / pow(b, 4.0))) + (0.5 * (t_5 / (b * b)))))))) / ((b * b) * ((3.0 + (-4.0 * (t_2 / pow(b, 6.0)))) - ((2.0 * t_0) + ((2.0 * (t_1 / pow(b, 4.0))) + (4.0 * t_0)))))) / (2.0 * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    t_0 = (a * c) / (b * b)
    t_1 = (a * c) ** 2.0d0
    t_2 = (a * c) ** 3.0d0
    t_3 = ((-8.0d0) * a) + ((-4.0d0) * a)
    t_4 = ((-8.0d0) * (a * c)) + ((-4.0d0) * (a * c))
    t_5 = ((16.0d0 * (a * a)) + (32.0d0 * (a * a))) - (0.25d0 * (t_3 ** 2.0d0))
    t_6 = ((16.0d0 * t_1) + (32.0d0 * t_1)) - (0.25d0 * (t_4 ** 2.0d0))
    code = ((b * (((-0.5d0) * (((0.25d0 * (t_6 ** 2.0d0)) + (0.5d0 * (t_4 * (((-64.0d0) * t_2) - (0.5d0 * (t_4 * t_6)))))) / (b ** 6.0d0))) + ((0.5d0 * t_4) + ((c * c) * ((0.5d0 * ((c * (((-64.0d0) * (a ** 3.0d0)) - (0.5d0 * (t_3 * t_5)))) / (b ** 4.0d0))) + (0.5d0 * (t_5 / (b * b)))))))) / ((b * b) * ((3.0d0 + ((-4.0d0) * (t_2 / (b ** 6.0d0)))) - ((2.0d0 * t_0) + ((2.0d0 * (t_1 / (b ** 4.0d0))) + (4.0d0 * t_0)))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) / (b * b);
	double t_1 = Math.pow((a * c), 2.0);
	double t_2 = Math.pow((a * c), 3.0);
	double t_3 = (-8.0 * a) + (-4.0 * a);
	double t_4 = (-8.0 * (a * c)) + (-4.0 * (a * c));
	double t_5 = ((16.0 * (a * a)) + (32.0 * (a * a))) - (0.25 * Math.pow(t_3, 2.0));
	double t_6 = ((16.0 * t_1) + (32.0 * t_1)) - (0.25 * Math.pow(t_4, 2.0));
	return ((b * ((-0.5 * (((0.25 * Math.pow(t_6, 2.0)) + (0.5 * (t_4 * ((-64.0 * t_2) - (0.5 * (t_4 * t_6)))))) / Math.pow(b, 6.0))) + ((0.5 * t_4) + ((c * c) * ((0.5 * ((c * ((-64.0 * Math.pow(a, 3.0)) - (0.5 * (t_3 * t_5)))) / Math.pow(b, 4.0))) + (0.5 * (t_5 / (b * b)))))))) / ((b * b) * ((3.0 + (-4.0 * (t_2 / Math.pow(b, 6.0)))) - ((2.0 * t_0) + ((2.0 * (t_1 / Math.pow(b, 4.0))) + (4.0 * t_0)))))) / (2.0 * a);
}

Python

def code(a, b, c):
	t_0 = (a * c) / (b * b)
	t_1 = math.pow((a * c), 2.0)
	t_2 = math.pow((a * c), 3.0)
	t_3 = (-8.0 * a) + (-4.0 * a)
	t_4 = (-8.0 * (a * c)) + (-4.0 * (a * c))
	t_5 = ((16.0 * (a * a)) + (32.0 * (a * a))) - (0.25 * math.pow(t_3, 2.0))
	t_6 = ((16.0 * t_1) + (32.0 * t_1)) - (0.25 * math.pow(t_4, 2.0))
	return ((b * ((-0.5 * (((0.25 * math.pow(t_6, 2.0)) + (0.5 * (t_4 * ((-64.0 * t_2) - (0.5 * (t_4 * t_6)))))) / math.pow(b, 6.0))) + ((0.5 * t_4) + ((c * c) * ((0.5 * ((c * ((-64.0 * math.pow(a, 3.0)) - (0.5 * (t_3 * t_5)))) / math.pow(b, 4.0))) + (0.5 * (t_5 / (b * b)))))))) / ((b * b) * ((3.0 + (-4.0 * (t_2 / math.pow(b, 6.0)))) - ((2.0 * t_0) + ((2.0 * (t_1 / math.pow(b, 4.0))) + (4.0 * t_0)))))) / (2.0 * a)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) / Float64(b * b))
	t_1 = Float64(a * c) ^ 2.0
	t_2 = Float64(a * c) ^ 3.0
	t_3 = Float64(Float64(-8.0 * a) + Float64(-4.0 * a))
	t_4 = Float64(Float64(-8.0 * Float64(a * c)) + Float64(-4.0 * Float64(a * c)))
	t_5 = Float64(Float64(Float64(16.0 * Float64(a * a)) + Float64(32.0 * Float64(a * a))) - Float64(0.25 * (t_3 ^ 2.0)))
	t_6 = Float64(Float64(Float64(16.0 * t_1) + Float64(32.0 * t_1)) - Float64(0.25 * (t_4 ^ 2.0)))
	return Float64(Float64(Float64(b * Float64(Float64(-0.5 * Float64(Float64(Float64(0.25 * (t_6 ^ 2.0)) + Float64(0.5 * Float64(t_4 * Float64(Float64(-64.0 * t_2) - Float64(0.5 * Float64(t_4 * t_6)))))) / (b ^ 6.0))) + Float64(Float64(0.5 * t_4) + Float64(Float64(c * c) * Float64(Float64(0.5 * Float64(Float64(c * Float64(Float64(-64.0 * (a ^ 3.0)) - Float64(0.5 * Float64(t_3 * t_5)))) / (b ^ 4.0))) + Float64(0.5 * Float64(t_5 / Float64(b * b)))))))) / Float64(Float64(b * b) * Float64(Float64(3.0 + Float64(-4.0 * Float64(t_2 / (b ^ 6.0)))) - Float64(Float64(2.0 * t_0) + Float64(Float64(2.0 * Float64(t_1 / (b ^ 4.0))) + Float64(4.0 * t_0)))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (a * c) / (b * b);
	t_1 = (a * c) ^ 2.0;
	t_2 = (a * c) ^ 3.0;
	t_3 = (-8.0 * a) + (-4.0 * a);
	t_4 = (-8.0 * (a * c)) + (-4.0 * (a * c));
	t_5 = ((16.0 * (a * a)) + (32.0 * (a * a))) - (0.25 * (t_3 ^ 2.0));
	t_6 = ((16.0 * t_1) + (32.0 * t_1)) - (0.25 * (t_4 ^ 2.0));
	tmp = ((b * ((-0.5 * (((0.25 * (t_6 ^ 2.0)) + (0.5 * (t_4 * ((-64.0 * t_2) - (0.5 * (t_4 * t_6)))))) / (b ^ 6.0))) + ((0.5 * t_4) + ((c * c) * ((0.5 * ((c * ((-64.0 * (a ^ 3.0)) - (0.5 * (t_3 * t_5)))) / (b ^ 4.0))) + (0.5 * (t_5 / (b * b)))))))) / ((b * b) * ((3.0 + (-4.0 * (t_2 / (b ^ 6.0)))) - ((2.0 * t_0) + ((2.0 * (t_1 / (b ^ 4.0))) + (4.0 * t_0)))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-8.0 * a), $MachinePrecision] + N[(-4.0 * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(-8.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(16.0 * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(32.0 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(16.0 * t$95$1), $MachinePrecision] + N[(32.0 * t$95$1), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(b * N[(N[(-0.5 * N[(N[(N[(0.25 * N[Power[t$95$6, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$4 * N[(N[(-64.0 * t$95$2), $MachinePrecision] - N[(0.5 * N[(t$95$4 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * t$95$4), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(N[(0.5 * N[(N[(c * N[(N[(-64.0 * N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$3 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$5 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(N[(3.0 + N[(-4.0 * N[(t$95$2 / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * t$95$0), $MachinePrecision] + N[(N[(2.0 * N[(t$95$1 / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 53.4%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. flip3-+ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 53.2%

