Quadratic roots, medium range

Percentage Accurate: 30.9% → 95.7%

Time: 1.6min

Alternatives: 13

Speedup: 3.6×

Specification

\[\left(\left(\frac{2220446049250313}{20000000000000000000000000000000} < a \land a < 9007199254740992\right) \land \left(\frac{2220446049250313}{20000000000000000000000000000000} < b \land b < 9007199254740992\right)\right) \land \left(\frac{2220446049250313}{20000000000000000000000000000000} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 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 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 30.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 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 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  (-
 (/ (- (* (* (* (* a a) c) c) (* c (* -2 (pow b -4)))) c) b)
 (/
  (-
   (* (* (/ a (* b b)) c) c)
   (* (/ -5 (* (pow b 6) a)) (pow (* c a) 4)))
  b)))

C

double code(double a, double b, double c) {
	return ((((((a * a) * c) * c) * (c * (-2.0 * pow(b, -4.0)))) - c) / b) - (((((a / (b * b)) * c) * c) - ((-5.0 / (pow(b, 6.0) * a)) * pow((c * a), 4.0))) / 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 = ((((((a * a) * c) * c) * (c * ((-2.0d0) * (b ** (-4.0d0))))) - c) / b) - (((((a / (b * b)) * c) * c) - (((-5.0d0) / ((b ** 6.0d0) * a)) * ((c * a) ** 4.0d0))) / b)
end function

Java

public static double code(double a, double b, double c) {
	return ((((((a * a) * c) * c) * (c * (-2.0 * Math.pow(b, -4.0)))) - c) / b) - (((((a / (b * b)) * c) * c) - ((-5.0 / (Math.pow(b, 6.0) * a)) * Math.pow((c * a), 4.0))) / b);
}

Python

def code(a, b, c):
	return ((((((a * a) * c) * c) * (c * (-2.0 * math.pow(b, -4.0)))) - c) / b) - (((((a / (b * b)) * c) * c) - ((-5.0 / (math.pow(b, 6.0) * a)) * math.pow((c * a), 4.0))) / b)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * c) * Float64(c * Float64(-2.0 * (b ^ -4.0)))) - c) / b) - Float64(Float64(Float64(Float64(Float64(a / Float64(b * b)) * c) * c) - Float64(Float64(-5.0 / Float64((b ^ 6.0) * a)) * (Float64(c * a) ^ 4.0))) / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((((((a * a) * c) * c) * (c * (-2.0 * (b ^ -4.0)))) - c) / b) - (((((a / (b * b)) * c) * c) - ((-5.0 / ((b ^ 6.0) * a)) * ((c * a) ^ 4.0))) / b);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(c * N[(-2 * N[Power[b, -4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision] - N[(N[(N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] - N[(N[(-5 / N[(N[Power[b, 6], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.7%

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

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \color{blue}{\frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot \frac{-1}{4} - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b}} \]

8. Applied rewrites 95.7%

   \[\leadsto \frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot \left(c \cdot \left(-2 \cdot {b}^{-4}\right)\right) - c}{b} - \color{blue}{\frac{\left(\frac{a}{b \cdot b} \cdot c\right) \cdot c - \frac{-5}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}}{b}} \]
9. Add Preprocessing

Alternative 2: 95.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  (-
 (/ (- (* (* (* (* (pow b -4) -2) c) c) (* (* a a) c)) c) b)
 (-
  (* c (* (/ c (* (* b b) b)) a))
  (/ (* (pow (* c a) 4) -5) (* (* (pow b 6) a) b)))))

C

double code(double a, double b, double c) {
	return ((((((pow(b, -4.0) * -2.0) * c) * c) * ((a * a) * c)) - c) / b) - ((c * ((c / ((b * b) * b)) * a)) - ((pow((c * a), 4.0) * -5.0) / ((pow(b, 6.0) * a) * 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 = (((((((b ** (-4.0d0)) * (-2.0d0)) * c) * c) * ((a * a) * c)) - c) / b) - ((c * ((c / ((b * b) * b)) * a)) - ((((c * a) ** 4.0d0) * (-5.0d0)) / (((b ** 6.0d0) * a) * b)))
end function

Java

public static double code(double a, double b, double c) {
	return ((((((Math.pow(b, -4.0) * -2.0) * c) * c) * ((a * a) * c)) - c) / b) - ((c * ((c / ((b * b) * b)) * a)) - ((Math.pow((c * a), 4.0) * -5.0) / ((Math.pow(b, 6.0) * a) * b)));
}

