Quadratic roots, narrow range

Percentage Accurate: 55.3% → 91.0%

Time: 5.4s

Alternatives: 9

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: 55.3% 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: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(b \cdot b\right)\\ \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{t\_0}, -c\right)}{b} + \frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \frac{c \cdot c}{t\_0}, -5, -a\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) (* b b))))
   (+
    (/ (fma (* a a) (/ (* -2.0 (* (* c c) c)) t_0) (- c)) b)
    (/
     (* (fma (* (* (* a a) a) (/ (* c c) t_0)) -5.0 (- a)) (* c c))
     (* (* b b) b)))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * (b * b);
	return (fma((a * a), ((-2.0 * ((c * c) * c)) / t_0), -c) / b) + ((fma((((a * a) * a) * ((c * c) / t_0)), -5.0, -a) * (c * c)) / ((b * b) * b));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * Float64(b * b))
	return Float64(Float64(fma(Float64(a * a), Float64(Float64(-2.0 * Float64(Float64(c * c) * c)) / t_0), Float64(-c)) / b) + Float64(Float64(fma(Float64(Float64(Float64(a * a) * a) * Float64(Float64(c * c) / t_0)), -5.0, Float64(-a)) * Float64(c * c)) / Float64(Float64(b * b) * b)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(-2.0 * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + (-c)), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * -5.0 + (-a)), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 91.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

7. Applied rewrites 91.0%

   \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot \left(\left(c \cdot c\right) \cdot a\right)\right) \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b} \cdot \frac{20}{a}, -0.25, \frac{-\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

8. Taylor expanded in b around inf

   \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \frac{-5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{4}} + -1 \cdot \left(a \cdot {c}^{2}\right)}{\color{blue}{{b}^{3}}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \frac{-5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{4}} + -1 \cdot \left(a \cdot {c}^{2}\right)}{{b}^{\color{blue}{3}}} \]

10. Applied rewrites 91.0%

    \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \frac{\mathsf{fma}\left(-c \cdot c, a, \frac{\left(-5 \cdot \left(\left(a \cdot a\right) \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}\right)}{\color{blue}{\left(b \cdot b\right) \cdot b}} \]

11. Taylor expanded in c around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

13. Applied rewrites 91.0%

    \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \frac{\mathsf{fma}\left(\left(\left(a \cdot a\right) \cdot a\right) \cdot \frac{c \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -5, -a\right) \cdot \left(c \cdot c\right)}{\left(b \cdot b\right) \cdot b} \]
14. Add Preprocessing

Alternative 2: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\mathsf{fma}\left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{-5 \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot c\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b}\right) \cdot c - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (*
   (-
    (*
     (-
      (*
       (fma
        (/ (* a a) (* (* b b) (* b b)))
        -2.0
        (/ (* -5.0 (* (* (* a a) a) c)) (* (* (* (* b b) b) (* b b)) b)))
       c)
      (/ a (* b b)))
     c)
    1.0)
   c)
  b))

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(N[(-5.0 * N[(N[(N[(a * a), $MachinePrecision] * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 91.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

6. Taylor expanded in c around 0

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

8. Applied rewrites 90.9%

   \[\leadsto \frac{\left(\left(\mathsf{fma}\left(\frac{a \cdot a}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, \frac{-5 \cdot \left(\left(\left(a \cdot a\right) \cdot a\right) \cdot c\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b}\right) \cdot c - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b} \]
9. Add Preprocessing

Alternative 3: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \frac{\left(c \cdot c\right) \cdot \left(-a\right)}{\left(b \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.6)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (+
    (/ (fma (* a a) (/ (* -2.0 (* (* c c) c)) (* (* b b) (* b b))) (- c)) b)
    (/ (* (* c c) (- a)) (* (* b b) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.6) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (fma((a * a), ((-2.0 * ((c * c) * c)) / ((b * b) * (b * b))), -c) / b) + (((c * c) * -a) / ((b * b) * b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(a * a), Float64(Float64(-2.0 * Float64(Float64(c * c) * c)) / Float64(Float64(b * b) * Float64(b * b))), Float64(-c)) / b) + Float64(Float64(Float64(c * c) * Float64(-a)) / Float64(Float64(b * b) * b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(-2.0 * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-c)), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5999999999999996

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

   if 5.5999999999999996 < b

   1. Initial program 49.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 93.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   7. Applied rewrites 93.9%

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot \left(\left(c \cdot c\right) \cdot a\right)\right) \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b} \cdot \frac{20}{a}, -0.25, \frac{-\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   8. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 91.5