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

5. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]

7. Applied rewrites 92.7%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-0.5 \cdot \frac{0.25 \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(0.5 \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(0.5 \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + 0.5 \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\color{blue}{{b}^{2} \cdot \left(\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}}}{2 \cdot a} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{{b}^{2} \cdot \color{blue}{\left(\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}}}{2 \cdot a} \]

   2. pow2 N/A

      \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\color{blue}{\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)} - \left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}}{2 \cdot a} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\color{blue}{\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)} - \left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}}{2 \cdot a} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right) - \color{blue}{\left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}\right)}}{2 \cdot a} \]

10. Applied rewrites 92.9%

    \[\leadsto \frac{\frac{b \cdot \left(-0.5 \cdot \frac{0.25 \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(0.5 \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(0.5 \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + 0.5 \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\color{blue}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{b \cdot b} + \left(2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{b \cdot b}\right)\right)\right)}}}{2 \cdot a} \]

11. Taylor expanded in c around 0

    \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + {c}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{c \cdot \left(-64 \cdot {a}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot a + -4 \cdot a\right) \cdot \left(\left(16 \cdot {a}^{2} + 32 \cdot {a}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot a + -4 \cdot a\right)}^{2}\right)\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {a}^{2} + 32 \cdot {a}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot a + -4 \cdot a\right)}^{2}}{{b}^{2}}\right)}\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{b \cdot b} + \left(2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{b \cdot b}\right)\right)\right)}}{2 \cdot a} \]

13. Applied rewrites 92.9%

    \[\leadsto \frac{\frac{b \cdot \left(-0.5 \cdot \frac{0.25 \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(0.5 \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(c \cdot c\right) \cdot \color{blue}{\left(0.5 \cdot \frac{c \cdot \left(-64 \cdot {a}^{3} - 0.5 \cdot \left(\left(-8 \cdot a + -4 \cdot a\right) \cdot \left(\left(16 \cdot \left(a \cdot a\right) + 32 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(-8 \cdot a + -4 \cdot a\right)}^{2}\right)\right)\right)}{{b}^{4}} + 0.5 \cdot \frac{\left(16 \cdot \left(a \cdot a\right) + 32 \cdot \left(a \cdot a\right)\right) - 0.25 \cdot {\left(-8 \cdot a + -4 \cdot a\right)}^{2}}{b \cdot b}\right)}\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{b \cdot b} + \left(2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{b \cdot b}\right)\right)\right)}}{2 \cdot a} \]
14. Add Preprocessing

Alternative 3: 91.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c}{b \cdot b}\\ t_1 := -8 \cdot c + -4 \cdot c\\ t_2 := \left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {t\_1}^{2}\\ t_3 := -64 \cdot {c}^{3} - 0.5 \cdot \left(t\_1 \cdot t\_2\right)\\ \frac{\frac{a \cdot \left(0.5 \cdot \left(b \cdot t\_1\right) + a \cdot \left(0.5 \cdot \frac{t\_2}{b} + a \cdot \left(-0.5 \cdot \frac{a \cdot \left(0.25 \cdot {t\_2}^{2} + 0.5 \cdot \left(t\_1 \cdot t\_3\right)\right)}{{b}^{5}} + 0.5 \cdot \frac{t\_3}{{b}^{3}}\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \left(2 \cdot t\_0 + \left(2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{4}} + 4 \cdot t\_0\right)\right)\right)}}{2 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (* a c) (* b b)))
        (t_1 (+ (* -8.0 c) (* -4.0 c)))
        (t_2 (- (+ (* 16.0 (* c c)) (* 32.0 (* c c))) (* 0.25 (pow t_1 2.0))))
        (t_3 (- (* -64.0 (pow c 3.0)) (* 0.5 (* t_1 t_2)))))
   (/
    (/
     (*
      a
      (+
       (* 0.5 (* b t_1))
       (*
        a
        (+
         (* 0.5 (/ t_2 b))
         (*
          a
          (+
           (*
            -0.5
            (/
             (* a (+ (* 0.25 (pow t_2 2.0)) (* 0.5 (* t_1 t_3))))
             (pow b 5.0)))
           (* 0.5 (/ t_3 (pow b 3.0)))))))))
     (*
      (* b b)
      (-
       (+ 3.0 (* -4.0 (/ (pow (* a c) 3.0) (pow b 6.0))))
       (+
        (* 2.0 t_0)
        (+ (* 2.0 (/ (pow (* a c) 2.0) (pow b 4.0))) (* 4.0 t_0))))))
    (* 2.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (a * c) / (b * b);
	double t_1 = (-8.0 * c) + (-4.0 * c);
	double t_2 = ((16.0 * (c * c)) + (32.0 * (c * c))) - (0.25 * pow(t_1, 2.0));
	double t_3 = (-64.0 * pow(c, 3.0)) - (0.5 * (t_1 * t_2));
	return ((a * ((0.5 * (b * t_1)) + (a * ((0.5 * (t_2 / b)) + (a * ((-0.5 * ((a * ((0.25 * pow(t_2, 2.0)) + (0.5 * (t_1 * t_3)))) / pow(b, 5.0))) + (0.5 * (t_3 / pow(b, 3.0))))))))) / ((b * b) * ((3.0 + (-4.0 * (pow((a * c), 3.0) / pow(b, 6.0)))) - ((2.0 * t_0) + ((2.0 * (pow((a * c), 2.0) / pow(b, 4.0))) + (4.0 * t_0)))))) / (2.0 * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (a * c) / (b * b)
    t_1 = ((-8.0d0) * c) + ((-4.0d0) * c)
    t_2 = ((16.0d0 * (c * c)) + (32.0d0 * (c * c))) - (0.25d0 * (t_1 ** 2.0d0))
    t_3 = ((-64.0d0) * (c ** 3.0d0)) - (0.5d0 * (t_1 * t_2))
    code = ((a * ((0.5d0 * (b * t_1)) + (a * ((0.5d0 * (t_2 / b)) + (a * (((-0.5d0) * ((a * ((0.25d0 * (t_2 ** 2.0d0)) + (0.5d0 * (t_1 * t_3)))) / (b ** 5.0d0))) + (0.5d0 * (t_3 / (b ** 3.0d0))))))))) / ((b * b) * ((3.0d0 + ((-4.0d0) * (((a * c) ** 3.0d0) / (b ** 6.0d0)))) - ((2.0d0 * t_0) + ((2.0d0 * (((a * c) ** 2.0d0) / (b ** 4.0d0))) + (4.0d0 * t_0)))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * c) / (b * b);
	double t_1 = (-8.0 * c) + (-4.0 * c);
	double t_2 = ((16.0 * (c * c)) + (32.0 * (c * c))) - (0.25 * Math.pow(t_1, 2.0));
	double t_3 = (-64.0 * Math.pow(c, 3.0)) - (0.5 * (t_1 * t_2));
	return ((a * ((0.5 * (b * t_1)) + (a * ((0.5 * (t_2 / b)) + (a * ((-0.5 * ((a * ((0.25 * Math.pow(t_2, 2.0)) + (0.5 * (t_1 * t_3)))) / Math.pow(b, 5.0))) + (0.5 * (t_3 / Math.pow(b, 3.0))))))))) / ((b * b) * ((3.0 + (-4.0 * (Math.pow((a * c), 3.0) / Math.pow(b, 6.0)))) - ((2.0 * t_0) + ((2.0 * (Math.pow((a * c), 2.0) / Math.pow(b, 4.0))) + (4.0 * t_0)))))) / (2.0 * a);
}