Python

def code(a, b, c):
	return ((((((math.pow(b, -4.0) * -2.0) * c) * c) * ((a * a) * c)) - c) / b) - ((c * ((c / ((b * b) * b)) * a)) - ((math.pow((c * a), 4.0) * -5.0) / ((math.pow(b, 6.0) * a) * b)))

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64((b ^ -4.0) * -2.0) * c) * c) * Float64(Float64(a * a) * c)) - c) / b) - Float64(Float64(c * Float64(Float64(c / Float64(Float64(b * b) * b)) * a)) - Float64(Float64((Float64(c * a) ^ 4.0) * -5.0) / Float64(Float64((b ^ 6.0) * a) * b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((((((b ^ -4.0) * -2.0) * c) * c) * ((a * a) * c)) - c) / b) - ((c * ((c / ((b * b) * b)) * a)) - ((((c * a) ^ 4.0) * -5.0) / (((b ^ 6.0) * a) * b)));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[Power[b, -4], $MachinePrecision] * -2), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision] - N[(N[(c * N[(N[(c / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[(c * a), $MachinePrecision], 4], $MachinePrecision] * -5), $MachinePrecision] / N[(N[(N[Power[b, 6], $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.7%

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

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \color{blue}{\frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot \frac{-1}{4} - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b}} \]

8. Applied rewrites 95.7%

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

10. Applied rewrites 95.7%

    \[\leadsto \frac{\left(\left(\left({b}^{-4} \cdot -2\right) \cdot c\right) \cdot c\right) \cdot \left(\left(a \cdot a\right) \cdot c\right) - c}{b} - \color{blue}{\left(c \cdot \left(\frac{c}{\left(b \cdot b\right) \cdot b} \cdot a\right) - \frac{{\left(c \cdot a\right)}^{4} \cdot -5}{\left({b}^{6} \cdot a\right) \cdot b}\right)} \]
11. Add Preprocessing

Alternative 3: 95.7% accurate, 0.1× speedup

Math

\[\frac{\left(-c\right) - \left(\left(\frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) + \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (/
 (-
  (- c)
  (-
   (+
    (* 1/4 (* (pow (* c a) 4) (/ 20 (* (pow b 6) a))))
    (* (* c c) (/ a (* b b))))
   (* (* (* (* a a) c) (* c c)) (* (pow b -4) -2))))
 b))

C

double code(double a, double b, double c) {
	return (-c - (((0.25 * (pow((c * a), 4.0) * (20.0 / (pow(b, 6.0) * a)))) + ((c * c) * (a / (b * b)))) - ((((a * a) * c) * (c * c)) * (pow(b, -4.0) * -2.0)))) / b;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-c - (((0.25d0 * (((c * a) ** 4.0d0) * (20.0d0 / ((b ** 6.0d0) * a)))) + ((c * c) * (a / (b * b)))) - ((((a * a) * c) * (c * c)) * ((b ** (-4.0d0)) * (-2.0d0))))) / b
end function

Java

public static double code(double a, double b, double c) {
	return (-c - (((0.25 * (Math.pow((c * a), 4.0) * (20.0 / (Math.pow(b, 6.0) * a)))) + ((c * c) * (a / (b * b)))) - ((((a * a) * c) * (c * c)) * (Math.pow(b, -4.0) * -2.0)))) / b;
}

Python

def code(a, b, c):
	return (-c - (((0.25 * (math.pow((c * a), 4.0) * (20.0 / (math.pow(b, 6.0) * a)))) + ((c * c) * (a / (b * b)))) - ((((a * a) * c) * (c * c)) * (math.pow(b, -4.0) * -2.0)))) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-c) - Float64(Float64(Float64(0.25 * Float64((Float64(c * a) ^ 4.0) * Float64(20.0 / Float64((b ^ 6.0) * a)))) + Float64(Float64(c * c) * Float64(a / Float64(b * b)))) - Float64(Float64(Float64(Float64(a * a) * c) * Float64(c * c)) * Float64((b ^ -4.0) * -2.0)))) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-c - (((0.25 * (((c * a) ^ 4.0) * (20.0 / ((b ^ 6.0) * a)))) + ((c * c) * (a / (b * b)))) - ((((a * a) * c) * (c * c)) * ((b ^ -4.0) * -2.0)))) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[((-c) - N[(N[(N[(1/4 * N[(N[Power[N[(c * a), $MachinePrecision], 4], $MachinePrecision] * N[(20 / N[(N[Power[b, 6], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(N[Power[b, -4], $MachinePrecision] * -2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.7%