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

   10. Applied rewrites 91.5%

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

Alternative 4: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-2 \cdot \left(\left(\left(c \cdot c\right) \cdot c\right) \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{c \cdot c}{b \cdot b}\right) \cdot a - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.6)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/
    (-
     (*
      (-
       (/ (* -2.0 (* (* (* c c) c) a)) (* (* b b) (* b b)))
       (/ (* c c) (* b b)))
      a)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.6) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (((((-2.0 * (((c * c) * c) * a)) / ((b * b) * (b * b))) - ((c * c) / (b * b))) * a) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(Float64(Float64(c * c) * c) * a)) / Float64(Float64(b * b) * Float64(b * b))) - Float64(Float64(c * c) / Float64(b * b))) * a) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-2.0 * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5999999999999996

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

   if 5.5999999999999996 < b

   1. Initial program 49.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 93.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   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 91.5%

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

Alternative 5: 89.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.6:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{-2 \cdot \left(\left(a \cdot a\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.6)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/
    (*
     (-
      (* (- (/ (* -2.0 (* (* a a) c)) (* (* b b) (* b b))) (/ a (* b b))) c)
      1.0)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.6) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = ((((((-2.0 * ((a * a) * c)) / ((b * b) * (b * b))) - (a / (b * b))) * c) - 1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.6)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(Float64(a * a) * c)) / Float64(Float64(b * b) * Float64(b * b))) - Float64(a / Float64(b * b))) * c) - 1.0) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.6], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-2.0 * N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.5999999999999996

   1. Initial program 79.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 79.5

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

   4. Applied rewrites 79.5%

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

   if 5.5999999999999996 < b

   1. Initial program 49.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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 93.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.4%

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

Alternative 6: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 17:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot c\right) \cdot \left(-a\right)}{\left(b \cdot b\right) \cdot b} - \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 17.0)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (- (/ (* (* c c) (- a)) (* (* b b) b)) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 17.0) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (((c * c) * -a) / ((b * b) * b)) - (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 17.0)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(c * c) * Float64(-a)) / Float64(Float64(b * b) * b)) - Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 17.0], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * (-a)), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 17

   1. Initial program 78.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. fp-cancel-sub-sign-inv N/A

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.2

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

   4. Applied rewrites 78.2%

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

   if 17 < b

   1. Initial program 48.2%

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

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   7. Applied rewrites 94.3%

      \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot \left(\left(c \cdot c\right) \cdot a\right)\right) \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b} \cdot \frac{20}{a}, -0.25, \frac{-\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

   8. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. pow3 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 87.1

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

   10. Applied rewrites 87.1%

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

Alternative 7: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 91.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right) + \mathsf{fma}\left(\frac{\left(\left(\left(a \cdot a\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(c \cdot c\right) \cdot \left(c \cdot c\right)\right)\right) \cdot 20}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)\right) \cdot a}, -0.25, -\frac{\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

7. Applied rewrites 91.0%

   \[\leadsto \frac{\mathsf{fma}\left(a \cdot a, \frac{-2 \cdot \left(\left(c \cdot c\right) \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -c\right)}{b} + \color{blue}{\frac{\mathsf{fma}\left(\frac{\left(a \cdot \left(\left(c \cdot c\right) \cdot a\right)\right) \cdot \left(a \cdot \left(\left(c \cdot c\right) \cdot a\right)\right)}{\left(\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot b} \cdot \frac{20}{a}, -0.25, \frac{-\left(c \cdot c\right) \cdot a}{b \cdot b}\right)}{b}} \]

8. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. lower-/.f64 N/A

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

   4. mul-1-neg N/A

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

   5. *-commutative N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. pow3 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 81.7

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

10. Applied rewrites 81.7%

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

Alternative 8: 81.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-neg.f64 81.7

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

5. Applied rewrites 81.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. *-commutative N/A

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

   8. pow2 N/A

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

   9. lift-neg.f64 N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. +-commutative N/A

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

   13. mul-1-neg N/A

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

   14. distribute-lft-out N/A

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

   15. associate-*r/ N/A

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

   16. div-add N/A

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

7. Applied rewrites 81.7%

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

   1. count-2-rev 81.7

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

   2. lift-+.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-frac-neg N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

   8. associate-*r/ N/A

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

   9. lift-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. associate-/r* N/A

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

9. Applied rewrites 81.7%

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

Alternative 9: 64.5% 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 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
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 64.5

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

5. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2025092 
(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)))