Python

def code(a, b, c):
	t_0 = (a * c) / (b * b)
	t_1 = (-8.0 * c) + (-4.0 * c)
	t_2 = ((16.0 * (c * c)) + (32.0 * (c * c))) - (0.25 * math.pow(t_1, 2.0))
	t_3 = (-64.0 * math.pow(c, 3.0)) - (0.5 * (t_1 * t_2))
	return ((a * ((0.5 * (b * t_1)) + (a * ((0.5 * (t_2 / b)) + (a * ((-0.5 * ((a * ((0.25 * math.pow(t_2, 2.0)) + (0.5 * (t_1 * t_3)))) / math.pow(b, 5.0))) + (0.5 * (t_3 / math.pow(b, 3.0))))))))) / ((b * b) * ((3.0 + (-4.0 * (math.pow((a * c), 3.0) / math.pow(b, 6.0)))) - ((2.0 * t_0) + ((2.0 * (math.pow((a * c), 2.0) / math.pow(b, 4.0))) + (4.0 * t_0)))))) / (2.0 * a)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * c) / Float64(b * b))
	t_1 = Float64(Float64(-8.0 * c) + Float64(-4.0 * c))
	t_2 = Float64(Float64(Float64(16.0 * Float64(c * c)) + Float64(32.0 * Float64(c * c))) - Float64(0.25 * (t_1 ^ 2.0)))
	t_3 = Float64(Float64(-64.0 * (c ^ 3.0)) - Float64(0.5 * Float64(t_1 * t_2)))
	return Float64(Float64(Float64(a * Float64(Float64(0.5 * Float64(b * t_1)) + Float64(a * Float64(Float64(0.5 * Float64(t_2 / b)) + Float64(a * Float64(Float64(-0.5 * Float64(Float64(a * Float64(Float64(0.25 * (t_2 ^ 2.0)) + Float64(0.5 * Float64(t_1 * t_3)))) / (b ^ 5.0))) + Float64(0.5 * Float64(t_3 / (b ^ 3.0))))))))) / Float64(Float64(b * b) * Float64(Float64(3.0 + Float64(-4.0 * Float64((Float64(a * c) ^ 3.0) / (b ^ 6.0)))) - Float64(Float64(2.0 * t_0) + Float64(Float64(2.0 * Float64((Float64(a * c) ^ 2.0) / (b ^ 4.0))) + Float64(4.0 * t_0)))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (a * c) / (b * b);
	t_1 = (-8.0 * c) + (-4.0 * c);
	t_2 = ((16.0 * (c * c)) + (32.0 * (c * c))) - (0.25 * (t_1 ^ 2.0));
	t_3 = (-64.0 * (c ^ 3.0)) - (0.5 * (t_1 * t_2));
	tmp = ((a * ((0.5 * (b * t_1)) + (a * ((0.5 * (t_2 / b)) + (a * ((-0.5 * ((a * ((0.25 * (t_2 ^ 2.0)) + (0.5 * (t_1 * t_3)))) / (b ^ 5.0))) + (0.5 * (t_3 / (b ^ 3.0))))))))) / ((b * b) * ((3.0 + (-4.0 * (((a * c) ^ 3.0) / (b ^ 6.0)))) - ((2.0 * t_0) + ((2.0 * (((a * c) ^ 2.0) / (b ^ 4.0))) + (4.0 * t_0)))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-8.0 * c), $MachinePrecision] + N[(-4.0 * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(16.0 * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(32.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-64.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(a * N[(N[(0.5 * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(0.5 * N[(t$95$2 / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5 * N[(N[(a * N[(N[(0.25 * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$3 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(N[(3.0 + N[(-4.0 * N[(N[Power[N[(a * c), $MachinePrecision], 3.0], $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(2.0 * t$95$0), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[N[(a * c), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 53.4%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. flip3-+ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 53.2%

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

5. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]

7. Applied rewrites 92.7%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-0.5 \cdot \frac{0.25 \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(0.5 \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(0.5 \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + 0.5 \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\color{blue}{{b}^{2} \cdot \left(\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}}}{2 \cdot a} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{{b}^{2} \cdot \color{blue}{\left(\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}}}{2 \cdot a} \]

   2. pow2 N/A

      \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\color{blue}{\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)} - \left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}}{2 \cdot a} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\color{blue}{\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right)} - \left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)\right)}}{2 \cdot a} \]