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

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(-c\right) - \left(\left(\frac{1}{4} \cdot \left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) + \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]
7. Add Preprocessing

Alternative 4: 95.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  (/
 (-
  (+
   (- (* (* (* (* a a) c) c) (* c (* -2 (pow b -4)))) c)
   (* (/ -5 (* (pow b 6) a)) (pow (* c a) 4)))
  (* (* (/ a (* b b)) c) c))
 b))

C

double code(double a, double b, double c) {
	return (((((((a * a) * c) * c) * (c * (-2.0 * pow(b, -4.0)))) - c) + ((-5.0 / (pow(b, 6.0) * a)) * pow((c * a), 4.0))) - (((a / (b * b)) * c) * 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 = (((((((a * a) * c) * c) * (c * ((-2.0d0) * (b ** (-4.0d0))))) - c) + (((-5.0d0) / ((b ** 6.0d0) * a)) * ((c * a) ** 4.0d0))) - (((a / (b * b)) * c) * c)) / b
end function

Java

public static double code(double a, double b, double c) {
	return (((((((a * a) * c) * c) * (c * (-2.0 * Math.pow(b, -4.0)))) - c) + ((-5.0 / (Math.pow(b, 6.0) * a)) * Math.pow((c * a), 4.0))) - (((a / (b * b)) * c) * c)) / b;
}

Python

def code(a, b, c):
	return (((((((a * a) * c) * c) * (c * (-2.0 * math.pow(b, -4.0)))) - c) + ((-5.0 / (math.pow(b, 6.0) * a)) * math.pow((c * a), 4.0))) - (((a / (b * b)) * c) * c)) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * c) * Float64(c * Float64(-2.0 * (b ^ -4.0)))) - c) + Float64(Float64(-5.0 / Float64((b ^ 6.0) * a)) * (Float64(c * a) ^ 4.0))) - Float64(Float64(Float64(a / Float64(b * b)) * c) * c)) / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((((((a * a) * c) * c) * (c * (-2.0 * (b ^ -4.0)))) - c) + ((-5.0 / ((b ^ 6.0) * a)) * ((c * a) ^ 4.0))) - (((a / (b * b)) * c) * c)) / b;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(c * N[(-2 * N[Power[b, -4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] + N[(N[(-5 / N[(N[Power[b, 6], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[(c * a), $MachinePrecision], 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.7%

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

6. Applied rewrites 95.7%

   \[\leadsto \frac{\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right) - c}{b} + \color{blue}{\frac{\left({\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}\right) \cdot \frac{-1}{4} - \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}}{b}} \]

8. Applied rewrites 95.7%

   \[\leadsto \frac{\left(\left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot \left(c \cdot \left(-2 \cdot {b}^{-4}\right)\right) - c\right) + \frac{-5}{{b}^{6} \cdot a} \cdot {\left(c \cdot a\right)}^{4}\right) - \left(\frac{a}{b \cdot b} \cdot c\right) \cdot c}{\color{blue}{b}} \]
9. Add Preprocessing

Alternative 5: 94.3% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
  :precision binary64
  (+
 (*
  c
  (* (- (* (* -2 (* a a)) (* c (pow b -5))) (/ a (* (* b b) b))) c))
 (/ c (- b))))

C

double code(double a, double b, double c) {
	return (c * ((((-2.0 * (a * a)) * (c * pow(b, -5.0))) - (a / ((b * b) * b))) * c)) + (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 * (((((-2.0d0) * (a * a)) * (c * (b ** (-5.0d0)))) - (a / ((b * b) * b))) * c)) + (c / -b)
end function

Java

public static double code(double a, double b, double c) {
	return (c * ((((-2.0 * (a * a)) * (c * Math.pow(b, -5.0))) - (a / ((b * b) * b))) * c)) + (c / -b);
}

Python

def code(a, b, c):
	return (c * ((((-2.0 * (a * a)) * (c * math.pow(b, -5.0))) - (a / ((b * b) * b))) * c)) + (c / -b)

Julia

function code(a, b, c)
	return Float64(Float64(c * Float64(Float64(Float64(Float64(-2.0 * Float64(a * a)) * Float64(c * (b ^ -5.0))) - Float64(a / Float64(Float64(b * b) * b))) * c)) + Float64(c / Float64(-b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (c * ((((-2.0 * (a * a)) * (c * (b ^ -5.0))) - (a / ((b * b) * b))) * c)) + (c / -b);
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(N[(N[(N[(-2 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * N[Power[b, -5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] + N[(c / (-b)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