   4. lower--.f64 N/A

      \[\leadsto \frac{\frac{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{a}^{3} \cdot {c}^{3}}{{b}^{6}}\right) - \color{blue}{\left(2 \cdot \frac{a \cdot c}{{b}^{2}} + \left(2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{{b}^{2}}\right)\right)}\right)}}{2 \cdot a} \]

10. Applied rewrites 92.9%

    \[\leadsto \frac{\frac{b \cdot \left(-0.5 \cdot \frac{0.25 \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(0.5 \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(0.5 \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + 0.5 \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}{\color{blue}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{b \cdot b} + \left(2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{b \cdot b}\right)\right)\right)}}}{2 \cdot a} \]

11. Taylor expanded in a around 0

    \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot \left(-8 \cdot c + -4 \cdot c\right)\right) + a \cdot \left(\frac{1}{2} \cdot \frac{\left(16 \cdot {c}^{2} + 32 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}}{b} + a \cdot \left(\frac{-1}{2} \cdot \frac{a \cdot \left(\frac{1}{4} \cdot {\left(\left(16 \cdot {c}^{2} + 32 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(-64 \cdot {c}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot {c}^{2} + 32 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + \frac{1}{2} \cdot \frac{-64 \cdot {c}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot {c}^{2} + 32 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{b \cdot b} + \left(2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{b \cdot b}\right)\right)\right)}}{2 \cdot a} \]

13. Applied rewrites 92.9%

    \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(0.5 \cdot \left(b \cdot \left(-8 \cdot c + -4 \cdot c\right)\right) + a \cdot \left(0.5 \cdot \frac{\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}}{b} + a \cdot \left(-0.5 \cdot \frac{a \cdot \left(0.25 \cdot {\left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(-64 \cdot {c}^{3} - 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + 0.5 \cdot \frac{-64 \cdot {c}^{3} - 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}}{\left(b \cdot b\right) \cdot \left(\left(3 + -4 \cdot \frac{{\left(a \cdot c\right)}^{3}}{{b}^{6}}\right) - \left(2 \cdot \frac{a \cdot c}{b \cdot b} + \left(2 \cdot \frac{{\left(a \cdot c\right)}^{2}}{{b}^{4}} + 4 \cdot \frac{a \cdot c}{b \cdot b}\right)\right)\right)}}{2 \cdot a} \]
14. Add Preprocessing

Alternative 4: 91.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -8 \cdot c + -4 \cdot c\\ t_1 := \left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {t\_0}^{2}\\ t_2 := -64 \cdot {c}^{3} - 0.5 \cdot \left(t\_0 \cdot t\_1\right)\\ t_3 := \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\\ \frac{\frac{a \cdot \left(0.5 \cdot \left(b \cdot t\_0\right) + a \cdot \left(0.5 \cdot \frac{t\_1}{b} + a \cdot \left(-0.5 \cdot \frac{a \cdot \left(0.25 \cdot {t\_1}^{2} + 0.5 \cdot \left(t\_0 \cdot t\_2\right)\right)}{{b}^{5}} + 0.5 \cdot \frac{t\_2}{{b}^{3}}\right)\right)\right)}{b \cdot b + \left(t\_3 \cdot t\_3 + b \cdot t\_3\right)}}{2 \cdot a} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (* -8.0 c) (* -4.0 c)))
        (t_1 (- (+ (* 16.0 (* c c)) (* 32.0 (* c c))) (* 0.25 (pow t_0 2.0))))
        (t_2 (- (* -64.0 (pow c 3.0)) (* 0.5 (* t_0 t_1))))
        (t_3 (sqrt (- (* b b) (* (* a 4.0) c)))))
   (/
    (/
     (*
      a
      (+
       (* 0.5 (* b t_0))
       (*
        a
        (+
         (* 0.5 (/ t_1 b))
         (*
          a
          (+
           (*
            -0.5
            (/
             (* a (+ (* 0.25 (pow t_1 2.0)) (* 0.5 (* t_0 t_2))))
             (pow b 5.0)))
           (* 0.5 (/ t_2 (pow b 3.0)))))))))
     (+ (* b b) (+ (* t_3 t_3) (* b t_3))))
    (* 2.0 a))))

C

double code(double a, double b, double c) {
	double t_0 = (-8.0 * c) + (-4.0 * c);
	double t_1 = ((16.0 * (c * c)) + (32.0 * (c * c))) - (0.25 * pow(t_0, 2.0));
	double t_2 = (-64.0 * pow(c, 3.0)) - (0.5 * (t_0 * t_1));
	double t_3 = sqrt(((b * b) - ((a * 4.0) * c)));
	return ((a * ((0.5 * (b * t_0)) + (a * ((0.5 * (t_1 / b)) + (a * ((-0.5 * ((a * ((0.25 * pow(t_1, 2.0)) + (0.5 * (t_0 * t_2)))) / pow(b, 5.0))) + (0.5 * (t_2 / pow(b, 3.0))))))))) / ((b * b) + ((t_3 * t_3) + (b * t_3)))) / (2.0 * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = ((-8.0d0) * c) + ((-4.0d0) * c)
    t_1 = ((16.0d0 * (c * c)) + (32.0d0 * (c * c))) - (0.25d0 * (t_0 ** 2.0d0))
    t_2 = ((-64.0d0) * (c ** 3.0d0)) - (0.5d0 * (t_0 * t_1))
    t_3 = sqrt(((b * b) - ((a * 4.0d0) * c)))
    code = ((a * ((0.5d0 * (b * t_0)) + (a * ((0.5d0 * (t_1 / b)) + (a * (((-0.5d0) * ((a * ((0.25d0 * (t_1 ** 2.0d0)) + (0.5d0 * (t_0 * t_2)))) / (b ** 5.0d0))) + (0.5d0 * (t_2 / (b ** 3.0d0))))))))) / ((b * b) + ((t_3 * t_3) + (b * t_3)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (-8.0 * c) + (-4.0 * c);
	double t_1 = ((16.0 * (c * c)) + (32.0 * (c * c))) - (0.25 * Math.pow(t_0, 2.0));
	double t_2 = (-64.0 * Math.pow(c, 3.0)) - (0.5 * (t_0 * t_1));
	double t_3 = Math.sqrt(((b * b) - ((a * 4.0) * c)));
	return ((a * ((0.5 * (b * t_0)) + (a * ((0.5 * (t_1 / b)) + (a * ((-0.5 * ((a * ((0.25 * Math.pow(t_1, 2.0)) + (0.5 * (t_0 * t_2)))) / Math.pow(b, 5.0))) + (0.5 * (t_2 / Math.pow(b, 3.0))))))))) / ((b * b) + ((t_3 * t_3) + (b * t_3)))) / (2.0 * a);
}