7. Applied rewrites 94.0%

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

9. Applied rewrites 94.3%

   \[\leadsto c \cdot \left(\left(\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot {b}^{-5}\right) - \frac{a}{\left(b \cdot b\right) \cdot b}\right) \cdot c\right) + \frac{c}{\color{blue}{-b}} \]
10. Add Preprocessing

Alternative 6: 94.0% accurate, 0.3× speedup

Math

\[\left(\left(\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot {b}^{-5}\right) - \frac{a}{\left(b \cdot b\right) \cdot b}\right) \cdot c - \frac{1}{b}\right) \cdot c \]

FPCore

(FPCore (a b c)
  :precision binary64
  (*
 (-
  (* (- (* (* -2 (* a a)) (* c (pow b -5))) (/ a (* (* b b) b))) c)
  (/ 1 b))
 c))

C

double code(double a, double b, double c) {
	return (((((-2.0 * (a * a)) * (c * pow(b, -5.0))) - (a / ((b * b) * b))) * c) - (1.0 / b)) * c;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((((((-2.0d0) * (a * a)) * (c * (b ** (-5.0d0)))) - (a / ((b * b) * b))) * c) - (1.0d0 / b)) * c
end function

Java

public static double code(double a, double b, double c) {
	return (((((-2.0 * (a * a)) * (c * Math.pow(b, -5.0))) - (a / ((b * b) * b))) * c) - (1.0 / b)) * c;
}

Python

def code(a, b, c):
	return (((((-2.0 * (a * a)) * (c * math.pow(b, -5.0))) - (a / ((b * b) * b))) * c) - (1.0 / b)) * c

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(a * a)) * Float64(c * (b ^ -5.0))) - Float64(a / Float64(Float64(b * b) * b))) * c) - Float64(1.0 / b)) * c)
end

MATLAB

function tmp = code(a, b, c)
	tmp = (((((-2.0 * (a * a)) * (c * (b ^ -5.0))) - (a / ((b * b) * b))) * c) - (1.0 / b)) * c;
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(-2 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(c * N[Power[b, -5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(1 / b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]

Derivation

1. Initial program 30.9%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 95.7%

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

5. Taylor expanded in c around 0

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

7. Applied rewrites 94.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 94.0%

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

9. Applied rewrites 94.0%

   \[\leadsto \left(\left(\left(-2 \cdot \left(a \cdot a\right)\right) \cdot \left(c \cdot {b}^{-5}\right) - \frac{a}{\left(b \cdot b\right) \cdot b}\right) \cdot c - \frac{1}{b}\right) \cdot c \]
10. Add Preprocessing

Alternative 7: 91.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -200000:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - b \cdot b}{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right) \cdot c + \frac{c}{-b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))
     -200000)
  (/
   (/
    (- (- (* b b) (* c (* a 4))) (* b b))
    (+ (4-sqrtz0z0z1z24 b c a) b))
   (* 2 a))
  (+ (* (* (- a) (/ c (* (* b b) b))) c) (/ 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)) < -2e5

   1. Initial program 30.9%

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

   3. Applied rewrites 31.9%

      \[\leadsto \frac{\color{blue}{\frac{\left(b \cdot b - c \cdot \left(a \cdot 4\right)\right) - b \cdot b}{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) + b}}}{2 \cdot a} \]

   if -2e5 < (/.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 30.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 90.9%

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

   7. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

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

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-+.f64 N/A

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

   9. Applied rewrites 91.2%

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

Alternative 8: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -200000:\\ \;\;\;\;\mathsf{134\_z0z1z2z3z4}\left(1, \left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \frac{1}{a}\right), \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right) \cdot c + \frac{c}{-b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))
     -200000)
  (134-z0z1z2z3z4
   1
   (* (4-sqrtz0z0z1z24 b c a) (/ 1 a))
   1/2
   b
   (/ 1/2 a))
  (+ (* (* (- a) (/ c (* (* b b) b))) c) (/ 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)) < -2e5

   1. Initial program 30.9%

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

   3. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(1, \left(\frac{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right)}{a}\right), \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \color{blue}{\left(\frac{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right)}{a}\right)}, \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

      2. mult-flip N/A

         \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \color{blue}{\left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \frac{1}{a}\right)}, \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \color{blue}{\left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \frac{1}{a}\right)}, \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

      4. lower-/.f64 32.5%

         \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \color{blue}{\frac{1}{a}}\right), \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