Python

def code(a, b, c):
	t_0 = (-8.0 * c) + (-4.0 * c)
	t_1 = ((16.0 * (c * c)) + (32.0 * (c * c))) - (0.25 * math.pow(t_0, 2.0))
	t_2 = (-64.0 * math.pow(c, 3.0)) - (0.5 * (t_0 * t_1))
	t_3 = math.sqrt(((b * b) - ((a * 4.0) * c)))
	return ((a * ((0.5 * (b * t_0)) + (a * ((0.5 * (t_1 / b)) + (a * ((-0.5 * ((a * ((0.25 * math.pow(t_1, 2.0)) + (0.5 * (t_0 * t_2)))) / math.pow(b, 5.0))) + (0.5 * (t_2 / math.pow(b, 3.0))))))))) / ((b * b) + ((t_3 * t_3) + (b * t_3)))) / (2.0 * a)

Julia

function code(a, b, c)
	t_0 = Float64(Float64(-8.0 * c) + Float64(-4.0 * c))
	t_1 = Float64(Float64(Float64(16.0 * Float64(c * c)) + Float64(32.0 * Float64(c * c))) - Float64(0.25 * (t_0 ^ 2.0)))
	t_2 = Float64(Float64(-64.0 * (c ^ 3.0)) - Float64(0.5 * Float64(t_0 * t_1)))
	t_3 = sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c)))
	return Float64(Float64(Float64(a * Float64(Float64(0.5 * Float64(b * t_0)) + Float64(a * Float64(Float64(0.5 * Float64(t_1 / b)) + Float64(a * Float64(Float64(-0.5 * Float64(Float64(a * Float64(Float64(0.25 * (t_1 ^ 2.0)) + Float64(0.5 * Float64(t_0 * t_2)))) / (b ^ 5.0))) + Float64(0.5 * Float64(t_2 / (b ^ 3.0))))))))) / Float64(Float64(b * b) + Float64(Float64(t_3 * t_3) + Float64(b * t_3)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = (-8.0 * c) + (-4.0 * c);
	t_1 = ((16.0 * (c * c)) + (32.0 * (c * c))) - (0.25 * (t_0 ^ 2.0));
	t_2 = (-64.0 * (c ^ 3.0)) - (0.5 * (t_0 * t_1));
	t_3 = sqrt(((b * b) - ((a * 4.0) * c)));
	tmp = ((a * ((0.5 * (b * t_0)) + (a * ((0.5 * (t_1 / b)) + (a * ((-0.5 * ((a * ((0.25 * (t_1 ^ 2.0)) + (0.5 * (t_0 * t_2)))) / (b ^ 5.0))) + (0.5 * (t_2 / (b ^ 3.0))))))))) / ((b * b) + ((t_3 * t_3) + (b * t_3)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(-8.0 * c), $MachinePrecision] + N[(-4.0 * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(16.0 * N[(c * c), $MachinePrecision]), $MachinePrecision] + N[(32.0 * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-64.0 * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(a * N[(N[(0.5 * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(0.5 * N[(t$95$1 / b), $MachinePrecision]), $MachinePrecision] + N[(a * N[(N[(-0.5 * N[(N[(a * N[(N[(0.25 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$2 / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] + N[(N[(t$95$3 * t$95$3), $MachinePrecision] + N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 53.4%

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

   1. lift-neg.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. flip3-+ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 53.2%

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

5. Taylor expanded in b around inf

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(\frac{-1}{2} \cdot \frac{\frac{1}{4} \cdot {\left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(\frac{1}{2} \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(\frac{1}{2} \cdot \frac{-64 \cdot \left({a}^{3} \cdot {c}^{3}\right) - \frac{1}{2} \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + \frac{1}{2} \cdot \frac{\left(16 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 32 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) - \frac{1}{4} \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{{b}^{2}}\right)\right)\right)}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]

7. Applied rewrites 92.7%

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(-0.5 \cdot \frac{0.25 \cdot {\left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{{b}^{6}} + \left(0.5 \cdot \left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) + \left(0.5 \cdot \frac{-64 \cdot {\left(a \cdot c\right)}^{3} - 0.5 \cdot \left(\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right) \cdot \left(\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)}{{b}^{4}} + 0.5 \cdot \frac{\left(16 \cdot {\left(a \cdot c\right)}^{2} + 32 \cdot {\left(a \cdot c\right)}^{2}\right) - 0.25 \cdot {\left(-8 \cdot \left(a \cdot c\right) + -4 \cdot \left(a \cdot c\right)\right)}^{2}}{b \cdot b}\right)\right)\right)}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(b \cdot \left(-8 \cdot c + -4 \cdot c\right)\right) + a \cdot \left(\frac{1}{2} \cdot \frac{\left(16 \cdot {c}^{2} + 32 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}}{b} + a \cdot \left(\frac{-1}{2} \cdot \frac{a \cdot \left(\frac{1}{4} \cdot {\left(\left(16 \cdot {c}^{2} + 32 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(-64 \cdot {c}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot {c}^{2} + 32 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + \frac{1}{2} \cdot \frac{-64 \cdot {c}^{3} - \frac{1}{2} \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot {c}^{2} + 32 \cdot {c}^{2}\right) - \frac{1}{4} \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]

10. Applied rewrites 92.7%

    \[\leadsto \frac{\frac{a \cdot \color{blue}{\left(0.5 \cdot \left(b \cdot \left(-8 \cdot c + -4 \cdot c\right)\right) + a \cdot \left(0.5 \cdot \frac{\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}}{b} + a \cdot \left(-0.5 \cdot \frac{a \cdot \left(0.25 \cdot {\left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(-64 \cdot {c}^{3} - 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + 0.5 \cdot \frac{-64 \cdot {c}^{3} - 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]

11. Final simplification 92.7%

    \[\leadsto \frac{\frac{a \cdot \left(0.5 \cdot \left(b \cdot \left(-8 \cdot c + -4 \cdot c\right)\right) + a \cdot \left(0.5 \cdot \frac{\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}}{b} + a \cdot \left(-0.5 \cdot \frac{a \cdot \left(0.25 \cdot {\left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)}^{2} + 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(-64 \cdot {c}^{3} - 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)\right)\right)\right)}{{b}^{5}} + 0.5 \cdot \frac{-64 \cdot {c}^{3} - 0.5 \cdot \left(\left(-8 \cdot c + -4 \cdot c\right) \cdot \left(\left(16 \cdot \left(c \cdot c\right) + 32 \cdot \left(c \cdot c\right)\right) - 0.25 \cdot {\left(-8 \cdot c + -4 \cdot c\right)}^{2}\right)\right)}{{b}^{3}}\right)\right)\right)}{b \cdot b + \left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a} \]
12. Add Preprocessing

Alternative 5: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(a \cdot 4\right) \cdot c\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_1}^{3}}{b \cdot b + \left(t\_0 + b \cdot t\_1\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* a 4.0) c))) (t_1 (sqrt t_0)))
   (if (<= b 0.075)
     (/
      (/ (+ (pow (- b) 3.0) (pow t_1 3.0)) (+ (* b b) (+ t_0 (* b t_1))))
      (* 2.0 a))
     (/
      (-
       (*
        a
        (-
         (*
          a
          (+
           (* -5.0 (/ (* a (pow c 4.0)) (pow b 6.0)))
           (* -2.0 (/ (pow c 3.0) (pow b 4.0)))))
         (ratio-of-squares c b)))
       c)
      b))))

Derivation

1. Split input into 2 regimes

2. if b < 0.0749999999999999972

   1. Initial program 91.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 92.1%

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

   6. Applied rewrites 92.4%

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

   if 0.0749999999999999972 < b

   1. Initial program 51.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b}} \]

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]

   8. Applied rewrites 94.3%

      \[\leadsto \frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}^{3}}{b \cdot b + \left(\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(a \cdot \left(-5 \cdot \frac{a \cdot {c}^{4}}{{b}^{6}} + -2 \cdot \frac{{c}^{3}}{{b}^{4}}\right) - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(a \cdot 4\right) \cdot c\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_1}^{3}}{b \cdot b + \left(t\_0 + b \cdot t\_1\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\left(c + a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}} \cdot -2}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* a 4.0) c))) (t_1 (sqrt t_0)))
   (if (<= b 0.075)
     (/
      (/ (+ (pow (- b) 3.0) (pow t_1 3.0)) (+ (* b b) (+ t_0 (* b t_1))))
      (* 2.0 a))
     (/
      (+
       (+
        (- (+ c (* a (ratio-of-squares c b))))
        (/ (* -0.25 (* (pow (* c a) 4.0) 20.0)) (* (pow b 6.0) a)))
       (* (/ (* (* a a) (* (* c c) c)) (pow b 4.0)) -2.0))
      b))))

Derivation

1. Split input into 2 regimes

2. if b < 0.0749999999999999972

   1. Initial program 91.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 92.1%

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

   6. Applied rewrites 92.4%

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

   if 0.0749999999999999972 < b

   1. Initial program 51.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{\frac{-1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b} \]

      2. unpow3 N/A

         \[\leadsto \frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{\frac{-1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}} \cdot -2}{b} \]

      3. pow2 N/A

         \[\leadsto \frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{\frac{-1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}} \cdot -2}{b} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{\frac{-1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot \left({c}^{2} \cdot c\right)}{{b}^{4}} \cdot -2}{b} \]

      5. pow2 N/A

         \[\leadsto \frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{\frac{-1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}} \cdot -2}{b} \]

      6. lower-*.f64 94.3

         \[\leadsto \frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}} \cdot -2}{b} \]

   7. Applied rewrites 94.3%

      \[\leadsto \frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}} \cdot -2}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}^{3}}{b \cdot b + \left(\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(-\left(c + a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot \left(\left(c \cdot c\right) \cdot c\right)}{{b}^{4}} \cdot -2}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(a \cdot 4\right) \cdot c\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {t\_1}^{3}}{b \cdot b + \left(t\_0 + b \cdot t\_1\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{a \cdot a}{{b}^{4}}\right) - \frac{a}{b \cdot b}\right) - 1\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* a 4.0) c))) (t_1 (sqrt t_0)))
   (if (<= b 0.075)
     (/
      (/ (+ (pow (- b) 3.0) (pow t_1 3.0)) (+ (* b b) (+ t_0 (* b t_1))))
      (* 2.0 a))
     (/
      (*
       c
       (-
        (*
         c
         (-
          (*
           c
           (+
            (* -5.0 (/ (* (pow a 3.0) c) (pow b 6.0)))
            (* -2.0 (/ (* a a) (pow b 4.0)))))
          (/ a (* b b))))
        1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((a * 4.0) * c);
	double t_1 = sqrt(t_0);
	double tmp;
	if (b <= 0.075) {
		tmp = ((pow(-b, 3.0) + pow(t_1, 3.0)) / ((b * b) + (t_0 + (b * t_1)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((c * ((-5.0 * ((pow(a, 3.0) * c) / pow(b, 6.0))) + (-2.0 * ((a * a) / pow(b, 4.0))))) - (a / (b * b)))) - 1.0)) / b;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * b) - ((a * 4.0d0) * c)
    t_1 = sqrt(t_0)
    if (b <= 0.075d0) then
        tmp = (((-b ** 3.0d0) + (t_1 ** 3.0d0)) / ((b * b) + (t_0 + (b * t_1)))) / (2.0d0 * a)
    else
        tmp = (c * ((c * ((c * (((-5.0d0) * (((a ** 3.0d0) * c) / (b ** 6.0d0))) + ((-2.0d0) * ((a * a) / (b ** 4.0d0))))) - (a / (b * b)))) - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) - ((a * 4.0) * c);
	double t_1 = Math.sqrt(t_0);
	double tmp;
	if (b <= 0.075) {
		tmp = ((Math.pow(-b, 3.0) + Math.pow(t_1, 3.0)) / ((b * b) + (t_0 + (b * t_1)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((c * ((-5.0 * ((Math.pow(a, 3.0) * c) / Math.pow(b, 6.0))) + (-2.0 * ((a * a) / Math.pow(b, 4.0))))) - (a / (b * b)))) - 1.0)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) - ((a * 4.0) * c)
	t_1 = math.sqrt(t_0)
	tmp = 0
	if b <= 0.075:
		tmp = ((math.pow(-b, 3.0) + math.pow(t_1, 3.0)) / ((b * b) + (t_0 + (b * t_1)))) / (2.0 * a)
	else:
		tmp = (c * ((c * ((c * ((-5.0 * ((math.pow(a, 3.0) * c) / math.pow(b, 6.0))) + (-2.0 * ((a * a) / math.pow(b, 4.0))))) - (a / (b * b)))) - 1.0)) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (b <= 0.075)
		tmp = Float64(Float64(Float64((Float64(-b) ^ 3.0) + (t_1 ^ 3.0)) / Float64(Float64(b * b) + Float64(t_0 + Float64(b * t_1)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(c * Float64(Float64(-5.0 * Float64(Float64((a ^ 3.0) * c) / (b ^ 6.0))) + Float64(-2.0 * Float64(Float64(a * a) / (b ^ 4.0))))) - Float64(a / Float64(b * b)))) - 1.0)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) - ((a * 4.0) * c);
	t_1 = sqrt(t_0);
	tmp = 0.0;
	if (b <= 0.075)
		tmp = (((-b ^ 3.0) + (t_1 ^ 3.0)) / ((b * b) + (t_0 + (b * t_1)))) / (2.0 * a);
	else
		tmp = (c * ((c * ((c * ((-5.0 * (((a ^ 3.0) * c) / (b ^ 6.0))) + (-2.0 * ((a * a) / (b ^ 4.0))))) - (a / (b * b)))) - 1.0)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[b, 0.075], N[(N[(N[(N[Power[(-b), 3.0], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] + N[(t$95$0 + N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(c * N[(N[(-5.0 * N[(N[(N[Power[a, 3.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0749999999999999972