   5. Applied rewrites 32.5%

      \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \color{blue}{\left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \frac{1}{a}\right)}, \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

   if -2e5 < (/.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 30.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 90.9%

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

   7. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

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

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-+.f64 N/A

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

   9. Applied rewrites 91.2%

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

Alternative 9: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -200000:\\ \;\;\;\;\frac{1}{\frac{a + a}{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) - b}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b}\right) \cdot c + \frac{c}{-b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))
     -200000)
  (/ 1 (/ (+ a a) (- (4-sqrtz0z0z1z24 b c a) b)))
  (+ (* (* (- a) (/ c (* (* b b) b))) c) (/ 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)) < -2e5

   1. Initial program 30.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 30.9%

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 30.9%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{a + a}{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) - b}}} \]

   if -2e5 < (/.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 30.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 90.9%

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

   7. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

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

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-+.f64 N/A

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

   9. Applied rewrites 91.2%

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

Alternative 10: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -200000:\\ \;\;\;\;\frac{1}{\frac{a + a}{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) - b}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-a\right) \cdot \frac{c}{\left(b \cdot b\right) \cdot b} - \frac{1}{b}\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))
     -200000)
  (/ 1 (/ (+ a a) (- (4-sqrtz0z0z1z24 b c a) b)))
  (* (- (* (- a) (/ c (* (* b b) b))) (/ 1 b)) c)))

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)) < -2e5

   1. Initial program 30.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 30.9%

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 30.9%

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{a + a}{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) - b}}} \]

   if -2e5 < (/.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 30.9%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 90.9%

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

   7. Applied rewrites 90.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 90.9%

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 90.9%

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

      13. lift-pow.f64 N/A

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

      14. unpow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-*.f64 90.9%

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

   9. Applied rewrites 90.9%

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

Alternative 11: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq \frac{-5534023222112865}{4611686018427387904}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))
     -5534023222112865/4611686018427387904)
  (* (/ 1/2 a) (- (4-sqrtz0z0z1z24 b c a) 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)) < -0.0011999999999999999

   1. Initial program 30.9%

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

   3. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(1, \left(\frac{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right)}{a}\right), \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right)} \]

   5. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{a} \cdot \left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) - b\right)} \]

   if -0.0011999999999999999 < (/.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 30.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.7%

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

   4. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 81.7%

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 81.7%

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

   6. Applied rewrites 81.7%

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

Alternative 12: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq \frac{-5534023222112865}{4611686018427387904}:\\ \;\;\;\;\frac{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

FPCore

(FPCore (a b c)
  :precision binary64
  (if (<=
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))
     -5534023222112865/4611686018427387904)
  (/ (- (4-sqrtz0z0z1z24 b c a) b) (+ a a))
  (/ (- 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)) < -0.0011999999999999999

   1. Initial program 30.9%

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

   3. Applied rewrites 32.3%

      \[\leadsto \color{blue}{\mathsf{134\_z0z1z2z3z4}\left(1, \left(\frac{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right)}{a}\right), \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \color{blue}{\left(\frac{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right)}{a}\right)}, \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

      2. mult-flip N/A

         \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \color{blue}{\left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \frac{1}{a}\right)}, \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \color{blue}{\left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \frac{1}{a}\right)}, \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

      4. lower-/.f64 32.5%

         \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \color{blue}{\frac{1}{a}}\right), \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

   5. Applied rewrites 32.5%

      \[\leadsto \mathsf{134\_z0z1z2z3z4}\left(1, \color{blue}{\left(\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) \cdot \frac{1}{a}\right)}, \frac{1}{2}, b, \left(\frac{\frac{1}{2}}{a}\right)\right) \]

   7. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{\mathsf{4\_sqrtz0z0z1z24}\left(b, c, a\right) - b}{a + a}} \]

   if -0.0011999999999999999 < (/.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 30.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 81.7%

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

   4. Applied rewrites 81.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 81.7%

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

      6. lift-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 81.7%

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

   6. Applied rewrites 81.7%

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

Alternative 13: 81.7% accurate, 3.6× speedup

Math

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

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 30.9%

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

2. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 81.7%

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

4. Applied rewrites 81.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 81.7%

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

   6. lift-*.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 81.7%

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

6. Applied rewrites 81.7%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 2220446049250313/20000000000000000000000000000000 a) (< a 9007199254740992)) (and (< 2220446049250313/20000000000000000000000000000000 b) (< b 9007199254740992))) (and (< 2220446049250313/20000000000000000000000000000000 c) (< c 9007199254740992)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))