   1. Initial program 91.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 92.1%

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

   6. Applied rewrites 92.4%

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

   if 0.0749999999999999972 < b

   1. Initial program 51.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b}} \]

   6. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{{a}^{2}}{{b}^{4}}\right) - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]

   8. Applied rewrites 94.1%

      \[\leadsto \frac{c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{a \cdot a}{{b}^{4}}\right) - \frac{a}{b \cdot b}\right) - 1\right)}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{{\left(-b\right)}^{3} + {\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}^{3}}{b \cdot b + \left(\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(c \cdot \left(-5 \cdot \frac{{a}^{3} \cdot c}{{b}^{6}} + -2 \cdot \frac{a \cdot a}{{b}^{4}}\right) - \frac{a}{b \cdot b}\right) - 1\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(a \cdot 4\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -3.3:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) \cdot b\right) \cdot b + {t\_0}^{1.5}}{b \cdot b + \left(t\_0 + b \cdot \sqrt{t\_0}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* a 4.0) c))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -3.3)
     (/
      (/
       (+ (* (* (- b) b) b) (pow t_0 1.5))
       (+ (* b b) (+ t_0 (* b (sqrt t_0)))))
      (* 2.0 a))
     (/
      (-
       (*
        a
        (- (* -2.0 (/ (* a (pow c 3.0)) (pow b 4.0))) (ratio-of-squares c b)))
       c)
      b))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -3.2999999999999998

   1. Initial program 83.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right)}^{1.5}}{b \cdot b + \left(\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a}} \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. sqr-neg-rev N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 84.5

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

   8. Applied rewrites 84.5%

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

   if -3.2999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

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

   8. Applied rewrites 92.7%

      \[\leadsto \frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -3.3:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) \cdot b\right) \cdot b + {\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right)}^{1.5}}{b \cdot b + \left(\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \mathsf{ratio\_of\_squares}\left(c, b\right)\right) - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot b - \left(a \cdot 4\right) \cdot c\\ \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -3.3:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) \cdot b\right) \cdot b + {t\_0}^{1.5}}{b \cdot b + \left(t\_0 + b \cdot \sqrt{t\_0}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot c}{{b}^{4}} - \frac{a}{b \cdot b}\right) - 1\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (* b b) (* (* a 4.0) c))))
   (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -3.3)
     (/
      (/
       (+ (* (* (- b) b) b) (pow t_0 1.5))
       (+ (* b b) (+ t_0 (* b (sqrt t_0)))))
      (* 2.0 a))
     (/
      (*
       c
       (- (* c (- (* -2.0 (/ (* (* a a) c) (pow b 4.0))) (/ a (* b b)))) 1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) - ((a * 4.0) * c);
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -3.3) {
		tmp = ((((-b * b) * b) + pow(t_0, 1.5)) / ((b * b) + (t_0 + (b * sqrt(t_0))))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (b * b) - ((a * 4.0d0) * c)
    if (((-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)) <= (-3.3d0)) then
        tmp = ((((-b * b) * b) + (t_0 ** 1.5d0)) / ((b * b) + (t_0 + (b * sqrt(t_0))))) / (2.0d0 * a)
    else
        tmp = (c * ((c * (((-2.0d0) * (((a * a) * c) / (b ** 4.0d0))) - (a / (b * b)))) - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (b * b) - ((a * 4.0) * c);
	double tmp;
	if (((-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -3.3) {
		tmp = ((((-b * b) * b) + Math.pow(t_0, 1.5)) / ((b * b) + (t_0 + (b * Math.sqrt(t_0))))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / Math.pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (b * b) - ((a * 4.0) * c)
	tmp = 0
	if ((-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -3.3:
		tmp = ((((-b * b) * b) + math.pow(t_0, 1.5)) / ((b * b) + (t_0 + (b * math.sqrt(t_0))))) / (2.0 * a)
	else:
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / math.pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -3.3)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-b) * b) * b) + (t_0 ^ 1.5)) / Float64(Float64(b * b) + Float64(t_0 + Float64(b * sqrt(t_0))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(Float64(a * a) * c) / (b ^ 4.0))) - Float64(a / Float64(b * b)))) - 1.0)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (b * b) - ((a * 4.0) * c);
	tmp = 0.0;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -3.3)
		tmp = ((((-b * b) * b) + (t_0 ^ 1.5)) / ((b * b) + (t_0 + (b * sqrt(t_0))))) / (2.0 * a);
	else
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / (b ^ 4.0))) - (a / (b * b)))) - 1.0)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -3.3], N[(N[(N[(N[(N[((-b) * b), $MachinePrecision] * b), $MachinePrecision] + N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] + N[(t$95$0 + N[(b * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -3.2999999999999998

   1. Initial program 83.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip3-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 83.0%

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

   6. Applied rewrites 84.2%

      \[\leadsto \color{blue}{\frac{\frac{{\left(-b\right)}^{3} + {\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right)}^{1.5}}{b \cdot b + \left(\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) - \left(-b\right) \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a}} \]
   7. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow3 N/A

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

      4. sqr-neg-rev N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 84.5

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

   8. Applied rewrites 84.5%

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

   if -3.2999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 48.7%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 92.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -3.3:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) \cdot b\right) \cdot b + {\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right)}^{1.5}}{b \cdot b + \left(\left(b \cdot b - \left(a \cdot 4\right) \cdot c\right) + b \cdot \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot c}{{b}^{4}} - \frac{a}{b \cdot b}\right) - 1\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot 4\right) \cdot c\\ \mathbf{if}\;b \leq 0.078:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\frac{{b}^{4} - t\_0 \cdot t\_0}{b \cdot b + t\_0}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot c}{{b}^{4}} - \frac{a}{b \cdot b}\right) - 1\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* a 4.0) c)))
   (if (<= b 0.078)
     (/
      (+ (- b) (sqrt (/ (- (pow b 4.0) (* t_0 t_0)) (+ (* b b) t_0))))
      (* 2.0 a))
     (/
      (*
       c
       (- (* c (- (* -2.0 (/ (* (* a a) c) (pow b 4.0))) (/ a (* b b)))) 1.0))
      b))))

C

double code(double a, double b, double c) {
	double t_0 = (a * 4.0) * c;
	double tmp;
	if (b <= 0.078) {
		tmp = (-b + sqrt(((pow(b, 4.0) - (t_0 * t_0)) / ((b * b) + t_0)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (a * 4.0d0) * c
    if (b <= 0.078d0) then
        tmp = (-b + sqrt((((b ** 4.0d0) - (t_0 * t_0)) / ((b * b) + t_0)))) / (2.0d0 * a)
    else
        tmp = (c * ((c * (((-2.0d0) * (((a * a) * c) / (b ** 4.0d0))) - (a / (b * b)))) - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (a * 4.0) * c;
	double tmp;
	if (b <= 0.078) {
		tmp = (-b + Math.sqrt(((Math.pow(b, 4.0) - (t_0 * t_0)) / ((b * b) + t_0)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / Math.pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (a * 4.0) * c
	tmp = 0
	if b <= 0.078:
		tmp = (-b + math.sqrt(((math.pow(b, 4.0) - (t_0 * t_0)) / ((b * b) + t_0)))) / (2.0 * a)
	else:
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / math.pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(a * 4.0) * c)
	tmp = 0.0
	if (b <= 0.078)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64((b ^ 4.0) - Float64(t_0 * t_0)) / Float64(Float64(b * b) + t_0)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(Float64(a * a) * c) / (b ^ 4.0))) - Float64(a / Float64(b * b)))) - 1.0)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (a * 4.0) * c;
	tmp = 0.0;
	if (b <= 0.078)
		tmp = (-b + sqrt((((b ^ 4.0) - (t_0 * t_0)) / ((b * b) + t_0)))) / (2.0 * a);
	else
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / (b ^ 4.0))) - (a / (b * b)))) - 1.0)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 0.078], N[(N[((-b) + N[Sqrt[N[(N[(N[Power[b, 4.0], $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0779999999999999999

   1. Initial program 91.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

      7. flip-- N/A

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

      8. associate-*r* N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 91.9%

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

   if 0.0779999999999999999 < b

   1. Initial program 51.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 91.4%

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

Alternative 11: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.078:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(\frac{-4 \cdot \left(c \cdot a\right)}{b \cdot b} + 1\right) \cdot \left(b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{\left(a \cdot a\right) \cdot c}{{b}^{4}} - \frac{a}{b \cdot b}\right) - 1\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.078)
   (/
    (+ (- b) (sqrt (* (+ (/ (* -4.0 (* c a)) (* b b)) 1.0) (* b b))))
    (* 2.0 a))
   (/
    (*
     c
     (- (* c (- (* -2.0 (/ (* (* a a) c) (pow b 4.0))) (/ a (* b b)))) 1.0))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.078) {
		tmp = (-b + sqrt(((((-4.0 * (c * a)) / (b * b)) + 1.0) * (b * b)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b;
	}
	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 <= 0.078d0) then
        tmp = (-b + sqrt((((((-4.0d0) * (c * a)) / (b * b)) + 1.0d0) * (b * b)))) / (2.0d0 * a)
    else
        tmp = (c * ((c * (((-2.0d0) * (((a * a) * c) / (b ** 4.0d0))) - (a / (b * b)))) - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.078) {
		tmp = (-b + Math.sqrt(((((-4.0 * (c * a)) / (b * b)) + 1.0) * (b * b)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / Math.pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.078:
		tmp = (-b + math.sqrt(((((-4.0 * (c * a)) / (b * b)) + 1.0) * (b * b)))) / (2.0 * a)
	else:
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / math.pow(b, 4.0))) - (a / (b * b)))) - 1.0)) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.078)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(Float64(Float64(-4.0 * Float64(c * a)) / Float64(b * b)) + 1.0) * Float64(b * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64(Float64(a * a) * c) / (b ^ 4.0))) - Float64(a / Float64(b * b)))) - 1.0)) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.078)
		tmp = (-b + sqrt(((((-4.0 * (c * a)) / (b * b)) + 1.0) * (b * b)))) / (2.0 * a);
	else
		tmp = (c * ((c * ((-2.0 * (((a * a) * c) / (b ^ 4.0))) - (a / (b * b)))) - 1.0)) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.078], N[(N[((-b) + N[Sqrt[N[(N[(N[(N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0779999999999999999

   1. Initial program 91.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if 0.0779999999999999999 < b

   1. Initial program 51.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\left(\left(\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)\right) + \frac{-0.25 \cdot \left({\left(c \cdot a\right)}^{4} \cdot 20\right)}{{b}^{6} \cdot a}\right) + \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{4}} \cdot -2}{b}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 91.4%

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

Alternative 12: 85.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0008:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(\frac{b \cdot b}{c} - a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.0008)
   (/ (+ (- b) (sqrt (* (- (/ (* b b) c) (* a 4.0)) c))) (* 2.0 a))
   (/ (+ c (* a (ratio-of-squares c b))) (- b))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -8.00000000000000038e-4

   1. Initial program 78.0%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if -8.00000000000000038e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 41.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-+.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. pow2 N/A

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

      11. lower-ratio-of-squares.f64 91.7

          \[\leadsto \frac{\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)}{b} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0008:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(\frac{b \cdot b}{c} - a \cdot 4\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;\frac{t\_0}{2 \cdot a} \leq -0.0008:\\ \;\;\;\;\frac{t\_0}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))))
   (if (<= (/ t_0 (* 2.0 a)) -0.0008)
     (/ t_0 (+ a a))
     (/ (+ c (* a (ratio-of-squares c b))) (- b)))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -8.00000000000000038e-4

   1. Initial program 78.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.0

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

   4. Applied rewrites 78.0%

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

   if -8.00000000000000038e-4 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 41.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-+.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. pow2 N/A

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

      11. lower-ratio-of-squares.f64 91.7

          \[\leadsto \frac{\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)}{b} \]

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\frac{\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.0008:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c + a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{c + a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)}{-b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ c (* a (ratio-of-squares c b))) (- b)))

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-+.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. pow2 N/A

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

   11. lower-ratio-of-squares.f64 83.0

       \[\leadsto \frac{\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)}{b} \]

5. Applied rewrites 83.0%

   \[\leadsto \color{blue}{\frac{\left(-c\right) + \left(-a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)\right)}{b}} \]

6. Final simplification 83.0%

   \[\leadsto \frac{c + a \cdot \mathsf{ratio\_of\_squares}\left(c, b\right)}{-b} \]
7. Add Preprocessing

Alternative 15: 64.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(c\right)}{b} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(c\right)}{\color{blue}{b}} \]

   4. lower-neg.f64 66.3

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

5. Applied rewrites 66.3%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025058 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))